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Deconfinement of gauge theories at high temperature Yeh, Huai-Che 2011

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Deconfinement of gauge theories at high temperature by Huai-Che Yeh  B.Sc., National Tsing Hua University, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2011 c Huai-Che Yeh 2011  Abstract The deconfinement phase transition of large N gauge theories has been recognized as the Hawking-Page transition through AdS/CFT correspondence. To derive the phase transition, the calculation of the partition function is restricted to the singlet sector. This thesis examines whether the singlet constraint of gauge theory at the deconfinement phase was a physical consequence. A model of multiple matrices quantum harmonic oscillators is considered in this thesis as a toy model of large N, four dimensional SU(N) gauge theory. This thesis aims to compare the partition function restricted to singlet sector to the known partition function of a group of non-interacting harmonic oscillators, which represent the model without imposing the singlet constraint. We compute the singlet partition function analytically in the large m limit for a model with m matrices, and we can correct the result to higher orders in the large m expansion by the resolvent method in a random matrix model. We also develop numerical methods using a quadratic programming algorithm to solve the partition function at small m. We assure that the eigenvalues are frozen at the saddle point at high temperature. However the singlet partition function has the free energy converging to (m − 1)N 2 ln(1 − q) in the large m limit while the free energy of harmonic oscillators is mN 2 ln(1 − q). Changing the integration variables from group elements to eigenvalues explains how to derive such discrepancy of the free energy between imposing and relaxing the single constraint. Gauge theory cannot simply relax the singlet constraint at high temperature without adding a normalization factor.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 Background . . . . . . . . . . . . . 2.1 Particles on a circle . . . . . . . 2.2 Singlet irreducible representation 2.3 Phase transition . . . . . . . . .  . . . .  4 4 7 9  3 Analytic methods and results . . . . . . . . . . . . . . . . . . 3.1 Method A1: Truncated Fourier expansion . . . . . . . . . . . 3.1.1 Minimizing the action in Fourier modes . . . . . . . . 3.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Method A2: Resolvent solution . . . . . . . . . . . . . . . . . 3.2.1 Expanding the potential in the large m limit with the high T approximation . . . . . . . . . . . . . . . . . . 3.2.2 Comparing with free oscillators . . . . . . . . . . . . 3.2.3 The standard resolvent method . . . . . . . . . . . . 3.2.4 The resolvent of expanded potential of adjoint matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . .  12 12 12 15 19  4 Numerical methods and results . . . . . . 4.1 Method N1: Truncated Fourier expansion 4.1.1 Quantized minimization . . . . . . 4.1.2 Results . . . . . . . . . . . . . . .  37 37 37 38  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  19 23 24 25 30  iii  Table of Contents 4.2  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1, A2, and . . . . . . . . . . . . . . . .  44 52  . . . .  . . . .  57 57 58 61  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  62  4.3  Method N2: Quadratic programming . . . . . 4.2.1 Finite element method . . . . . . . . . 4.2.2 Error estimation . . . . . . . . . . . . . Comparison of numeric results with analytics . 4.3.1 Comparison of method N2 with methods N1 . . . . . . . . . . . . . . . . . . . . 4.3.2 Comparing with free oscillators . . . .  5 Discussion and Conclusion . . . . . . . 5.1 Summary . . . . . . . . . . . . . . . . 5.2 Singlet condition at high temperature 5.3 Conclusion . . . . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  41 41 44 44  Appendices Codes .1 .2 .3 .4  . . . . . . . Method A1 Method N1 Method N2 Method A2  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  63 63 64 64 66  iv  List of Figures 3.1  3.2  3.3 3.4  3.5  3.6  Density of eigenvalues as a function of the angle around the circle, (computed by method A1 for m=20). The top graph, at q=0.1, shows that the first two Fourier modes are sufficient to approximate the density well (no improvement is seen when the first five mode are used). The bottom graph, at q=0.3, demonstrates the need to use a higher number of modes with a narrower eigenvalue distribution. . . . . . . . . . . . . . . . Effective action, (computed by method A1 for m=20). The values of action computed with different cutoffs are not obviously dissimilar for different cutoff at low q, but only the one with the largest cutoff keeps overlapping with others to higher q. Judging by the overlapping region, it suggests how high the temperature is method A1 valid for the given cutoff. ρ(x)dx on (m,q) plane; color= integral values. ρ(x)dx = 1 for m ≤ 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2nd , and V 4th and relative denPotentials graph for Vexact , Vapp app sity distributions. For the distance between eigenvalues above 0.05, each approximate potentials have significant difference 4th starts diverging to negative with the exact potential. Vapp infinity at u = 0.1. According to ρapp and ρexact (N 2), eigenvalues at two ends are separated at least further than 0.1. 4th strongly repulse others, and we can Thus eigenvalues in Vapp not find the stationary point. . . . . . . . . . . . . . . . . . . Roots of Q(S) on (m,q) plane. Roots 1, 2, 3 in 3.52 correspond to B 2 , ξ, η. Root1 is always real. root2 and root3 are complex conjugate pair, and they are not real only at q < 0.8 and m < 12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Re[ξ] − B 2 on (m,q) plane; It is negative only at q < 0.65 and m < 10, and therefore −B ↔ B is always the branch cut crossing the center. . . . . . . . . . . . . . . . . . . . . . . . .  17  18 30  32  33  34  v  List of Figures 3.7  4.1  4.2  4.3  4.4  Correction terms of action at q=0.9. As the number of matrices increases, correction terms become much smaller, and therefore the validity of the large m expansion is faithful. Except the first data set at m = 5, the most important correction with the greatest absolute value is δSV as expected. Considering the inconsistency in the resolvent method at small m, it is reasonable to exclude the first data set at m = 5. . . . . Method N1 with different cutoffs solving for qn as a function of n. The Fourier transformed results from ρexact (N 2) shows that the higher order Fourier modes can be ignored at this temperature. At q=0.3, ρn>100 does not matter because q n>100 can be ignored in the truncated Fourier expansion. In the lower graph, dot lines are solved by method N1, and solid lines are solved by method A1. . . . . . . . . . . . . . . . . . Method N1 with different cutoff solves actions as a function of q. Diamond data are solved analytically by method A1, and solid dots data are solved by method N1, and solid lines is the approximated action Sapp . We clearly observe method A1 is invalid at q > 0.8, and method N1 improves the accuracy till q = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density distributions obtained from different manual boundaries. B means the manual boundary set up in method N2. 4th repulse every If B is set too big, the negative divergent Vapp eigenvalues to the boundary. If B is set too small, the potential well does not have enough space for eigenvalues, and they are squeezed on the boundary. . . . . . . . . . . . . . . . . . . The error in ρ is defined as the root mean square of density dif-  36  39  40  45  2  4.5 4.6  2nd (N 2) − ρ ference, ∆ρ = app i , and ρi = ρ(i∆θ), i ρi ρapp i = ρapp (i∆θ). . . . . . . . . . . . . . . . . . . . . . . . . The error in the action is defined as S 2nd (N 2) − Sapp . . . . . ρ(θ) vs x at high q and m. Black and yellow lines are solved by analytic methods with high q and large m expansion. Dash lines are solved by method N1 for different cutoff. Dash-dot line is solved by method A1. Red dots with error bars are solved by method N2. . . . . . . . . . . . . . . . . . . . . . .  46 46  47  vi  List of Figures 4.7  4.8  4.9 4.10  4.11 4.12  4.13  4.14  Method N1 with different cutoff solves qn as a function of n. The results from quadratic programming with finite elements method N2 shows that the higher order Fourier modes can not be ignored at this temperature. At q=0.9, ρn>100 are still crucial because q n>100 can not be ignored in the truncated Fourier expansion. . . . . . . . . . . . . . . . . . . . . . . . . Comparison of data of the action obtained from every method. For m=20, the large m expansion already works very well at q=0.6. 2nd and 4th order approximations overlap with the N2 data of the exact potential, but the truncated method gives bad results at high temperature. . . . . . . . . . . . . . Sexact and Sapp as functions of q, for m=2,3,4,5. The error bars in Sexact are small enough to be ignored. . . . . . . . . . δSV , S 4th (A2) − Sapp , and Sexact (N 2) − Sapp as functions of q. S 4th (A2) − Sapp represents 4th order potential correction, and δSV represents the most important parts of it. Sexact −S2 represents all orders of the corrections of large m expansion. . Sexact and Sf ree as functions of q, for m=2,3,4,5. The error bars over Sexact are small enough to be ignored. . . . . . . . . Sf ree Sexact (N 2)−Sf ree and Sexact (N 2) as functions of q, for m=2,3,4,5. The error bars in Sexact are small enough to be ignored. The discrepancy between Sexact (N 2) and Sf ree slightly tilts upward at high q, and the ratio converges to 1 at high q. We then check more data within a smaller region at high q in fig4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sf ree Sexact (N 2) − Sf ree and Sexact (N 2) as functions of q graphs, for m=2,3,4,5 at high q = 0.95 ∼ 0.9999. The constant gap of Sf ree ln(1−q) Sexact − Sf ree suggests fitting Sexact , where C is by ln(1−q)+C a constant related to m. . . . . . . . . . . . . . . . . . . . . . The gap between Sexact and Sf ree as a function of m at q=0.99. For small m at high temperature( m < 6), the gap is like a constant independent to q according to fig4.13. At larger m, the gap converges to the approximate solution, Sapp − Sf ree = −m ln 2 + 21 ln mq. (see eq3.34) We can divide the discrepancy between Sexact and Sf ree , which equals to Sapp − Sf ree at large m or a constant gap at small m, by Sf ree = (m − 1) ln(1 − q), which diverges at q → 1, and the discrepancy always vanishes at enough high temperature for any value of m. Thus the relative error between Sexact and Sf ree will always disappear, and Sexact converges to Sf ree . . .  48  49 50  51 52  54  55  56 vii  Acknowledgements I would like to offer my sincerest gratitude to my supervisor, Dr. Joanna Karczmarek, who has supported me throughout my thesis with her patience and knowledge. Without her friendly and continuous help this thesis would not have been completed or written. I would like to thank Dr. Mark Van Raamsdonk for reviewing my thesis and all his valuable suggestions. I would like to thank Dr. Gordon Semenoff for pointing out the resolvent method. I would also like to thank my wife, Zoe Yu Wang, for all the support, the encouragement and the effort on providing a home in which to complete my writing up.  viii  Chapter 1  Introduction In gauge theory, quantum states are restricted to the singlet sector. It is known that the large N gauge theory with SU(N) symmetry on the ddimentional sphere undergoes the phase transition, and color singlets are deconfined at high temperature. In the case of N = 4 supersymmetric Yang-Mills theory, the deconfinement transition is interpreted as the formation of black hole in the dual string theory by gauge/gravity duality, which indicates an excited state in gauge theory is dual to a matter distribution in anti-de Sitter space, (see [9]). In other words, according to AdS/CFT correspondence, the deconfinement transition in the large N gauge theory is dual to the Hagedorn behavior in the weakly coupled string theory, and the phase transition is also recognized as Hawking-Page transition of a black hole, and the thermalization of the excited state is identified as the gravitational collapse. To study gauge theory, the thermodynamics of matrix quantum mechanics is an important research topic. Matrix quantum mechanics attracts research interest in string theory originally in the study of 2D gravity. One dimensional bosonic string theory and 2D quantum gravity share the same character that the integral on two dimensional surfaces. The partition function in such theories, which can also be interpreted as the generating functional for the correlation functions, integrates over the internal geometry of 2D surfaces, and this integral can be discretized as a sum over randomly triangulated surfaces representing all possible Feynman diagrams. The Feynman diagrams of such discretization is dual to the double line Feynman diagrams of U(N) matrix quantum mechanics. Summing over these double line Feynman diagrams in the large N limit results in the partition function of matrix model. Thus the thermodynamics of matrix quantum mechanics corresponds to 2D gravity and string theory, [3]. The researches of the matrix model mainly focus on the partition function restricted to the singlet sector. It is pointed out in [1], the interacting matrix model eventually becomes free at sufficiently high temperature, and the singlet constraint is relaxed. Therefore the ordinary ungauged partition function at high temperature is represented by the singlet gauged partition 1  Chapter 1. Introduction function. For the matrix-black-hole-model as a unitary quantum theory, the time irreversibility of the star collapsing process is studied in [4] neglecting the singlet condition at highly excited state. It is an important strategy to calculate the partition function of gauge theory in the singlet sector and then ignore the singlet constraint in deconfinement phase at high temperature. The main purpose of this thesis is to exam the convergence from the singlet partition function to the ordinary partition function without imposing singlet condition at high temperature. As a toy model of N = 4 supersymmetric Yang-Mills theory, this thesis studies a simple free SU(N) m matrices model in the large N limit, which describes mN 2 quantum harmonic oscillators. We compute its partition function projected in singlet sector when there is no interactions in the model except for the singlet constraint from confinement, and we compare the result to the known partition function for free harmonic oscillators. The dimension of symmetric n-th power of the irreducible representation of the unitary group stands for the degeneracy of harmonic oscillators at energy level n. The singlet partition function is a generating function of the number of copies of the singlet representation, which has dimension equals to one. To calculate the number of singlet representations over every energy level, we need an integral over group elements according to the orthogonality of characters. We change the integration variables of group elements to eigenvalues, and the integral turns out to be a dynamical problem of the eigenvalues. The partition function becomes a one dimensional many body system of particles with pairwise interaction. As shown in [1], all the eigenvalues are frozen around a saddle point at high temperature, thus the partition function is solved by the equilibrium positions of these eigen-particles. The rest of the thesis is about solving the equilibrium distribution of eigen-particles above deconfinement temperature. This thesis is organized as the following. In Chapter 2, we illustrate in one matrix model how the integral of partition function becomes 1D many body system of pairwisely interacting particles, and we introduce generally how to use representation theory to project partition functions into an irreducible representation sector. Then we derive the effective action of eigenvalue distributions and calculate the partition function below deconfinement temperature at large N. We demonstrate the phase transition in the singlet sector in a multi-matrix model. In Chapter 3, we describe several analytical methods minimizing the effective action above deconfinement temperature, each of which is valid in some approximation. First, we minimize the action in Fourier space and truncate the Fourier expansion since higher order modes can be neglected in low temperature, (referring to [1]). Second, we expand the effective potential for a large number of matrices in the model 2  Chapter 1. Introduction and find the zero order solution explicitly. Third, we use the standard resolvent method in a random matrix model to correct the expanded solution to the next order. In Chapter 4, we develop two numerical methods minimizing the action. Based on the truncated Fourier expansion, the first numerical method can easily include many higher order modes, but it is still not valid at higher temperatures. The second numerical method directly minimizes the action in angular space using a quadratic programming algorithm. It is the most powerful method, and it is valid without any approximations. Then we compare results from all the methods and make conclusions about the behavior of singlet partition function. The last chapter includes the summary of our results and the discussion about the origin of the discrepancy between the singlet partition function and the ordinary partition function.  3  Chapter 2  Background The partition function of a matrix quantum mechanics is commonly studied using representation theory. For a matrix model with SU(N) symmetry, all the computations can be done by considering eigenvalues of the group elements. Since the eigenvalues of a matrix in the SU(N) group lie on a unit circle, the common strategy to compute the partition function is to treat eigenvalues as a dynamical problem of particles moving on a circle under a pairwise interaction. This chapter illustrates the above idea and discusses the possible phase transition by analyzing the partition function.  2.1  Particles on a circle  Hamiltonian We start from a free one matrix model, which contains N 2 identical free harmonic oscillators.The model is described by a field, N × N Hermitian matrix Xij , and its conjugate momentum, Πij , with a commutation relation, [Xij , Πkl ] = iδil δjk . The Hamiltonian is 1 ω2 H = T r Π2 + T r X2 . 2 2  (2.1)  With the commutation relation, we can rewrite the Hamiltonian by creation and annihilation operators as  where  H = ωT r A† A ,  (2.2)  1 A = √ (Π − iωX) . 2ω  (2.3)  The N 2 elements of A describe N 2 decoupled harmonic oscillators. Trivially † 2 a multiple matrix model has H = m i=1 ωT r Ai Ai for mN noninteracting oscillators. We focus on the partition function in the Hilbert space generated by A†i . The partition function and free energy F are defined by Z = T re−βH = e−βF .  (2.4) 4  2.1. Particles on a circle Partition function in a representation In a general U(N) invariant model, the symmetry of the Hamiltonian has various irreducible representations, and the Hilbert space is divided into sectors, each of which corresponds to infinite number of copies of one particular irreducible representation. Different sectors of irreducible representation do not couple together, thus we can define the partition function separately for each sector corresponding to an irreducible representation, R, as ZR (β) =  1 T rR e−βHR , dim(R)  (2.5)  where dim(R) is the dimension of irreducible representation R, (see[2]). Given a group element U in U(N), a generalized partition function for the entire Hilbert space can be defined as Z(β, U ) =  ZR (β)χR (U ),  (2.6)  R  where χR (U ) is the character of U in the representation R, (referring to [2]). Z(β), the ordinary partition function, is the sum of all ZR in every sectors, and this is exactly the generalized partition function for the identity element, Z(β, 1) = R dim(R)ZR (β), since χ(1) = dim(R). Because of the orthogonality of characters, [dU ]χ∗R1 (U )χR2 (U ) = δR1 R2 , ZR in a sector can be expressed from Z(β, U ) as ZR (β) =  [dU ]Z(β, U )χR (U ).  (2.7)  Here we follow the procedure in [1] to derive the partition function specifically for an irreducible representation R. The creation operator A† corresponds to the tensor product of fundamental and antifundamental representations, V ⊗ V ∗ . The n-th symmetric power of A† at energy level n corresponds to Symn (V ⊗ V ∗ ). For the gauged model, we only need to count the number of singlets in the direct sum, [7]. Therefore the partition function in a sector of the irreducible representation R is ∞  ZR (q) =  q n χSymn (V ⊗V ∗ ) (U )χR (U ), where q ≡ e−βω .  [dU ]  (2.8)  n=0  Referring to another formula of characters in [5], ∞  ∞  q n χSymn (R) (U ) = exp n=0  m=1  qm χR (U m ) , m  (2.9) 5  2.1. Particles on a circle the partition function can be further simplified as the result in [1], ∞  ZR (q) =  [dU ]exp m=1  1 m q χR˜ (U m ) χR (U ), m  (2.10)  ˜ = V ⊗ V ∗. where R Pairwise interaction between eigenvalues We can decompose the tensor product of fundamental and antifundamental into direct sum of singlet and adjoint irreducible representations. For SU(N), we know the charactors of singlet and adjoint irreducible representations are χsing = 1 and χadj = tr(U )tr(U † ) − 1, and the eigenvalues of any elements in SU(N) group can be written as eiθ for θ ∈ [−π, π]. Therefore we have the partition function explicitly as ∞  ZR (q) =  dU exp m=1  1 m q + m  ∞ m=1  1 m q m  e−imθk −1  eimθj j  χR (U ).  k  (2.11) Refering to [1], we can write the Haar measure in terms of eigenvalues instead of the group elements. π  dθk 2π  dU → −π k  θi − θj = 2  sin2 i<j  π −π k  dθk 2π  i<j  1 ≡ 4 The Vandermonde ∆ is defined by ∆ = Thus the partition function becomes ZR (q) =  1 4  l  dθl |∆(eiθ )|2 exp 2π 1 = 4  l  iθk k<m (e  ∞ j,k m=1  dθl |∆(eiθ )|2 2π  k  1 iθi |e − eiθj |2 4 dθk |∆(eiθ )|2(2.12) 2π  − eiθm ).  1 m im(θj −θk ) q e χR (U ) m  j,k  1 χR (U ). (2.13) 1 − qei(θj −θk )  In order to evaluate above equation, we can imagine an effective potential as the function of eigenvalues appearing in the integral and rewrite the  6  2.2. Singlet irreducible representation partition function as ZR (q) = (2 )  −N  π N π j=1  V (u) = − ln  dθj − 1 e 2 2π  k,m  V (θk −θm )  χR (eiθ ),  where the effective potential is sin2 ( u2 ) + 2 y = ln +q q sin2 ( u2 ) + y sin2 ( u2 ) + 2 with y =  (1 − q)2 . 4  (2.14)  (2.15) (2.16)  is a regulator that allows us to include k=m terms in the double sum, and it will be taken to zero later. The partition function (2.14) can be interpreted as a partition function of N particles on a circle under a pairwise interaction. For different models, there will be different corresponding effective potentials. We then treat eigenvalues as the angular coordinate of the particles on the circle and find the distribution with the smallest energy by minimizing the effective action, S = 12 k,m V (θk − θm ). Thus the partition function is evaluated as a mechanic problem.  2.2  Singlet irreducible representation  In the following discussion, we focus on the singlet (trivial) representation, because all states are constrained to be in this representation in a gauged matrix model. We derive the exact partition function at large N at low temperature, where the classical distribution of eigenvalues is uniform. In the singlet irreducible representation, χR = 1, and the partition function is Zsing (q) = (2 )−N  π  N  −π k=1  dθk − 1 e 2 2π  k,m  V (θk −θm )  .  (2.17)  At low temperature, the potential is mostly repulsive and these static particles are expected to be distributed evenly on the circle. Define the particle density as ρ(θ) = N −1 j δ(θ − θj ). The effective action is then S=  1 2  V (θk − θm ) → k,m  N2 2  π  dθ1 dθ2 V (θ1 − θ2 )ρ(θ1 )ρ(θ2 ).  (2.18)  −π  7  2.2. Singlet irreducible representation In the large N limit at low temperature, evenly distributed eigenvalues 1 have the classical solution of density function, ρ(θ) = 2π . In order to calculate the partition function, we consider the subleading quantum fluctuations of the equilibrium distribution. The density function will be perturbed, and it can be Fourier expanded as ∞  1 1 ρ(θ) = + 2π π  n=1  1 ρn cos nθ + π  ∞  σn sin nθ.  (2.19)  n=1  Inserting this into the action gives N −2 S = S0 +  π  1 2π 2  1 + 2 2π  dθ1 dθ2 V (θ1 − θ2 ) cos nθ1 cos mθ2  ρn ρm −π  n,m  π  dθ1 dθ2 V (θ1 − θ2 ) sin nθ1 sin mθ2 , (2.20)  σn σm −π  n,m  1 where S0 = 4π duV (u), and the linear terms in ρn and the crossing terms with factors like sin nθ1 cos mθ2 both vanish in the integral. Changing variables to u = θ1 − θ2 and θ = (θ1 + θ2 )/2, and evaluating the θ integral shows  N  −2  1 S = S0 + 2π  ∞  2π  ρ2n + σn2  duV (u) cos nu  (2.21)  0  n=1  With little calculation, 2π  1 →0 4π  S0 = lim  With Vn = lim  2π →0 0  du ln 0  du ln  (1 − q)2 4 sin2 (u/2) +  (1−q)2 4 sin2 (u/2)+  2  2  +q  = 0.  + q cos(nu) =  2π n (1  (2.22) − q n ), the  action can be rewritten as S=  N2 2π  ρ2n + σn2 Vn .  (2.23)  By changing the integration variable from dθk to the Fourier coefficients, dρn and dσn , the partition function becomes ∞ −S(ρn ,σn )  Zsing (q) = C  dρn dσn e n=1  1 =C (2 )N  ∞ n=1  2π 2 , N 2 Vn  (2.24)  8  2.3. Phase transition where C is a q-independent normalization factor. At zero temperature, q = 0, the fact that the partition function equals to one gives Zsing (q = 0) =  C (2 )N  ∞ n=1  πn = 1. N2  (2.25)  With the normalization factor C derived above, the partition function at q in singlet representation is ∞  Zsing (q) = n=1  1 , 1 − qn  (2.26)  in agreement with the result for finite N in [2].  2.3  Phase transition  Multiple matrix model It is more interesting to investigate multiple matrices in the model. Unlike single matrix model, which has regular partition functions at any temperature, Z m (x) diverges at finite temperature for a multi matrix model. Such phase transition at a Hagedorn temperature and the behavior of Z m (x) beyond TH are the main interest of the following calculation in this thesis. For m matrices in the model, each with frequency ωξ for qξ = e−βωξ , the partition function is given simply by multiplying each partition function for one matrix model together in the integral. m ZR I (x)  1 = 4  l  =  1 4  l  ∞  m  dθl |∆(eiθ )|2 exp 2π  j,k n=1  ξ m  dθl |∆(eiθ )|2 2π  ξ ξ 1 (qξ )n ein(θj −θk ) χRI (U ) n  1 ξ  ξ  j,k  ξ  1 − qξ ei(θj −θk )  χRI (U )  (2.27)  As before, the system can be described by particles on a circle, 2π N  ZR (q) = (2 )−N  with S =  1 2  V (θk − θm ) → k,m  0  k=1  N2 2  π  dθk −S e χR (eiθ ) 2π  dθ1 dθ2 V (θ1 − θ2 )ρ(θ1 )ρ(θ2 ). (2.28) −π  9  2.3. Phase transition In the singlet representation, the effective potential for m matrices is m 2  V (u) = − ln sin (u/2) +  2  ln(qi sin2 (u/2) + y).  +  (2.29)  i=1  The Fourier modes of the effective potential are 2π  Vn = 0  2π duV (u) cos(nu) = 1− n  m  qin for n ≥ 1 and q ∈ [0, 1], (2.30) i=1  and the effective action is S 1 = S0 + 2 N 2π  ∞  ρ2n + σn2 Vn (T ),  (2.31)  n=1  where ρn = ρ(θ) cos nθdθ and σn = ρ(θ) sin nθdθ. There is one difference from the one matrix model, S0 does not vanish here. 1 →0 4π  2π  duV (u) = (1 − m) ln 2  S0 = lim  (2.32)  0  In the simplest case, we assume all matrices are identical, i.e. ∀i, qi = q. 1 At low temperature state, q < m , we have Vn≥1 > 0, which implies ρn≥1 = 0 and σn≥1 = 0 in order to minimize the effective action, and 1 therefore these eigenvalues distribute on the circle uniformly, ρ(θ) = 2π . Similarly to the one matrix model, the action can still be computed by considering the quantum fluctuation on the classic distribution. ∞ m Zsing (q)) = n=1  1 1 − mqin  (2.33)  The different part with one matrix model is the partition function diverges 1 in multiple matrix model at a finite Hagedorn temperature, qH = m . Hagedorn temperature Referring to the argument in [1], at a temperature higher than TH , the effective potential becomes attractive, and the eigenvalues are compressed into a clump, leaving a gap on the circle. The classical distribution of eigenvalues looks like a delta function because the attractive potential becomes much stronger with increasing temperature. Without lost of generality, we assume such classical eigenvalues distribution is symmetric about θ = 0 and expand it in a form given in 2.19 with σn = 0, similar to [1]. 10  2.3. Phase transition Due to the fact that Vn is monotonically increasing with n, we need only to consider the first few Fourier modes of ρn , which correspond to negative Fourier terms of Vn (T > TH ), because the rest positive potential terms tend to result in smaller Fourier coefficients of density in the minimized action. In the following section, we derive that ρn converges to zero according to a factor q n 1 at large n. At Hagedorn temperature, V1 = 0 and Vn>1 > 0. Therefore ρ1 = 0 and ρn>1 = 0. The integration of ρ(θ) over the circle is regulated to be one. 1 Thus ρ(θ) = 2π (1 + t cos θ), and 0 ≤ t ≤ 1 because density must be positive everywhere on the circle. At slightly higher temperature when the remainder of the potential terms 1 are still positive, V1 < 0 and Vn>1 > 0, we have t = 1 for ρ(θ) = 2π (1+cos θ), because the extremal of action must be on the boundary for its quadratic form. The least action value is N2 1 2 N2 ( ) V1 (Th + ∆T ) = N 2 S0 + ∆T V1 (TH ) , (see [1].) 2π 2 8π (2.34) Therefore the free energy defining as F = T S at temperature from zero to slightly hotter than Hagedorn temperature is S = N 2 S0 +  lim  N →∞  =  1 T FT →TH (T ) = S0 + N2 2π S0 +  ρ2n Vn n  S0 , for T < TH 1 8π T ∆T V1 (TH ) , for T  > TH  .  (2.35)  This phase transition in general is independent of chosen model as referring qH . to [1]. To the 1st order in our model at T > TH , F/N 2 = −∆T 4T H  11  Chapter 3  Analytic methods and results We present two different approaches leading to an analytic computation of the partition function above Hagedorn temperature. First approach is based on the fact that the minimized action in the Fourier expansion has small higher order modes ρn for the corresponding positive potential terms Vn . Thus the optimization of the action can be solved analytically with finite number of Fourier modes, ρn . However, too many modes at high temperature makes the calculation too massive to be solved. By introducing a large m expansion, where m means the number of matrices, second approach finds the analytic solution at higher temperature for a simplified potential.  3.1 3.1.1  Method A1: Truncated Fourier expansion Minimizing the action in Fourier modes  Above Hagedorn temperature, the attractive effective potential squeezes eigenvalues together in a small region. We are interested in the saddle point of eigenvalues, which means the classical equilibrium distribution of particles. Without loss of generality, the distributing function is assumed to be symmetric to the center θ = 0, and we expand the action and the density in the Fourier series, S = N 2 S0 +  N2 2π  ∞  |ρn |2 Vn and ρ(θ) = n=1  1 1 + 2π π  ρn cos nθ, n≥1  n where Vn (q) = 2π n (1−mq ). Because of minimization of the action, there is a tendancy for the density to accumilate at low frequency modes, and the rest modes ρn>t are small. In other words, higher order Fourier coefficients can be ignored as an approximation. The following introduction of truncated Fourier expansion starts from the whole non-truncated expansion. Due to the strongly attractive potential, we expect the eigenvalues to accumulate within a small interval [−θ0 , θ0 ], with ρ(θ) = 0 outside the interval.  12  3.1. Method A1: Truncated Fourier expansion In the minimum configration, the net force is zero. θ0  V (α − θ)ρ(θ)dθ = 0 ∀α ∈ [−θ0 , θ0 ]  (3.1)  −θ0  Also, V (θ) = V0 +  1 π  Vn cos nθ → V (α − θ) = − n  =−  1 π  1 π  nVn sin n(α − θ) n  nVn (sin nα cos nθ − sin nθ cos nα) n  and the assumption of even distribution in [1] demands ρ(θ) sin nθdθ = 0. For ρn = ρ(θ) cos nθdθ, the net force equation (3.1) can be written as θ0  V (α − θ)ρ(θ)dθ = 0 = − −θ0  In the m matrix model, Vn =  2π n (1  1 π  nVn ρn sin nα.  (3.2)  n≥1  − mq n ), we thus have mq n ρn sin nα.  ρn sin nα = n  n  Inserting the Fourier transformation  1 4π  π −π  cot  α−θ 2  cos nθdθ = sin nα  into the left hand side, θ0  cot −θ0  α−θ ρ(θ)dθ = 2 2  mq n ρn sin nα.  (3.3)  n  This equation happens to be the equilibrium condition in another matrix model with an action ∞  S=N n=1  an ρn (tr(U n ) + tr(U †n )), n  (3.4)  which is solved in [6]. The solution is 1 ρ(θ) = π  θ0 θ sin ( ) − sin2 ( ) 2 2  ∞  2  n=1  1 Qn cos((n − )θ) 2  (3.5)  for θ ∈ [−θ0 , θ0 ], ρ(θ) = 0 otherwise. 13  3.1. Method A1: Truncated Fourier expansion Here  ∞  mq n+l ρn+l Pl (cos(θ0 )),  Qn ≡ 2  (3.6)  l=1  and Pl is the Legendre polynomials, ∞ 1  Pl (x)z l = (1 − 2xz + z 2 )− 2 .  (3.7)  l=1  Therefore, (3.5) can be treated as a linear equation of ρn vector. 1 1 + 2π 2π  ρn cos nθ = n≥1  1 π  sin2 (  θ0 θ ) − sin2 ( ) 2 2  ∞ n=1  1 Qn cos((n − )θ) 2  In the vectors form of equations, →R ·ρ = ρ  (3.8) c 1  1  (B r+k− 2 (s2 ) + B |r+k− 2 | (s2 ))Pc−k (1 − 2s2 ).  with Rrc ≡ mq c k=1  In addition, the boundary condition ρ(θ > θ0 ) results in ∞  ∞  mq 1+l ρ1+l Pl (cos(θ0 ))  mq l ρl Pl (cos(θ0 )) + 2 =  Q0 + 2 = Q1 →  l=1  l=1  In the vectors form of equations, →A·ρ = 1  (3.9) c  2  2  with Ac ≡ mq (Pc−1 (1 − 2s ) − Pc (1 − 2s )). In the above equations, s2 ≡ sin2 ( θ20 ), and 1  1 θ0 θ0 θ 1 dθ sin2 ( ) − sin2 ( ) cos((n − )θ) π −θ0 2 2 2 are polynomials defined by the generating function  B n− 2 (s2 ) = ∞  1  B n+ 2 (x)z n = n=0  1 2z  (1 − z)2 + 4zx + z − 1  (3.10)  (3.11)  Finally, solving (3.8) and (3.9) is equivalent to solving the net force equation. These linear equations can be solved by the following process. First the 14  3.1. Method A1: Truncated Fourier expansion boundary position, θ0 , can be found by recognizing that det(R−1) = 0 from (3.8). The next step is to define a new coefficient matrix M by replacing the first row of the zero matrix, 1 − R ≡ M, and it results in ρ = M−1 e1 where e1 = (1, 0, 0, . . . ). R ·ρ=ρ det(R − 1) = 0 → ρ = M−1 e1 A·ρ=1  (3.12)  Furthermore, above linear equations with infinite dimensions, (3.12), can be approximately treated as finite dimension. Since over certain order of T , the factor of the temperature with power T is negligible, q n>T ∼ 0 because of q ∈ [0, 1], and therefore we truncate the infinite Fourier series at T and neglect Rr>T , c>T and Ac>T . The coefficient matrix looks like MT ×T L  M=  0 1  .  (3.13)  Therefore ρn can be solved up to kth order solely by MT ×T . ρ=  M−1 T ×T −LM−1 T ×T  0 1  e1  (3.14)  Terms of ρn>T are truncated. Therefore (ρ1 , ρ2 , ..., ρT ) = M−1 T ×T e1 → ρTtruncated (θ) =  1 π  sin2 (  θ0 θ ) − sin2 ( ) 2 2  T n=1  1 Qn cos((n − )θ) 2  (3.15)  T −n  mq n+l ρn+l Pl (cos(θ0 )).  where Qn ≡ 2 l=1  3.1.2  Results  The simplest case is for T = 1. The coefficient matrix and the boundary equation are M = 2mqs2 , det(1 − R) = mq(2s2 − s4 ) − 1 = 0.  (3.16)  By solving for s, we find θ0 . s2 = sin2  θ0 2  =1−  1−  1 . mq  (3.17) 15  3.1. Method A1: Truncated Fourier expansion Thus, the density function is =1 ρTtrunc (θ) =  1 π sin2  θ0 2  sin2  θ0 2  − sin2  θ 2  cos  θ 2  .  (3.18)  Let’s demonstrate a more complicated example, truncated solution for T =2. According to (3.12), the coefficient matrix looks like M2×2 =  2mqs2 2mq 2 (2 − 3s2 )s2 2 2 2 2 2 −4qs (1 − s ) 1 + 2q s (−4 + 14s2 − 20s4 + 9s6 )  . (3.19)  The boundary equation, det(1 − R) = 0, is 1+m2 q 3 s8 (6−6s2 +s4 )+mqs2 (−2+s2 +q(−4+14s2 −20s4 +9s6 )) = 0. (3.20) Explicitly choosing m = 20 for example, the Hagedorn temperature is at q = 0.05. Therefore phase transition happens at small q, where the Fourier modes of density decays fast enough to truncate q n terms at small cutoff. In the figure 3.1 for q = 0.1, the boundary equations gives θ0 = 2 sin−1 (s) = 0.20202,  (3.21)  and the coefficient matrix gives the density function as =2 ρTtrunc (θ) =  0.26794 − sin2 (θ) θ 3θ (3.57318 cos( ) + 0.21703 cos( )). (3.22) π 2 2  The coefficient of ρ2 is smaller than ρ1 , as expected. If we calculate higher order terms, the subsequent terms shall be smaller than the previous term approximately by a factor of q, ρ3 ∼ ρ2 × q for example. At higher temperature, the density distribution is accumulated within a smaller region and resembles a delta function. However, delta func1 tion is expanded with infinite terms in Fourier expansion, δ(x) = 2π + ∞ 1 cos(nx), and therefore higher order Fourier modes, ρ , can not be n n=1 π ignored at high temperature. Truncation method is valid only for small q, where the solutions converge easier for small cutoff. At high q, we need a larger cutoff, which dramatically increases the complexity of calculation. An example of eigenvalue distribution computed by the method in this section is shown in fig 3.1.  16  3.2. Method A2: Resolvent solution  Figure 3.1: Density of eigenvalues as a function of the angle around the circle, (computed by method A1 for m=20). The top graph, at q=0.1, shows that the first two Fourier modes are sufficient to approximate the density well (no improvement is seen when the first five mode are used). The bottom graph, at q=0.3, demonstrates the need to use a higher number of modes with a narrower eigenvalue distribution.  17  3.2. Method A2: Resolvent solution  Figure 3.2: Effective action, (computed by method A1 for m=20). The values of action computed with different cutoffs are not obviously dissimilar for different cutoff at low q, but only the one with the largest cutoff keeps overlapping with others to higher q. Judging by the overlapping region, it suggests how high the temperature is method A1 valid for the given cutoff.  18  3.2. Method A2: Resolvent solution  3.2  Method A2: Resolvent solution  Assuming the matrix model contains a large number m of matrices in the following section, we solve for ρapp (θ) by expanding the effective potential at large m. With a standard resolvent technique in a random matrix model [8], higher order approximate solutions are derived explicitly in section 3.2.4.  3.2.1  Expanding the potential in the large m limit with the high T approximation  Large m and q First of all, the interacting potential can be simplified in the high temperature limit. Thus, it is possible to have an analytic solution to compare with different numeric algorithms in the region when the approximations are valid. The interacting potential eq(2.29) is rewritten as (1 − q)2 u − ln sin2 ( ) + 4 2  4q u sin2 ( ) (1 − q)2 2 (3.23) At high temperature, eigenvalues are squeezed into a small region, and therefore u 1. It is important to add another limitation in order to expand 4q 2 u the last term, m ln 1 + (1−q) 2 sin ( 2 ) . Increasing q closer to 1 reduces the V (u) = m ln  range of u but does not guarantee  4q (1−q)2  2  + m ln 1 +  sin2 ( u2 ) to be small, which is meant  x2  to expand ln(1+x) ≈ x− 2 +. . . . The behavior of the effective action will be the main interest of this thesis in the situation where the expansion breaks down even at extreme high temperatures. Fortunately, large number m of matrices in the model also narrows down the distribution of eigenvalues, so the potential can be expanded in both the large m and high temperature limit. Naively replacing sin u2 → u2 for u 1 in the high temperature limit, V (u) ≈ m ln  u2 q (1 − q)2 − ln + m ln 1 + u2 . 4 4 (1 − q)2  The last term suggests us to rescale the variable to the dimensionless one, mq 2 u ˜2 = 2(1−q) 2 u , in order to isolate the role of m. The width of the density distribution is now independent of the temperature with the simplified potential, V (˜ u) = − ln u ˜2 + m ln 1 +  2˜ u2 (1 − q)2 mq + (m − 1) ln + ln( ). (3.24) m 4 2 19  3.2. Method A2: Resolvent solution Since u ˜ does not diverge in the large m limit, the potential can be further simplified with the analytic solution ρapp (x) by expanding the second term, k+1 xk , ln(1 + x) = ∞ k=1 (−1) k mq 2˜ u4 (1 − q)2 + ln( + O(m−2 ) + (m − 1) ln ). m 4 2 (3.25) Ignoring terms O(1/m) in the large m limit, the density function is derived as 2 ρapp (˜ x) = 1−x ˜2 , (3.26) π which can be checked by the net force equation for |x| ≤ 1, V (˜ u) = − ln u ˜2 + 2˜ u2 −  dy  dV (x − y) ρ(y) = dx  1  dy −1  −2 + 4(x − y) x−y  2 π  1 − y 2 = 0. (3.27)  The fact u ˜ must be within [−2, 2] also satisfies the condition that it does not diverge in the large m limit in order to expand the potential (3.25). The 2  approximate boundary for ρ(θ) is θ0 = 2(1−q) mq . Comparing ρapp to the one derived from a truncated Fourier expansion =1 (θ) ∼ (3.18), ρTtrunc  2 πθ0 (A1)  1−  θ2 , θ02 (A1)  where θ0 (A1) ∼  √2 , mq  we clearly rec-  ognize their similar form. However θ0 (A1) is only valid for small q, and =1 diverges at high q and m. Therefore the method of truncated Fourier ρTtrunc expansion fails to be a good approximate solution in the region we are interested. This approximate solution (3.26) can also be derived by the resolvent method, which is introduced in the next section. Relaxing the high q assumption Alternatively, by considering sin2 u2 = u2 u4 u6 8 ˜, we can expand the 4 − 48 + 1440 + O(u ) and the same scaled parameter u potential with respect to 1 − q as V (˜ u) = − ln u ˜2 − ln  +m ln 1 + +(m − 1) ln  (1 − q)2 u ˜ 2mq  2mq sin2 (1 − q)2 u ˜2 4q sin2 (1 − q)2  (1 − q)2 u ˜ 2mq  (1 − q)2 mq + ln( ) 4 2 20  3.2. Method A2: Resolvent solution (1 − q)2 2 (1 − q)4 4 (1 − q)6 6 u ˜ + u ˜ − u ˜ + O((1 − q)8 ) 6mq 90m2 q 2 2520m3 q 3 (1 − q)6 8 2 2 (1 − q)2 4 (1 − q)4 6 ˜ − u ˜ + u ˜ − u ˜ + O((1 − q)8 ) +m ln 1 + u m 3m2 q 45m3 q 2 1260m4 q 3 mq (1 − q)2 + ln( ). +(m − 1) ln 4 2  = − ln u ˜2 − ln 1 −  Without assuming 1 − q 1, we can recognize in the first line the last 6 expanded terms are actually orders of O( (1−q) ), and in the second line they 3 3 m q 6  are orders of O( (1−q) ). Therefore with only the large m limit, the terms like m4 q 3 ln(1 + O(1/m)) ≈ O(1/m) − 21 O(1/m)2 can be expanded as before. Keeping 1 terms up to m , the expanded potential for any temperature is V (˜ u) = − ln u ˜2 + 2 +  (1 − q)2 6mq  2 (1 − q)2 + u ˜4 m 3mq (1 − q)2 mq +(m − 1) ln + ln( ). 4 2 u ˜2 −  (3.28)  (3.28) corrects (3.25) by relaxing the high temperature limit, and they are exactly the same in the large m limit. Thus u ˜ must also be the order of one in (3.28), and it is safe to expand the log terms and to neglect terms with factors of O(1/m2 ). Fortunately even with large m limit along, we still get the same ρapp as eq(3.26) by ignoring terms with O(1/m). We continue to study the effective action under large m and high q approximations in the following. Approximate action Sapp is the approximate effective action arising from the approximate density distribution eq(3.26) in the large m limit, and it is calculated below. Sapp =  1 2  = m ln  dxdyVapp (x − y)ρapp (x)ρapp (y) 1−q 1 + ln 2 2  mq (1 − q)2  +  1 2  ln 2 +  4 · C + 1 (3.29) π2  √ 1 where C comes from −1 − ln(x − y)2 1 − x2 1 − y 2 dxdy ≈ 4.654244782 = C. With the help of the net force equation, we can further estimate the next order correction of Sapp without deriving ρapp to the next order.  21  3.2. Method A2: Resolvent solution Considering the possible difference of boundary, ∆, in the linear perturbation of Sapp , B+∆  B+∆  −B−∆  −B−∆  Sapp + δS =  dxdy  (ρapp (x) + δρ(x)) (ρapp (y) + δρ(y)) × (Vapp (x − y) + δV (x − y))  B  B  −B  −B  dxdyδV (x − y)ρapp (x)ρapp (y)  → Sapp + B+∆  +2  B  dyVapp (x − y)δρ(x)ρapp (y),  dx −B−∆  (3.30)  −B 2  2  2 where Vapp is (3.28) in the large m limit, and δV = (1−q) ˜2 − m + (1−q) u ˜4 , 6mq u 3mq and δρ is the correction of ρ due to δV . The last term of 3.30 disappears in the first order correction of Sapp , which can be shown by the help of integration by parts. B+∆  B  dyVapp (x − y)δρ(x)ρapp (y)  dx −B−∆  −B  B+∆  B  = −B  dyρapp (y) (Vapp (x − y)f (x)) |B+∆ −B−∆ −  dx∂x Vapp (x − y)f (x) −B−∆ x  where f (x) ≡  dyδρ(y). 0  For the even potential V (u), because ρ(y) and integration region are all symmetric for changing the variable y → −y, the first term can be rewritten as, B  [f (B + ∆) − f (−B − ∆)]  dyρapp (y)Vapp (B + ∆ − y) = 0. −B  In the above equation, it vanishes because of f (B + ∆) − f (−B − ∆) = B+∆ −B−∆ dxδρ(x) = 0. Recall the net force equation,  B −B  dy∂x Vapp (x−y)ρapp (y) = 0, the second  22  3.2. Method A2: Resolvent solution term vanishes in the interval x ∈ [−B, B], and the remainder is subleading. −B  B+∆  −  +  B  −B−∆  B  ≈ −∆ f (B +  dy∂x Vapp (x − y)ρapp (y)  dxf (x) −B  ∆ ∆ ) + f (−B − ) 2 2  B  dy −B  ∆ 2 ∂ Vapp (B − y)ρapp (y) 2 x  2  ∼ O(∆ )  (3.31)  Finally, the linear correction of action can be explicitly calculated as δSV  =  1 2  dxdyδV (x − y)ρapp (x)ρapp (y) = −  10q + (1 − q)2 (3.32) 16m  In the large m limit at high temperature, the linear order correction 1 decays as m , and Sapp is a valid approximate solution.  3.2.2  Comparing with free oscillators  In this section, reorganizing Sapp reveals the action is converging to the free oscillators action in the large m limit at high temperature, which means the adjoint matrix model will eventually be free and decoupled in this limit. A normal simple harmonic oscillator with a scalar Hamiltonian has par1 tition function as ZQHO (q) = 1−q , and the effective action is all irreps = − ln ZQHO = ln(1 − q). SQHO  (3.33)  The partition function of n free oscillators is simply the products of each, so the effective action will be m ln(1 − q) for free m matrices model, where the effective action is normalized, Sef f ective = NS2 . In the large m limit of our matrix model, the leading order of the action, referring to eq(3.29), is Sapp = (m − 1) ln(1 − q) +  1 ln mq − m ln 2. 2  (3.34)  In the high temperature limit, the effective action is Sf ree = (m − 1) ln(1 − q),  (3.35)  which looks like a matrix model for m − 1 free matrices. The missing freedom of a matrix must be due to the gauging of the SU(N) symmetry. Or, alternatively, it is due to restricting the Hilbert space of an ungauged model to the singlet sector. 23  3.2. Method A2: Resolvent solution  3.2.3  The standard resolvent method  In the large N limit, the random matrix model for one hermitian matrix with a single trace potential can be solved by a standard method of resolvent([3], [8]), and our model with multiple matrices in this thesis can also be solved by the same method with the simplified potential in large m limit at high temperature. We first review the general result in one matrix model. N  dλi ∆2 (λ)e−N  Z=  N j=1  V (λj )  (3.36)  i=1  Because of the stationary condition for the effective action, we vary the action according to eigenvalues, λi , and we derives the saddle point equation, i.e. equation of motion, 1 N  V (λi ) =  j<i  1 . λi − λj  (3.37)  The index of the eigenvalues is assigned for convenience such that λ1 < λ2 < ... < λN . In the large N limit, λi can be treated as a non-decreasing differentiable function λ(x) of x ∈ [0, 1] with λi = N λ(i/N ) for i = 1, ..., N . With the normalized spectral density of eigenvalues ρ(λ) =  1 N  N  δ(λ − λi ) = i=1  dx , dλ(x)  (3.38)  all the summation in the calculation can be represented as an integral over λ. The saddle point equation becomes V (λ) = 2  dx  ρ(x) , λ−x  (3.39)  and the solution of this equation, ρ(x), will minimize free energy at the saddle point, 1 ln Z = N →∞ N 2 lim  ρ(x)ρ(y) ln(x − y) −  ρ(x)V (x).  (3.40)  The saddle point equation is solved by introducing its Hilbert transform, i.e. the trace of the resolvent of the matrix M, ω(z) =  1 1 Tr N z−M  =  1 N  N i=1  1 = z − λi  ρ(x) dx. z−x  (3.41) 24  3.2. Method A2: Resolvent solution Thus, the saddle point density is just the real part of the discontinuity of ω(z) on the branch cuts. 1 ω(x ± i ) = V (x) ∓ iπρ(x) 2 Multiplying the saddle point equation by i gives ω 2 (z) +  1 λi −z  1 1 ω (z) − V (z)ω(z) = N N  N i=1  and summing over the index  V (z) − V (λi ) λi − z  V (z) − V (x) ≡ −P (z), dxρ(x) x−z  =  (3.42)  (3.43)  where the term N1 ω (z) can be neglected in the large N limit. Therefore above equation gives 1 1 ω(z) = V (z) − 2 2  (V (z))2 − 4P (z).  (3.44)  The density function is thus ρ(s) =  1 2π  4P (z) − (V (z))2 .  (3.45)  Generally for V (z) ∼ O(z l ), ω(z) has 2(l−1) branch points corresponding to the roots of V −4P , thus ρ(x) is not zero in separate regions corresponding to l − 1 branch cuts. In the case of the potential with a single minimum, eigenvalues are accumulated within the potential well, and there is only a single branch cut.  3.2.4  The resolvent of expanded potential of adjoint matrix model  For the matrix model in this thesis, the effective two body potential is 1 1 (1 − q)2 V (u) = − ln u2 + αu2 + βu4 + O(u6 ) + m ln + ln 4, (3.46) 2 4 4 2mq mq 1 where α = (1−q) 2 + 6 and β = − 3(1−q)2 − The net force equation is  0=  dV (x − y) ρ(y)dy = dx  2mq 2 . (1−q)4  −2 + α(x − y) + β(x − y)3 ρ(y)dy, x−y 25  3.2. Method A2: Resolvent solution and it can be rewritten as −2 ρ(y)dy + x−y  α + 3β  y 2 ρ(y)dy x + βx3 = 0  (3.47)  due to the fact, dyρ(y) = 1, and the assumption of even density distribution, where dyρ(y)y = 0 and dyρ(y)y 3 = 0. We can treat eq(3.47) as the equation of motion for a particle balanced by two effective potentials. The first term in eq(3.47) is a two body interaction, dVP (x − y) ρ(y)dy = dx  −2 ρ(y)dy, x−y  and the second term is a local potential, dVL (x) = (α + 3βA2 ) x + βx3 , dx where we denote Ak = y k ρ(y)dy as a new variable. Thus the expanded action can be rewritten into the form of the one with standard solutions in the random matrix model. S =  =−  N2 2  N2 2  VP (x − y)ρ(x)ρ(y)dxdy + N 2  ln(x − y)2 ρ(x)ρ(y)dxdy + N 2  VL (x)ρ(x)dx  α + 3βA2 2  x2 +  β 4 x ρ(x)dx. 4  This becomes a problem of the eigenvalue density for a hermitian matrix 2 model with a single trace potential V (M ) = T r α+3βA M 2 + β4 M 4 and 2 with a consistency condition T rM 2 = A2 . In terms of the resolvent of the matrix model, we have this P (z) = =  1 V (z) − V (M ) Tr N z−M  (3.48)  1 T r(α + 3βA2 + βz 2 ) + N  = α + 4βA2 + βz 2 , where  1 N T rM  1 2 N T rM (z)− 12 (V  = 0 and  Therefore by ω(z) = 12 V  1 T r(βzM ) + N  1 T r(βM 2 ) N (3.49)  = A2 . (z))2 − 4P (z), the density function can 26  3.2. Method A2: Resolvent solution be written 2πiρ(s) = ω(s + i ) − ω(s − i ) → 2πρ(s) =  4P (z) − (V (z))2  =  4(α + 4βA2 + βs2 ) − ((α + 3βA2 )s + βs3 )2 (3.50)  =  Q(s),  where Q(s) ≡ 4P (z)−(V (z))2 = −β 2 s6 −(2αβ +6β 2 A2 )s4 −((α +3βA2 )2 − 4β)s2 + (4α + 16βA2 ). Q(s) is a 6th order polynomial with 3 pairs of symmetric roots on the complex plane, and these roots as branch cuts result in the discontinuity of ω(z) and therefore ρ(s). With the consistency condition, A2 = x2 ρ(x)dx, density function can be found analytically. Similarly, higher order corrections of the interacting potential in the large m limit at high temperature expansion can be calculated including higher order terms in the polynomial, Q(s) ∼ O(sk ), and with more consistency equations, Ak = xk ρ(x)dx. Q(s) can be written as Q(s) = −β 2 (s2 − B 2 )(s2 − ξ)(s2 − η),  (3.51)  where B is the always the real root and represents the boundary of density function, ξ andη can be either real or imaginary complex conjugate roots. Three roots of Q(s) are explicitly as the following root1 = root2 = − root3 = +  −2γ f (A2 , β, γ) − − 3β 3β 2 −2γ 1 f (A2 , β, γ) + 3β 2 3β 2 √ 3 f (A2 , β, γ) − 2 3β 2 −2γ 1 f (A2 , β, γ) + 3β 2 3β 2 √ 3 f (A2 , β, γ) − 2 3β 2  γ2 3  1 f (A2 , β, γ) 2 γ 1 + 4β + 3 f (A2 , β, γ) 4β +  4β +  γ2 3  4β +  γ2 3  1 f (A2 , β, γ) γ2 1 + 4β + 3 f (A2 , β, γ) 1 , f (A2 , β, γ)  (3.52)  27  3.2. Method A2: Resolvent solution where γ ≡ α + 3βA2 and f (A2 , β, γ) ≡  − 54A2 β 5 − 18β 4 γ − β 3 γ 3 √  + 6 3  1/3  −16β 9  +  27A22 β 10  + 18A2  β9γ  −  β8γ2  + A2  β8γ3  Since f (A2 , β, γ) is not necessary a real number, B 2 can not be chosen among the three roots before solving A2 for given m and q. We can brutally pick a root as B 2 and numerically solve A2 by the consistent equation B 2 −B x ρ(x)dx = A2 three times, and we find out the general results of roots B 2 , ξ, and η in the section 3.2.5. Branch cuts We can verify that Q(s) has only a branch cut on the real axes from -B to B, and ξ = η on the real axes. Since we have replaced sin u by u at the high temperature limit, the eigenvalue distributes from −∞ to ∞ rather than from 0 to 2π. At the high temperature limit, only the group elements around the identity element contribute the integral for the partition function, and we are using the Hermitian matrix model instead of the original unitary matrix model, i.e. the integral variable is the Hermitian matrix M instead of the unitary matrix U, where U = 1+iM . Therefore, the eigenvalues of the Hermitian matrix must be real numbers, and the branch points of Q(s) must be on the real axes. The external local potential V˜ is β V˜ (x) = x4 + 4  α + 3βA2 2  x2 .  (3.53)  Because β is negative, the 4th order polynomial V˜ has only a local minimum or none, and no minimum case corresponds to the repulsive potential when eigenvalues do not have stationary distribution without the boundary, and the resolvent method gives no sensible answers. The local minimum is like a potential well contains all the eigenvalues on the branch cut from -B to B. The results in sec3.2.5 confirms the argument above. Discriminant Interestingly, it is possible to find further simplified solutions by applying the knowledge of the branch cuts. The discriminant of Q(s) = 0 for its coincident roots ξ and η. The discriminant of Q(s) is 216 β 10 (A2 β + γ)(−16β + 27A22 β 2 + 18A2 βγ − γ 2 + A2 γ 3 )2 = 0.  (3.54) 28  .  3.2. Method A2: Resolvent solution We can solve this discriminant equation for the positive A2 , which implies 1  f (A2 , β, γ) = −β(12β + γ 2 ) 2 ,  (3.55)  1 −2 γ − (12β + γ 2 ) 2 , 3β  (3.56)  and therefore B2 = and ξ=η=  −2 3β  1 1 γ + (12β + γ 2 ) 2 2  .  (3.57)  Thus, we can find the eigenvalue density function according to eq(3.50). Alternatively, we can also solve the relation between coefficients as an equation set by the coincident roots. Q(s) = −β 2 (s2 − B 2 )(s2 − ξ)2 = −β 2 s6 − (2αβ + 6β 2 A2 )s4 − ((α + 3βA2 )2 − 4β)s2 + (4α + 16βA2 ). Comparing each terms results a solvable set of three equations, 4α + 16A2 β = B 2 β 2 ξ 2 2  2  (3.58)  2  2αβ + 6β A2 = −β (B + 2ξ) 2  2  2  2  (α + 3βA2 ) − 4β = β (2B ξ + ξ ).  (3.59) (3.60)  Computing the first two equations, we have A2 and B 2 as 2α + αβξ 2 + β 2 ξ 3 β(8 + 3βξ 2 ) α + 4βξ . = −4 β(8 + 3βξ 2 )  A2 = −  (3.61)  B2  (3.62)  Thus the last equation appears to be 0  α2 − 64β + 16αβξ + 3αβ 2 ξ 3 + 3β 3 ξ 4 (8 + 3βξ 2 )2 → α2 − 64β + 16αβξ + 3αβ 2 ξ 3 + 3β 3 ξ 4 = 0. =  −4  (3.63)  Solving this 4th order polynomial for ξ leads to the solution of our matrix model with potential eq(3.46).  29  3.2. Method A2: Resolvent solution B  Ρx  x  B 1.0  0.9 0.95  0.9  0.85  1.1  q  0.8  0.7  1  0.6  1.05 1  0.5 6  8  10  12  14  16  m  Figure 3.3: for m ≤ 8.  3.2.5  ρ(x)dx on (m,q) plane; color= integral values.  ρ(x)dx = 1  Results  Inconsistency Following the resolvent method, the formula eq(3.51) does not always result in a consistent solution, i.e. ρ(x)dx may not equal to 1, which is contradict to eq(3.38). It is observed that inconsistent solutions happen at small m ≤ 8 as shown in figure 3.3. The inconsistency is due to the asymptotic behavior of the 4th order approximate potential, which diverges to negative infinity. Therefore the 4th density will rather spread out because of the strongly repulsive force. Vapp is a good approximation of the exact potential only within a small region where the distance between eigenvalues is small. In fig3.4, we display the 4th , V 2nd , and V difference between Vapp exact . We check the boundary of eigenapp values with ρapp and ρexact (N 2), which is solved by numerical method N2 in next chapter, and it shows the eigenvalues at two ends are separated too 4th diverges to negative infinity. Since the potential well of far, where Vapp  30  3.2. Method A2: Resolvent solution 4th is wider than V 2nd and V Vapp exact , the eigenvalues will spread even wider, app and they repulse each other out of the potential well. In the parameter region where ρapp or ρexact (N 2) distributes wider than the potential well of 4th , the negative divergence of V 4th breaks the assumption of the stationVapp app ary eigenvalue distribution. Therefore the resolvent method results in the inconsistent results. Generally for a larger m, eigenvalues are within a smaller region where 4th Vapp well overlaps on Vexact , and it does not diverge to negative infinity, thus the resolvent method works.  Branch cuts In figure 3.5, three roots from 3.52 are examined for some values of (m,q) with the condition, A2 = x2 ρ(x)dx. Root1 , root2 , and root3 happens to be B 2 , ξ, η exactly, since root1 is always real, and root2 is conjugate to root3 . The data in the region of q < 0.8 and m < 12 violets our analysis of the branch cuts in sec 3.2.4, where ξ and η are not on the real axes, and there are more branch cuts. In the rest of the parameter region, ξ and η are real and equal to the other, which result in a zero length branch cut outside −B ↔ B because ξ is bigger than B 2 as shown in the fig3.6. When we derive inconsistent solutions, which has ρdx = 1 or more than one cuts, it means the assumption of stationary condition does not hold, and therefore the resolvent method is failed. By replacing sin(u) to u, we actually relax the periodic boundary condition, and there is no boundary to bound eigenvalues. Whenever the solution is inconsistent, eigenvalues 4th , and eigenvalues at two far distribute wider than the potential well of Vapp ends repulse each other to infinity. Therefore the solutions is not stationary and consistent. Correction terms With the help of 4th order analytic solution, it is 1 able to check that the correction terms of 2nd order action decreases as m , and therefore we confirm the validity of large m expansion. The explicit 2nd or V 4th is provided as the discrepancy between the actions applying Vapp app following. We introduce few new notations here to make the following argument more clear and systematic. Explicitly, the second order potential V 2nd refers 2nd as eq(3.25) neglecting terms with a factor of 1/m, and the forth orto Vapp 4th in eq(3.28). Here ρ2nd and S 2nd correspond der potential V 4th refers to Vapp to the density and action in the potential V 2nd , and they are solved analytically as ρapp in eq(3.26) and Sapp in eq(3.29). Likewise ρ4th and S 4th are for V 4th in the large m expansion at any temperature. 31  3.2. Method A2: Resolvent solution  2nd , and V 4th and relative denFigure 3.4: Potentials graph for Vexact , Vapp app sity distributions. For the distance between eigenvalues above 0.05, each approximate potentials have significant difference with the exact potential. 4th starts diverging to negative infinity at u = 0.1. According to ρ Vapp app and ρexact (N 2), eigenvalues at two ends are separated at least further than 0.1. 4th strongly repulse others, and we can not find the Thus eigenvalues in Vapp 32 stationary point.  3.2. Method A2: Resolvent solution Re B  Im B  0.9  0.9  0.8  0.8 q  1.0  q  1.0  0.7  0.7 0.1  0.6  0.6  0.05  0.15  0.2 0.25 0.5  0.5 6  8  10  12  14  16  6  8  10  m  12  14  16  14  16  14  16  m  Re Ξ  Im Ξ  1.0  1.0  0.9  0.9  0.05  0.8  0.02  q  q  0.8  0.7  0.7 0.08 0.06  0.6  0.1  0.6  0.2 0.15  0 0.04  0.25 0.12  0.3 0.5  0.1  0.5 6  8  10  12  14  16  6  8  10  m  12 m  Re Η  Im Η  1.0  1.0  0.9  0.9  0.05  0.02  q  0.8  q  0.8  0.7  0.7 0.04 0.06  0.6  0.1  0.6  0.2 0.15  0.1  0  0.25  0.08 0.12  0.3 0.5  0.5 6  8  10  12 m  14  16  6  8  10  12 m  Figure 3.5: Roots of Q(S) on (m,q) plane. Roots 1, 2, 3 in 3.52 correspond to B 2 , ξ, η. Root1 is always real. root2 and root3 are complex conjugate pair, and they are not real only at q < 0.8 and m < 12. 33  3.2. Method A2: Resolvent solution  Re Ξ  B^2  1.0  0.9  0 0.05  q  0.8  0.7  0.1 0.6 0.05  0.5 6  8  10  12  14  16  m  Figure 3.6: Re[ξ] − B 2 on (m,q) plane; It is negative only at q < 0.65 and m < 10, and therefore −B ↔ B is always the branch cut crossing the center.  34  3.2. Method A2: Resolvent solution  S 4th ≡ and S 2nd ≡  dxdyV 4th (x − y)ρ4th (x)ρ4th (y) dxdyV 2nd (x − y)ρ2nd (x)ρ2nd (y).  With δV ≡ V 4th − V 2nd and δρ ≡ ρ4th − ρ2nd , we denote each correction terms clearly as the following. S 4th = S 2nd + δSV + 2δSρ + 2δSρV + δSρρ + δSρρV  (3.64)  1st order perturbation, δSV ≡  dxdyδV (x − y)ρ2nd (x)ρ2nd (y)  and δSρ ≡  dxdyV 2nd (x − y)ρ2nd (x)δρ(y).  2nd order perturbation, δSρV ≡  dxdyδV (x − y)ρ2nd (x)δρ(y)  and δSρρ ≡  dxdyV 2nd (x − y)δρ(x)δρ(y).  3rd order perturbation, δSρρV ≡  dxdyδV (x − y)δρ(x)δρ(y).  Figure 3.7 shows a comparison between the resolvent method solving the 4th order potential and the approximate action considering 2nd order potential. The most important term is δSV as claimed in equation(3.32), and the other terms correct the action to the next order of 1/m as explained in sec3.2.1. Correction terms converge to zero with increasing number of matrices verifying the validity of the approximate action and the large m expansion.  35  3.2. Method A2: Resolvent solution  Figure 3.7: Correction terms of action at q=0.9. As the number of matrices increases, correction terms become much smaller, and therefore the validity of the large m expansion is faithful. Except the first data set at m = 5, the most important correction with the greatest absolute value is δSV as expected. Considering the inconsistency in the resolvent method at small m, it is reasonable to exclude the first data set at m = 5.  36  Chapter 4  Numerical methods and results Several analytic methods have been introduced to tackle the effective action, but they are all adequate only within certain parameter regions, and none of them is valid for a small number of matrices. Truncated Fourier expansion works at small q, and this requirement asks for large m since q must be 1 above deconfinement temperature. Resolvent method works higher than m 1 for large m and high q, and the correction is proportional to m . The role of high q is to decrease the boundary of the eigenvalue distribution, which is approximately sin(θboundary ) ∼ θboundary . With the technique of quadratic programming, this chapter provides numerical methods valid in general cases and compares their results with previous analytic solutions.  4.1  Method N1: Truncated Fourier expansion  First of all, we introduce a numerical method for truncated Fourier expansion, which can easily take care of large amount of Fourier modes, and therefore it resolves action at high temperature.  4.1.1  Quantized minimization  It requires massive calculation to pursue a higher order analytic solution by truncated expansion, and the action series decays much slower at high temperature, so it is worth finding another algorithm to solve this numerically. The Fourier modes of the density distribution, ρn , are derived by min2 imizing the action, S = S0 + ∞ n=1 ρn Vn , where S0 = (1 − m) ln 2. These 1 + π1 ∞ Fourier modes must satisfy the positivity constraint, ρ(θ) = 2π n=1 ρn cos nθ ≥ 0 ∀θ ∈ [−π, π]. Then T , a cutoff of ρn in the summation, is introduced since highly oscillating ρn vanishes as q n 1, and θ axis is divided by a large number Nθ of small intervals.  37  4.1. Method N1: Truncated Fourier expansion Thus, we obtain a minimization problem of Nc variables with Nθ constrains. T  Minimizing  ρ2n Vn  S = S0 + n=1  with constrains  ρ(θi ) =  1 1 + 2π π  T  ρn cos nθi ≥ 0  for i = 1, . . . , Nθ  n=1  Fortunately, there are many reliable quadratic programs solving this quadratic optimization problem, and therefore the calculation can easily reach higher orders of truncation. We use Nθ = 600 and T up to 200, which is much higher than a feasible cutoff of around 10 in analytic truncation, method A1. Nevertheless, the truncation method only approaches the minimum of action from a larger value, and we can always have another smaller value of action by increasing the cutoff, T . So the cutoff strongly effects the value of action, and looking for an accurate results at high temperature, q ∼ 1, requires very massive works. In fact, even when the results of density function converges with different T , in the next section, we show the truncated Fourier expansion method resolves inaccurate density distribution at high temperature.  4.1.2  Results  Eigenvalue distribution When the cutoff T is truncated at sufficiently high order of ρn series, ρ(θ) with much higher cutoff will overlap on the previous one. With insufficient T , the density data will distribute much wider than the one with sufficient T . Figure 4.1 shows ρn with different T overlapping together and results in a consistent density distribution, ρ(θ) at low temperature, q = 3. The comparison data, ρexact (N 2), is derived by quadratic programming method with finite elements, (method N2), which appears in the section 4.2. ρexact (N 2) solves the exact potential as method A1 and N1, unlike method A2 solving the approximated potential in large m expansion. On the above part of the figure, it shows higher order of Fourier modes ρn diminish very fast, and therefore it is reasonable to truncate the Fourier series. Redrawing the density function in real space, ρ(θ) shows the results are consistent as long as T is large enough. Comparing to method A1, method N1 with quadratic programming effectively simplifies the calculation process and extends the parameter region to higher q with larger feasible cutoffs. 38  4.1. Method N1: Truncated Fourier expansion  Figure 4.1: Method N1 with different cutoffs solving for qn as a function of n. The Fourier transformed results from ρexact (N 2) shows that the higher order Fourier modes can be ignored at this temperature. At q=0.3, ρn>100 does not matter because q n>100 can be ignored in the truncated Fourier expansion. In the lower graph, dot lines are solved by method N1, and solid lines are solved by method A1.  39  4.1. Method N1: Truncated Fourier expansion  Figure 4.2: Method N1 with different cutoff solves actions as a function of q. Diamond data are solved analytically by method A1, and solid dots data are solved by method N1, and solid lines is the approximated action Sapp . We clearly observe method A1 is invalid at q > 0.8, and method N1 improves the accuracy till q = 0.9.  40  4.2. Method N2: Quadratic programming Effective action For m=20, Sapp represents faithful approximation in the large m expansion, and it demonstrate the inaccuracy region at high temperature of other methods. Figure 4.2 shows method A1 results in inaccurate values of action after q = 0.8, which deviate from the results of method N1 and Sapp . The expected bad behavior at high temperature can be adjusted in numerical truncation method N1 because more higher order modes are included easily. Since we are interested in high temperature region in order to clarify whether or not the matrix model will be free, a proper method at high temperature even for small number of m is studied in the following section.  4.2  Method N2: Quadratic programming  Directly minimizing the action in the angle space by quadratic programming turns out to be the most effective method.  4.2.1  Finite element method  Due to the form of the pair interaction, the action can be minimized by quadratic programming with finite elements method. In the large N limit, the amount of eigenvalues is large enough to treat its density function as a continuous function along angle axis. By discretization of the density function, the action becomes a quadratic summation and is readily minimized numerically. ρ(j∆θ) → ρj V ((i − j)∆θ) → Vi−j B  Vij ρi ρj ∆θ2 +  V (x − y)ρ(x)ρ(y)dxdy →  S= −B  i=j  V0 ρ2i ∆θ2 i  where B = n∆θ means the boundary of the eigenvalue region. The coordinate is set up that ρ(x) is a even function, and therefore only half of the variables of ρj need to be calculated in the minimization. Expressing ρ, and V in the matrix form, ρ=  ρn ρn−1 . . . ρ1 ρ1 ρ2 . . . ρn  41  4.2. Method N2: Quadratic programming   V0  V1    V1 V0   .. ..  . .   ← → Vn−1 Vn−2 V =  Vn Vn−1    Vn+1 Vn   .. ..  . . V2n−1 V2n−2  . . . Vn−1 Vn Vn+1 .. Vn−1 Vn ... . .. .. .. . V1 . . V1 V2 . . . V0 . . . V1 V0 V1 .. ... . V1 V0 .. .. .. . Vn−1 . . . . . Vn Vn−1 Vn−2  . . . V2n−1 .. ... . .. . Vn+1 ... Vn . . . Vn−1 .. ... . .. . V1 ... V0                   n  → T←  S=ρ V ρ=2  ρi ρj (Vi−j + Vi+j−1 ) .  (4.1)  i,j=1  It is worth pointing out that in the large N limit, V0 is not zero nor is it the divergent V (0). We provide the detailed derivation of V0 below. By equally dividing from the edge of the eigenvalue distribution to the other edge into 2n pieces, ρ(j∆θ+x) is treated as a constant ρj , in the region x ∈ [(j − 1/2)∆θ, (j + 1/2)∆θ]. Referring to the precise integral shows the diagonal part of the action is ∆θ/2  ρ2j V0 ∆θ2 = j  dxdyV (x − y)ρ(j∆θ + x)ρ(j∆θ + y). j  (4.2)  −∆θ/2  Within a small region, ρ(j∆θ + x) ∼ ρj , and the integral becomes ∆θ/2  ρj 2  dxdyV (x − y).  (4.3)  −∆θ/2  Interestingly, V0 has the same value obtained by an explicit calculation whether the exact potential or the approximated potential is used. ∆θ/2  dxdy − ln −∆θ/2  = −  3 − ln  ∆θ2 4  u2 + m ln 4  ∆θ2 + m 4  q (1 − q)2 ln 1 + ∆θ2 q (1 − q)2  qu2 (1 − q)2 + 4 4  (1 − q)2 ∆θ tan−1 q  + ∆θ2 −3 + ln  q ∆θ (1 − q)2  q 2 (1 − q)2 ∆θ + 4 4  (4.4) 42  4.2. Method N2: Quadratic programming Neglecting small terms, 3 − ln  ∆θ2 4  q ∆θ2 (1−q)2  << 1, it becomes  ∆θ2 + m 4∆θ2 − ∆θ2 − 3∆θ2 + ∆θ2 ln  =  3 − ln  ∆θ2 + m ln 4  (1 − q)2 4  (1 − q)2 4  +  q ∆θ4 (1 − q)2  ∆θ2  Therefore, ∆θ2 (1 − q)2 + m ln . (4.5) 4 4 V0 term also controls the strength of the pair interaction because all the factors controlling the strength like q and m can be moved into the V0 for Vapp . In other words, the density function is scale free of m and q for the approximate potential. By scaling x → x = Bx V0 = 3 − ln  Vapp (u) = α − ln u2 +  2 2 2 u = α − ln B 2 − ln u 2 + 2u , B2  (4.6)  where α = m ln  (1 − q)2 4  + ln 4 , and B =  We can derive V0 = 3 + ln  4n2 + α, B2  2(1 − q)2 . mq  (4.7)  (4.8)  and  2 1 − x 2. (4.9) πB Shifting an overall constant potential leaves a scale invariant potential independent of matrix number and temperature, ρ(x ) =  Vapp (u ) = − ln(u 2 ) + 2u 2 , and V0  = 3 + 2 ln 2n.  (4.10) (4.11)  There is a factor in the algorithm we need to set up manually, the boundary of numerical eigenvalue distribution. The density distribution is discretized on a grid within this boundary, and it has to be wider than the resulting boundary of eigenvalues, where the density vanishes. The manual input boundary is like the wall of a box confining the eigenvalues, and these 43  4.3. Comparison of numeric results with analytics eigenvalues are held by the pair interaction between each other. While particles are close together within a small region, different orders of potential expansion makes no difference in solving ρ(θ). However we need to choose the manual boundary carefully, otherwise the distance between particles at two different boundaries can be large, such that the behaviors of different order potentials are different, see the upper part in fig3.4. Especially when solving the 4th order expanded potential, in setting the manually boundary one has to be very careful. Because the asymptotic 4th diverges to minus infinity unlike V 2nd or V behavior of Vapp exact , eigenvalues app are repulsed to the wall if the manual boundary is set too wide. However the 4th with a suitable boundary for a large enough m does solution solving Vapp agree with the solution of the exact potential better than the one solving 2nd , figure 4.3. Vapp  4.2.2  Error estimation  Discretizing the continuous density function causes some error of the quadratic programming method. Since the 2nd order potential is very close to the exact potential in the large m limit at high temperature, the error can be 2nd and the estimated from the difference between the analytic solution of Vapp 2nd . By decreasing the volume of each quadratic programming solution of Vapp element in quadratic programming, the errors of both action and density decrease as well, see fig. 4.4 and fig. 4.5  4.3  Comparison of numeric results with analytics  4.3.1  Comparison of method N2 with methods A1, A2, and N1  Eigenvalue distribution As explained in the previous section, ignoring high frequency Fourier modes makes the method of truncated Fourier expansion results in poor solution. As shown in fig4.6, when ρ(θ) behaves like a delta function, data of different cutoffs barely overlap together at high temperature. In fig4.7, even when the truncated method N1 obtain results converge together using bigger cutoff, method N2 shows higher order modes can not be ignored at q = 0.9, and the result of method N2 is different from the one of method N1. Thus too many modes included in the calculation at high temperature make truncation not possible. Resolvent method gives a good result overlapping with method N2, but it is only a good approximation with very narrow boundary in the large m limit. 44  4.3. Comparison of numeric results with analytics  Figure 4.3: Density distributions obtained from different manual boundaries. B means the manual boundary set up in method N2. If B is set too big, the 4th repulse every eigenvalues to the boundary. If B is negative divergent Vapp set too small, the potential well does not have enough space for eigenvalues, and they are squeezed on the boundary.  45  4.3. Comparison of numeric results with analytics  Figure 4.4: The error in ρ is defined as the root mean square of density difference, ∆ρ =  2  i  ρi 2nd (N 2) − ρapp i , and ρi = ρ(i∆θ), ρapp i = ρapp (i∆θ).  Figure 4.5: The error in the action is defined as S 2nd (N 2) − Sapp  46  4.3. Comparison of numeric results with analytics  Figure 4.6: ρ(θ) vs x at high q and m. Black and yellow lines are solved by analytic methods with high q and large m expansion. Dash lines are solved by method N1 for different cutoff. Dash-dot line is solved by method A1. Red dots with error bars are solved by method N2.  47  4.3. Comparison of numeric results with analytics Thus QP method with finite elements is the most efficient way for any given number of matrices at any temperature.  Figure 4.7: Method N1 with different cutoff solves qn as a function of n. The results from quadratic programming with finite elements method N2 shows that the higher order Fourier modes can not be ignored at this temperature. At q=0.9, ρn>100 are still crucial because q n>100 can not be ignored in the truncated Fourier expansion.  Effective action In fig4.8, we show the action data obtained from every method with its highest feasible resolution. Defects of each methods are not as dramatic as shown in the density graph. Although truncation method still fails at high temperature, method N1 agrees with the approximation up to q over 0.9. The large m approximation is good for m = 20 such that Sapp and the resolvent method A2 agree with method N2 within all the calculated q region. For small m case as shown in fig. 4.9, when comparing to method N2 Sapp the m expansion is good even for m = 5. The ratio, Sexact (N 2) , approaches 1 very quickly for increasing m at high temperature. However, the 2nd approximation gives only 90% of the action value from method N2 for m=2, which suggests the model may not be free or converges to the free value very slowly even at extreme high temperature for small m as indicated in sec 3.2.2.  48  4.3. Comparison of numeric results with analytics  Figure 4.8: Comparison of data of the action obtained from every method. For m=20, the large m expansion already works very well at q=0.6. 2nd and 4th order approximations overlap with the N2 data of the exact potential, but the truncated method gives bad results at high temperature.  49  4.3. Comparison of numeric results with analytics  Figure 4.9: Sexact and Sapp as functions of q, for m=2,3,4,5. The error bars in Sexact are small enough to be ignored.  50  4.3. Comparison of numeric results with analytics  Figure 4.10: δSV , S 4th (A2) − Sapp , and Sexact (N 2) − Sapp as functions of q. S 4th (A2)−Sapp represents 4th order potential correction, and δSV represents the most important parts of it. Sexact − S2 represents all orders of the corrections of large m expansion.  51  4.3. Comparison of numeric results with analytics Correction terms To show when the large m expansion is appropriate, here we focus on corrections to the large m expansion by comparing the next order potential analytically in the resolvent method A2 and numerically in the quadratic programming method N2. In the figure 4.10, different orders of corrections are compared together. S 4th (A2)−Sapp represents the correction of considering second order of 1/m for the potential expansion, and δSV , which corresponds mostly important parts can be drawn analytically. With the numerical method(N2), Sexact (N 2) deals the general potential without expansion, so Sexact (N 2) − Sapp represents overall corrections in the large m expansion. S 4th (A2)−Sapp , Sexact (N 2)−Sapp , and δSV all merge together to zero with increasing m. This demonstrates the validity of the approximate solution ρ2th app and the large m expansion.  4.3.2  Comparing with free oscillators  By numerically calculating the action of the adjoint matrix model at high temperature, it is interesting to find out whether or not the action will be close to ordinary N 2 oscillators and at what temperature they are overlapped. Here the free action refers to (3.35) as calculated in the large m expansion, and we are mainly interested in small m cases because the action is known to be free at large m by eq(3.34).  Figure 4.11: Sexact and Sf ree as functions of q, for m=2,3,4,5. The error bars over Sexact are small enough to be ignored.  52  4.3. Comparison of numeric results with analytics The subtraction and division of actions in figure 4.12 shows that our matrix model will be mostly like the complete free model with m = 3 at high temperature since its difference to free action is almost zero. This is an accidental result in our specific matrix model. By exploring data at extreme high temperature in figure 4.13, there’s always a constant gap between Sexact and Sf ree in this region. While Sexact becomes larger and larger with increasing q, the ratio, Sf ree /Sexact , eventually converges to 1. So matrix model will be free at extreme hot temperature even for small m.  53  4.3. Comparison of numeric results with analytics  S  f ree Figure 4.12: Sexact (N 2) − Sf ree and Sexact (N 2) as functions of q, for m=2,3,4,5. The error bars in Sexact are small enough to be ignored. The discrepancy between Sexact (N 2) and Sf ree slightly tilts upward at high q, and the ratio converges to 1 at high q. We then check more data within a smaller region at high q in fig4.13  54  4.3. Comparison of numeric results with analytics  S  f ree Figure 4.13: Sexact (N 2) − Sf ree and Sexact (N 2) as functions of q graphs, for m=2,3,4,5 at high q = 0.95 ∼ 0.9999. The constant gap of Sexact − Sf ree Sf ree ln(1−q) suggests fitting Sexact by ln(1−q)+C , where C is a constant related to m.  55  4.3. Comparison of numeric results with analytics  Figure 4.14: The gap between Sexact and Sf ree as a function of m at q=0.99. For small m at high temperature( m < 6), the gap is like a constant independent to q according to fig4.13. At larger m, the gap converges to the approximate solution, Sapp − Sf ree = −m ln 2 + 12 ln mq. (see eq3.34) We can divide the discrepancy between Sexact and Sf ree , which equals to Sapp −Sf ree at large m or a constant gap at small m, by Sf ree = (m − 1) ln(1 − q), which diverges at q → 1, and the discrepancy always vanishes at enough high temperature for any value of m. Thus the relative error between Sexact and Sf ree will always disappear, and Sexact converges to Sf ree .  56  Chapter 5  Discussion and Conclusion 5.1  Summary  Partition function in matrix quantum mechanics This thesis is about the partition function in the matrix model with SU(N) symmetry. By decomposing the Hamiltonian into a direct sum of irreducible representations, the Hilbert space are divided into independent sectors of each irreducible representation, and therefore we can define the partition function in one sector for the corresponding irreducible representation. These partition functions involve the integral over the eigenvalues of group elements, eq(2.14), and the integral is computed by saddle point method. Thus the partition function is computed by finding the classical equilibrium distribution of eigenvalues. Various analytical and numerical methods are introduced to find the equilibrium distribution of eigenvalues minimizing the effective action. Each analytical method is valid with different approximations of the effective potential. Method A1, from [1], finds the density function ρ(A1) by neglecting higher order Fourier modes for q n 1 at low temperature. We are interested in high temperature region q ∼ 1 when too many Fourier modes are involved in the calculation. Method N1 can easily include more Fourier modes than method A1 in the truncated Fourier expansions by numerically minimizing the action with quadratic programming algorithm, and therefore method N1 improves the solutions in higher temperature. Looking for the solutions at high temperature limit, we expand the effective potential in large m, the number of matrices, and derive the simplified approximate potential. We solve for the approximate density function ρapp 1 analytically by ignoring terms with the factor m in the expanded poten2  tial. The boundary of ρapp = 0 is 2(1−q) mq , and ρapp behaves like a delta function at high temperature. Sapp at high temperature limit is just like mN 2 free quantum oscillators but with a different factor (m − 1)N 2 instead of mN 2 , and this difference is less important in the large m limit. Delta function-like eigenvalue distribution and decoupled effective action seem to relax the singlet condition for gauged model at high temperature for large m  57  5.2. Singlet condition at high temperature as claimed in [1] and [4]. Method A2 computes the density function ρ4th app (A2) 1 including terms in the potential to the order O( m ) by the standard resolvent method, [8]. However, the potential expanded to the next order is unstable for small m because the distance between eigenvalues is so long that the potential becomes repulsive, and the resolvent method fails to derive consistent solutions. When the potential is stable for m large enough, the branch cut analysis in method A2 shows there is only one clump of eigenvalues as expected. We keep examining further for the case of small m. Method N2 deals with the exact potential without any approximation, and it is a general numerical method valid for any effective potential for pairwise interaction. Method N2 uses quadratic programming algorithm to find the distribution which minimizes the action like method N1, but it calculates the action in angle space instead of Fourier space. We estimate the error of computing in the exact potential by calculating the approximate potential. The difference between the numerical value from method N2 and the analytical solution shall represent the numerical error from discretization of the angular variables and quadratic programming since Vapp is similar to Vexact at high temperature for large m. Method N2 agrees with truncated Fourier expansion at small q for any m, fig3.1, and it agrees with analytical solutions at any q for large m, fig4.6, and all our methods derive consistent results when their valid regions of m and q overlap. At high temperature, our matrix model has free oscillator-like action, Sf ree = (m − 1) ln(1 − q), which is verified by different methods for any m. In the large m case, the leading term of Sapp in eq(3.34) is Sf ree . With the results of actions for small m from method N2 in fig 4.13, we show Sexact (N 2) eventually approaches to Sf ree .  5.2  Singlet condition at high temperature  We have confirmed the eigenvalue distribution is like a delta function centered at θ = 0 at high temperature q → 1. In other words, the integral of the partition function is sharply peaked around the identity element, where eigenvalues are accumulated at θ = 0. This implies the partition function for the singlet irreducible representation can be computed by the saddle point method. The integral of the generalized partition function is approximately approached as Zsing (β) =  [dU ]Z(β, U ) ≈ Z(β, 1) = Z(β)  (5.1) 58  5.2. Singlet condition at high temperature since Z(β, 1) = R dim(R)ZR (β) = Z(β). Therefore the partition function for singlet irreducible representation converges to the ordinary partition function for the whole Hilbert space. [1] and [4] then argue the singlet condition does not have to be imposed for gauged model at high temperature limit. However, this deconfinement picture is not entirely correct. Otherwise we should observe the effective action as mN 2 ln(1 − q) instead of Sf ree . This different factor seems to result from a normalization factor in the saddle point method when we change the integral variables from group elements to eigenvalues. We define a delta function in the space of unitary matrices, dim(R)χR (U ). D(U ) = (5.2) R  Because of the orthogonality of characters, we have ˜ = χ ˜ (1) [dU ]D(U )χR˜ (U ) = dim(R) R  (5.3)  ˜ By changing the variables to eigenvalfor any irreducible representation R. ues of the matrix, the above integral becomes C i  dφi 2π  |∆(φi )|2 D(φi )χR˜ (φi ) = χR˜ (φi = 0),  (5.4)  where C is a normalization constant. Thus D(U) is expressed in terms of eigenvalues as 1 δ(φk ), (5.5) D(φ) = C|∆(φ)|2 k  k where δ(φk ) is normalized as dφ 2π δ(φk ) = 1. ˆ ˆ Denoting the saddle point of Z(β, U ) by {eiφ1 , . . . , eiφN }, we have the singlet partition function  Zsing (β) =  ˆ [dU ]Z(β, U ) ≈ Z(β, φ).  (5.6)  Using the delta function 5.5, the ordinary partition function is written in the variables of eigenvalues as Z(β) =  [dU ]D(U )Z(β, U ) ≈  1 ˆ Z(β, φ), ˆ 2 ˜ C|∆(φ)|  (5.7)  59  5.2. Singlet condition at high temperature where C˜ is another normalization constant. Therefore we find the factor difference because of changing variables from group elements to eigenvalues, ˆ 2 Z(β). ˜ Zsing (β) = C|∆( φ)|  (5.8)  ˜ which has the order of N while In the large N limit, we can ignore ln C, other terms have order of N 2 , since C˜ is the normalization constant in the integral of N independent eigenvalues. The Vandermonde determinant at the saddle point can be calculated as ˆ 2 = ln |∆(φ)|  ln sin2 i<j  =  1 2 N 2  φi − φj 2  dxdy ln  1 ≈ N2 2 x−y 2  dxdy ln sin2  x−y 2  ρ(x)ρ(y)  2  ρ(x)ρ(y) at high temperature, (5.9)  ˆ where ρ(x) is like a delta function at the saddle point φ. In the high temperature limit, the approximate density function suggests the temperature dependence of ρ(φ) is ρ(x, q) ∼ f (q)ρ0 (f (q)x), where ρ0 (x) has a finite value in the integral above and f (q) → ∞ as q → 1. Referring mq mq 2 to sec3.2.1, the first order solution ρapp (x) = π2 2(1−q) 1 − 2(1−q) 2 2x , which suggests f (q) =  mq . 2(1−q)2  In the higher order solution, we may have  different f (q), but it is always possible to isolate the dependence of q in f (q). Thus we change the variables in the integral, x → x = f (q)x, and the integral becomes 1 ln |∆(φˆi )|2 ≈ −N 2 ln f (q) + N 2 2  dx dy ln  x −y 2  ≈ −N 2 ln f (q) at high temperature.  2  ρ0 (x )ρ0 (y ) (5.10)  Finally, we derive the leading terms∼ O(N 2 ) of the difference between imposing the singlet constrain and relaxing it to be ln Zsing (q) = ln ZQHO (q) − N 2 ln f (q),  (5.11)  which corresponds to the difference between Sf ree and the action of mN 2 quantum harmonic oscillators, Sf ree = SQHO + ln f (q),  (5.12)  where SQHO = m ln(1 − q). 60  5.3. Conclusion Comparing to the approximate action at the high temperature limit in eq(3.35), Sf ree = (m − 1) ln(1 − q), we conclude f (q) =  1 , 1−q  (5.13)  which agrees with the specific f (q) choosen for the first order solution ρapp mq 1 since limq→1 ln 2(1−q) 2 = ln 1−q .  5.3  Conclusion  As a toy model of large N, 4D SU(N) gauge theory, we study a simple non-interacting model of matrix-harmonic-oscillators. Our purpose is to examine whether the singlet constraint still plays an important role at high temperature in gauge theory. We compare the partition function projected in the singlet sector to the ordinary partition function including every sector, and we discover a discrepancy between these two partition functions. In the argument of changing variables from group elements to eigenvalues, we discover the discrepancy of effective actions between imposing singlet constrain or not. The discrepancy agrees with the result explicitly derived from numerical and analytical methods, and it can not be neglected in the small m model. The singlet constrain on the matrix model is decoupled at high temperature limit and large m limit. Equation (5.11) is actually a general statement, which is not restricted in our toy model. Matrix model in the large N and high temperature limit has a discripancy ln f (q) between the effective actions applying the singlet condition or the one without the condition. The partition function Zsing of 2 the former times the factor f (q)N equals to the latter as eq(5.11), where f (q) depends on the model and diverges as at high temperature. It would be interesting to compute f (q) in a more complicate gauge model like super Yang-Mills field and investigate its phase transition as [1]. Or we could revise previous results, which is derived under the singlet constrain at deconfinement phase, [4] for example. When the partition function under the singlet constraint is treated as the general partition at high temperature, any different results could be due to an extra normalized factor from changing integral variables.  61  Bibliography [1] Ofer Aharony, Joseph Marsano, Shiraz Minwalla, Kyriakos Papadodimas, and Mark Van Raamsdonk. The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories. Adv.Theor.Math.Phys., 8:603–696, 2004. [2] Dmitri Boulatov and Vladimir Kazakov. One-dimensional string theory with vortices as the upside down matrix oscillator. Int.J.Mod.Phys., A8:809–852, 1993. [3] P. Di Francesco, Paul H. Ginsparg, and Jean Zinn-Justin. 2-D Gravity and random matrices. Phys. Rept., 254:1–133, 1995. [4] Guido Festuccia and Hong Liu. The Arrow of time, black holes, and quantum mixing of large N Yang-Mills theories. JHEP, 0712:027, 2007. [5] W. Fulton and J. Harris. Representation Theory. Springer-Velrag New York Inc., 1991. [6] J. Jurkiewicz and K. Zalewski. VACUUM STRUCTURE OF THE U(N→ ∞) GAUGE THEORY ON A TWO-DIMENSIONAL LATTICE FOR A BROAD CLASS OF VARIANT ACTIONS. Nucl.Phys., B220:167, 1983. [7] Alexios P. Polychronakos. Integrable systems from gauged matrix models. Physics Letters B, 266(1-2):29 – 34, 1991. [8] Gordon W. Semenoff and Richard J. Szabo. Fermionic matrix models. Int.J.Mod.Phys., A12:2135–2292, 1997. [9] Bo Sundborg. The Hagedorn transition, deconfinement and N=4 SYM theory. Nucl.Phys., B573:349–363, 2000.  62  Codes In the appendix, we provide the codes calculating the eigenvalue distributions and the values of effective action. Method A1, N1, and N2 are written in maple. Method A2 is written in Mathematica.  .1  Method A1  > restart; with(orthopoly); with(LinearAlgebra); with(plots); qtest := .3; dim := 5; mats := 20; gen := expand(series((sqrt((1-z)^2+4*z*x)+z-1)/(2*z), z, 2*dim+3)); for i from 0 to 2*dim do B[i] := coeff(gen, z, i) end do; Rmatrix := proc (j, k) options operator, arrow; expand(mats*q^k*(sum((B[j+l-1]+B[abs(j-l+1/2)-1/2])*P(k-l, 1-2*x), l = 1 .. k))) end proc; R := Matrix(dim, Rmatrix); fin := Determinant(R-IdentityMatrix(dim)); Avector := proc (j) options operator, arrow; expand(mats*q^j*(P(j-1, 1-2*x)-P(j, 1-2*x))) end proc; A := Vector(dim, Avector); subs(q = qtest, fin); fsolve(% = 0, x = 0 .. 1); squres := min(%); boundary := 2*arcsin(sqrt(squres)); M := IdentityMatrix(dim)-subs(q = qtest, x = squres, R); for j to dim do M[1, j] := subs(q = qtest, x = squres, A[j]) end do; IM := MatrixInverse(M); for j to dim do rho[j] := IM[j, 1] end do; for j to dim do Q[j] := expand(sum(2*mats*qtest^(j+k)*rho[j+k]* P(k, cos(boundary)), k = 0 .. dim-j)); V[j] := 2*Pi*(1-mats*qtest^j)/j end do; density := sqrt(squres-sin((1/2)*theta)^2)*(sum(Q[k]*cos((k-1/2)*theta), k = 1 .. dim))/Pi; S := convert(sum(rho[k]^2*V[k]/(2*Pi), k = 1 .. dim)+(1-mats)*log(2), float); ‘pic&rho;‘ := plot(density, theta = -boundary .. boundary, 63  .2. Method N1 legend = "cutoff="); display(‘pic&rho;‘);  .2  Method N1  > restart; with(plots); with(Optimization); dim := 6; Vector(dim, symbol = K); Vector(dim, symbol = pic); K := [100, 120, 140, 160, 180, 200]; pise := 600; ‘d&theta;‘ := Pi/pise; q := .9; mats := 20; picapp := plot(piecewise(abs(x) < sqrt(2/(mats*q/(1-q)^2+1/12)), (mats*q/(1-q)^2+1/12)*sqrt(2/(mats*q/(1-q)^2+1/12)-x^2)/Pi, 0), x = -sqrt(2/(mats*q/(1-q)^2+1/12)) .. sqrt(2/(mats*q/(1-q)^2+1/12)), color = black, legend = "app"); plotsetup(jpeg, plotoutput = ‘rho coefficients.jpeg‘, plotoptions = ‘portrait,noborder‘); for k to dim do unassign(’i’, ’rho’, ’action’, ’density’, ’temp’); Vector(K[k], symbol = rho); action := proc (rho) options operator, arrow; sum((1-mats*q^i)*rho[i]^2/i, i = 1 .. K[k]) end proc; density := proc (theta) options operator, arrow; (1/2)/Pi+(sum(rho[i]*cos(i*theta), i = 1 .. K[k]))/Pi end proc; temp := QPSolve(action(rho), {seq(density(j*‘d&theta;‘) >= 0, j = 0 .. pise)}); S[k] := convert(temp[1]+(1-mats)*log(2), float); picfourier[k] := pointplot({seq([l, eval(rho[l], temp[2])], l = 1 .. K[k])}, color = COLOR(RGB, k/dim, 0, (dim-k)/dim), symbol = solidcircle, legend = "cut off="); picrho[k] := plot(eval(density(x), temp[2]), x = -Pi .. Pi, color = COLOR(RGB, k/dim, 0, (dim-k)/dim), legend = "cut off=") end do; display({seq(picfourier[i], i = 1 .. dim)}); display({seq(picrho[i], i = 1 .. dim)}, picapp); seq(S[i], i = 1 .. dim);  .3  Method N2  > restart; with(plots); with(Optimization); with(LinearAlgebra); pise := 200; mats := 20; q := .9; boundary := 0.7e-1; ‘d&theta;‘ := boundary/(pise-.5); ‘&rho;app‘ := piecewise(x < sqrt(2/(mats*q/(1-q)^2+1/12)), (mats*q/(1-q)^2+1/12)*sqrt(2/(mats*q/(1-q)^2+1/12)-x^2)/Pi, 0); 64  .3. Method N2 ‘&rho;pic1‘ := plot(‘&rho;app‘, x = -boundary .. boundary, color = "Blue"); Vexact := -log(sin((1/2)*u)^2)+mats*log((1/4)*(1-q)^2)+mats*log(4*q* sin((1/2)*u)^2/(1-q)^2+1); Vapp := -log((1/4)*u^2)+mats*log((1/4)*(1-q)^2)+(mats*q/(1-q)^2+1/12)*u^2; Vappfine := -log((1/4)*u^2)+mats*log((1/4)*(1-q)^2)+(mats*q/(1-q)^2+1/12)* u^2+(-mats*q^2/(2*(1-q)^4)-mats*q/(12*(1-q)^2)+1/1440)*u^4; V0 := 3-log((1/4)*‘d&theta;‘^2)+mats*log((1/4)*(1-q)^2); V0fine := 3-log((1/4)*‘d&theta;‘^2)+mats*log((1/4)*(1-q)^2)+ (1/6*(-mats*q^2/(2*(1-q)^4)-mats*q/(12*(1-q)^2)+1/1440))*‘d&theta;‘^2; V := piecewise(u > (1/2)*‘d&theta;‘, Vappfine, u < -(1/2)*‘d&theta;‘, Vappfine, V0fine); Vector(pise, symbol = rho); actiontrapezoid := 2*‘d&theta;‘^2*add(add(rho[i]*rho[j]* evalf(subs(u = (i-j)*‘d&theta;‘, V)), i = 1 .. pise), j = 1 .. pise)+ 2*‘d&theta;‘^2*add(add(rho[i]*rho[j]*evalf(subs(u = (i+j-1)* ‘d&theta;‘, V)), i = 1 .. pise), j = 1 .. pise)+2*‘d&theta;‘^2*rho[pise]* add(rho[i]*(evalf(subs(u = (pise-i)*‘d&theta;‘, V))evalf(subs(u = (pise+i-1)*‘d&theta;‘, V))), i = 1 .. pise-1)+(1/2)* ‘d&theta;‘^2*rho[pise]^2*(evalf(subs(u = 0, V))-evalf(subs(u = (2*pise-1)* ‘d&theta;‘, V))); actionrectangle := 2*‘d&theta;‘^2*add(add(rho[i]*rho[j]* evalf(subs(u = (i-j)*‘d&theta;‘, V)), i = 1 .. pise), j = 1 .. pise)+ 2*‘d&theta;‘^2*add(add(rho[i]*rho[j]*evalf(subs(u = (i+j-1)* ‘d&theta;‘, V)), i = 1 .. pise), j = 1 .. pise); constr1 := {rho[pise] >= 0, seq(rho[j] >= rho[j+1], j = 1 .. pise-1), add(rho[j], j = 1 .. pise)*‘d&theta;‘ = .5}; constr2 := {seq(rho[j] >= 0, j = 1 .. pise), add(rho[j], j = 1 .. pise)* ‘d&theta;‘ = .5}; Stemp := QPSolve(actionrectangle, constr2); S := (1/2)*Stemp[1]; Sana := 1/2*((2*(mats-1))*log(1-q)-2*mats*log(2)+log(mats*q+(1/12)*(1-q)^2)+ log(2)+4/Pi^2*4.654244782+1); ‘&rho;pic2‘ := pointplot({seq([-(i-.5)*‘d&theta;‘, eval(rho[i], Stemp[2])], i = 1 .. pise)}, color = "Red", symbol = point); ‘&rho;pic3‘ := pointplot({seq([(l-.5)*‘d&theta;‘, eval(rho[l], Stemp[2])], l = 1 .. pise)}, color = "Red", symbol = point); display({‘&rho;pic1‘, ‘&rho;pic2‘, ‘&rho;pic3‘}); latex(‘&Delta;S‘ = convert(S-Sana, float)); ‘&delta;S‘ := (5/4)*(-mats*q^2/(2*(1-q)^4)-mats*q/(12*(1-q)^2)+1/1440)/ (mats*q/(1-q)^2+1/12)^2; 65  .4. Method A2 latex(convert(‘&delta;S‘, float)); latex(convert(Sana, float));  .4  Method A2  AppendTo[$Echo,"stdout"] root1[A_,\[Alpha]_,\[Beta]_]:=Simplify[1/3 (-6 A-(2 \[Alpha])/\[Beta]+ (-12 \[Beta]-(\[Alpha]+3 A \[Beta])^2)/(-54 A \[Beta]^518 \[Beta]^4 (\[Alpha]+3 A \[Beta])-\[Beta]^3 (\[Alpha]+3 A \[Beta])^3+ 6 Sqrt[3] \[Sqrt](\[Beta]^8 (-16 \[Beta]+27 A^2 \[Beta]^2+ 18 A \[Beta] (\[Alpha]+3 A \[Beta])-(\[Alpha]+3 A \[Beta])^2+A (\[Alpha]+ 3 A \[Beta])^3)))^(1/3)-1/\[Beta]^2 (-54 A \[Beta]^5-18 \[Beta]^4 (\[Alpha]+ 3 A \[Beta])-\[Beta]^3 (\[Alpha]+3 A \[Beta])^3+ 6 Sqrt[3] \[Sqrt](\[Beta]^8 (-16 \[Beta]+27 A^2 \[Beta]^2+18 A \[Beta] (\[Alpha]+ 3 A \[Beta])-(\[Alpha]+3 A \[Beta])^2+A (\[Alpha]+3 A \[Beta])^3)))^(1/3))] root2[A_,\[Alpha]_,\[Beta]_]:=Simplify[1/6 (-12 A-(4 \[Alpha])/\[Beta]((1+I Sqrt[3]) (-12 \[Beta]-(\[Alpha]+3 A \[Beta])^2))/(-54 A \[Beta]^518 \[Beta]^4 (\[Alpha]+3 A \[Beta])-\[Beta]^3 (\[Alpha]+3 A \[Beta])^3+ 6 Sqrt[3] \[Sqrt](\[Beta]^8 (-16 \[Beta]+27 A^2 \[Beta]^2+ 18 A \[Beta] (\[Alpha]+3 A \[Beta])-(\[Alpha]+3 A \[Beta])^2+A (\[Alpha]+ 3 A \[Beta])^3)))^(1/3)+1/\[Beta]^2(1-I Sqrt[3]) (-54 A \[Beta]^518 \[Beta]^4 (\[Alpha]+3 A \[Beta])-\[Beta]^3 (\[Alpha]+3 A \[Beta])^3+ 6 Sqrt[3] \[Sqrt](\[Beta]^8 (-16 \[Beta]+27 A^2 \[Beta]^2+ 18 A \[Beta] (\[Alpha]+3 A \[Beta])-(\[Alpha]+3 A \[Beta])^2+A (\[Alpha]+ 3 A \[Beta])^3)))^(1/3))] m=20 qstart=0.6 qend=0.99 qpise=20 dq=(qend-qstart)/qpise count=1 Do[i=ilp; q=qstart+i dq; \[Alpha]=(2 m q)/(1-q)^2+1/6; \[Beta]=-((2 m q^2)/(1-q)^4)-(m q)/(3 (1-q)^2)+1/360; d=(m q)/(1-q)^2+1/12; B[A_]:=root1[A,\[Alpha],\[Beta]]; \[Rho][A_,x_]:=1/(2 \[Pi])((-x \[Alpha]^2-2 \[Alpha] (-2+x (3 A+x) \[Beta])66  .4. Method A2 \[Beta] (-4 (4 A+x)+x (3 A+x)^2 \[Beta]))^(1/2)); AA[A_]:=NIntegrate[Sqrt[x] \[Rho][A,x],{x,0,B[A]}]; Atwo=Re[A/.FindRoot[AA[A]==A,{A,10^-9}]]; density[x_]:=\[Rho][Atwo,x^2]; boundary=Re[Sqrt[B[Atwo]]]; \[Xi][count]=root2[Atwo,\[Alpha],\[Beta]]; Sfour[count]=Re[N[1/2 NIntegrate[(-Log[(x-y)^2]+\[Alpha]/2 (x-y)^2+ \[Beta]/4 (x-y)^4) density[x] density[y],{x,-boundary,boundary}, {y,-boundary,boundary}]+m Log[(1-q)/2]+Log[2]]]; str=OpenAppend["resolfour.txt"]; Write[str,Sfour[count]]; Close[str]; count=count+1; ClearAll[i,q],{ilp,0,qpise}]  67  

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