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Quantum dimer model and its application in topologically protected qubit Wong, Mandy Man Chu 2006

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Quantum Dimer Model and its application in Topologically Protected Qubit by Mandy Man Chu Wong B . S c , The University of British Columbia, 2004 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Master of Science in The Faculty of Graduate Studies (Physics) The University Of British Columbia August 18, 2006 © Mandy Man Chu Wong 2006 Abstract Quantum dimer model (QDM) is a paradigm for the study of high-Tc su-perconductivity. We review the history of this model and its properties with enphasis on the critcal behaviour and excitation. We review the method of mapping the Q D M into three different models; the Ising model, the height model, and Polyakov's compact Q E D . The duality relation between the height model and Polyakov's compact Q E D in the Q D M limit is investi-gated. We review a new application of Q D M in quantum computing. The triangular dimer model can serve as a suitable candidate for topologically protected qubit. Finally, a specific implementation using Josephson junction array is discussed. Contents Abstract i i Contents i i i List of Figures v Acknowledgements ix 1 Introduction 1 2 Valence Bond state and Short-Range Valence Bond state . 3 3 Effective Hamiltonian of the Quantum Dimer Models . . . 8 3.1 Topological Sectors 9 3.2 Ground State in Various Limits 12 4 Quantum Dimer Model and Ising Gauge Theory 16 4.1 The Quantum Dimer Model as an Ising Gauge Theory . . . . 16 4.2 Ising Gauge Theory as a Generalized Dimer Model 19 4.2.1 Even Quantum Dimer Model 19 4.2.2 Odd Ising gauge theory and the Polyakov loop term . 22 4.2.3 Quantum dimer model limit of odd Ising gauge theory 22 5 Quantum Dimer Model and Polyakov's Compact QED . . 23 5.1 Quantum Dimer Model in the context of Quantum Electro-dynamics 26 6 Quantum Dimer Model with Height Representation . . . . 29 6.1 Height Representation 29 6.2 Dynamics 32 6.3 Continuous Theory of Dynamics 34 6.3.1 Continuum Equations and Fourier Modes 35 6.3.2 Dynamic Scaling 37 Contents iv 6.3.3 Fokker-Planck Mode Spectrum 37 6.4 Height-Shift Mode _. 39 6.4.1 Fokker-Planck Modes of the Mean Height h(t) . . . . 41 7 Duality between the Height-Representation and Polyakov's Compact QED 43 7.1 Duality Transformation of Polyakov's Compact Q E D 43 7.2 Gauge Transformation and Ground State Degeneracy . . . . 45 7.2.1 The Meaning of Local and Global Gauge Transformation 47 7.2.2 Ground State Degeneracy 48 7.3 Dual Form of the Hamiltonian 49 8 Topologically Protected Qubit in Triangular Quantum Dimer Model 52 8.1 Triangular Dimer System 53 8.1.1 The topological structure of the Hilbert space 54 8.1.2 Liquid ground state in triangular lattice dimer model 54 8.2 Basic operations of a qubit 56 8.3 Implementing the topologically protected qubit using Joseph-son junction array 56 9 Conclusion 61 Bibliography 62 V List of Figures 2.1 A Valence Bond (VB) state on a 4 X 4 square lattice. It is the product of 8 Valence Bonds. The valence bonds need not be nearest neighbor 4 2.2 A Short-Range V B state on a 4 X 4 square lattice. Valence Bonds only form between neighboring sites 4 2.3 (a) and (b) are two configurations of dimers which differ only in local arrangement of the dimers at the sites 1, 2, 3 and 4. . 5 2.4 The 'transition graph' of the dimer coverings (a) and (b) from Figure 3 is their superposition. The only non-trivial loop con-nects the sites 1, 2, 3 and 4 with 4 links(dimers). It represents the overlap of the non-orthogonal V B states | a) and | b) as-sociated with the dimer covering in (a) and (b). . . . . . . . . 6 3.1 Resonance process. Flipping parallel dimers along vertical direction to horizontal direction and vice versa 8 3.2 (a) shows a columnar state and (b) shows a staggered state. They belong to different topological sectors, meaning that the staggered state cannot be obtained from the columnar state by resonance(fiippirig) movement 9 3.3 (a) shows the columnar reference state with dimers pointing from red to black, (b) shows the staggered state with dimers pointing from black to red. (c) shows the overlap of (a) and (b); In this 4X4 lattice, it has two loops encircling the 2-torus in the x direction, which gives a winding number of (2,0) . . . 10 3.4 (a) shows the columnar reference state with dimers pointing from red to black, (b) shows another columnar state with dimers pointing from black to red. (c) shows the overlap of (a) and (b); There are no loop encircling the 2-torus in the x and y direction, which gives a winding number of (0,0) . . . . 11 List of Figures vi 3.5 A few examples of the ground states in the staggered phase for a 6X6 (N=6) lattice, (a) shows a staggered state with a winding number (0,^) a n d (b) has a winding number of (y,0) . (c) and (d) are staggered states that tilt in different directions with winding numbers of ( y , ^ ) and ( — y ) respectively 13 3.6 Equal amplitude ground state for a 2X3 lattice. At the R K point (t = v), this is an eigenstate of the quantum dimer model Hamiltonian with zero eigen energy. When applying H^m and Hv to the ground state they generate terms that cancel each other term wise 14 3.7 The four degenerate ground states in the limit j < —1. The boundary condition here is periodic in both x and y direction. A l l four states belong to the same topological sector with winding number (0,0) 15 3.8 In a plaquette state, every other plaquette has equal probabil-ity of being in || or in = and is independent of other plaquettes. 15 4.1 (a) shows the numbering of the link indices 1,2,3, and 4 on a plaquette. (b) shows the two of the overlap links between summing over plaquette and the neighboring links of a site (+) 18 5.1 The dimer (i^dimer) energy of a single link (x, x + ej) as a function of lj(x). For any value of the coupling constant k, configurations with lj = 0,1 have exactly zero energy while any other state will have energies growing like ^ 24 5.2 The above figure shows how an allowed state would satisfy Gauss's law described in Eqn. 5.18. Note that there is a back-ground charge density that equals to +1(-1) on red(black) lattice sites 27 6.1 The above figure shows the clockwise orientation assigned to each red site, x represents the position of the lattice site while f represents the position of the dual lattice site •'. 30 6.2 The above figure shows the height pattern {z(r)} for a dimer covering. The net height difference for the path around one plaquette is zero. Also, adding the same constant to each z(r) makes an equally valid height representation. 31 List of Figures vii 6.3 The above figures show two dimer coverings of a 6x6 system with periodic boundary condition, (a) The columnar state has a height difference of (wx,wy) = (0,0) at the boundary; (b) The staggered state has (wx,wy) — (0,12) 31 6.4 The above figure shows the change in the height variable when a dimer is flipped. The corresponding height variable is raised/lowered by 4 34 6.5 In the above figure, (a) shows the change in height about a lattice site satisfying the hard core dimer constraint, (b) A discrete height field for a monomer. A monomer is a site with either zero or two electrons 36 7.1 The sites of the direct lattice are labelled by x while the sites of the dual lattice are labelled by r 44 7.2 A square lattice with periodic boundary condition is isomor-phic to a torus. The two non-contractible loops must wrap around the boundary of the torus. 71 (r) is along the x\ di-rection while 72(r) is along the £2 direction and both loops go through the dual lattice site r. 46 7.3 In (a), the value of the line integrals in Eqn. 7.15 are / 7 l(f) [B] = 1 and I12^[B] = 2.The boundary are continuous. In (b), two plaquettes are flipped as indicated by grey circles but line integrals are still unchanged. .T 7 l(^[£r] = 1 and I12(^[B'] =2 . 48 8.1 One allowed configuration satisfying the constraint that every vertex must belong to only one dimer in a triangular lattice. 53 8.2 The Hamiltonian for triangular dimer system. The kinetic term rotates parallel dimers while the potential term allows frustration of parallel dimers. The Hamiltonian is a sum over three different kinds of plaquettes 53 8.3 (a) is a columnar state with the maximum number of flip-pable plaquette. (b) is a staggered state with the no flippable plaquette 54 8.4 The even and odd dimer count along the reference line 7 is invariant under the action of the Hamiltonian 55 8.5 The array is made of Josephson junctions to emulate the tri-angular dimer system • 57 8.6 The series of steps in a basic dimer flip process. The overall hopping amplitude is of order t ~ -fij- 58 List of Figures viii 8.7 A possible construction of a qubit with cylindrical bound-ary condition using Josephson junction array. The phase of a qubit state is controlled by a gated superconducting strip (GSS) placed along the reference line 7. The amplitude mix-ing is controlled by a tunable Josephson junction placed at the inner qubit boundary .59 ix Acknowledgements I am grateful to my supervisor Fei Zhou. His continuous support and guid-ance has made this a challenging and rewarding experience. I would like to thank Mona Berciu for giving me good academic advices. Last but not least, I thank God for giving me a loving circle of family, friends, and colleagues. 1 Chapter 1 Introduction Quantum dimer model were introduced by Rokhsar and Kivelson to capture the low energy dynamic of Heisenberg antiferromagnets. The model consider a prototype for the short range resonating valence bond (RVB) state in the limit where the spin gap is large, so that the manifold of low-energy states is spanned by linearly independent set of nearest-neighbor valence bond states. The valence bond phase was originally introduced in the context of spin ^ Heisenberg antiferromagnets by Fazekas and Anderson back in 1974. The conventional understanding of spin systems interacting by Heisenberg antiferromagnet Hamiltonian is that the ground state is close to the Neel state with a staggered sub-lattice magnetization. However, for spin | and low-dimensional lattices with significant frustration, this picture is incorrect. For this reason, Anderson proposed that in certain situations the ground state is actually closer to one in which pairs of spins are bound into singlet states. These pairs are visualized as valence bonds, and since in general there will be several alternative ways in which to pair the spins, the actual ground state will be a superposition over such products of singlet pairs -hence the resonating valence bond (RVB) state. In this paper, we present the Quantum Dimer Model as an unifying theory by studying its connections with three well developed theories; the height model, the Ising gauge theory and the Polyakov's compact Q E D . We wil l also look at a new application based on the quantum dimer model. Much of this thesis is review work on different published papers. We wish to provide an easy introduction for students or researchers beginning to study this topic. Many of the calculations go in greater detail than the original paper to help with the understanding of the specfic techniques used. In addition, we wish to draw interest to the perhaps less well-known application of Q D M in quantum computing. The remaining of the paper is organized as follow. In chapter 2, I will define the valence bond state which form the basis of the quantum dimer model. In chapter 3, the effective Hamiltonian is presented along with a discussion on the topological sectors in the Hilbert space and the ground state in various limits. In chapter 4, we wil l look at the mapping between Chapter 1. Introduction 2 Q D M and Ising gauge theory [1]. Chapter 5 is based on E . Fradkin and S. Kivelson's work which discuss short range resonating valence bond theories and superconductivity [3]. The central idea is the realization that dimer models are exactly equivalent to Polyakov's compact electrodynamic in 2+1 dimension. Chapter 6 talks about the results in Christopher Henley's paper [2] in which he maps the Q D M into height model at the R K point. In chapter 7, I will present the duality relation between the Height representation and Polyakov's compact Q E D . Finally, chapter 9 wil l take about how the Q D M is a suitable candidate for topologically protected qubit. 3 Chapter 2 Valence Bond state and Short-Range Valence Bond state Consider a S = l / 2 square lattice with nearest neighbor(J) and next-nearest neighbor(K) interactions at half filling. If K « 0, the ground state appears to be a Neel antiferromagnet. However, when J ~ K, competing interactions may create frustration and destroy Neel ordering. A n alternative way to describe the low energy dynamics for such a system is to pair up spins into valence bonds. [4] The basic building block in this representation is a singlet pairing (Valence Bond) of two spins at sites i and j of the lattice, not necessarily nearest neighbors. Denote with | (ij))-i m = ^ ( i T i i i ) - Ui t i ) ) (2.i) The valence bond above is only defined up to a sign. When calculating overlap later on, one must pick a convention to avoid ambiguity. If we partition the set of lattice sites into pairs, one can define a V B state as a tensor product of all these singlet pairs. Figure 2.1 shows a V B state | VB) on a 4X4 square lattice. | VB) = I (ikjk)) (2-2) pairs Let first consider the simple case where all residual order is extremely short ranged so that the manifold of the low-energy states is spanned by the linearly independent set of nearest-neighbor V B states (short-range V B state). Figure 2.2 shows a Short-Range V B state on a 4X4 square lattice. A Short-Range "Resonating" V B state is. a coherent superposition of all the nearest-neighbor V B states above. Its energy is further lowered as a result of the matrix elements connecting the different valence-bond con-figuration. It is much like a double well with tunnelling. Although the Chapter 2. Valence Bond state and Short-Range Valence Bond state 4 Figure 2.1: A Valence Bond (VB) state on a 4 X 4 square lattice. It is the product of 8 Valence Bonds. The valence bonds need not be nearest neighbor. G ^ ^ B - -e—-O O - - O - ^ ) ()-- --<>— - o - A — - A Figure 2.2: A Short-Range V B state on a 4 X 4 square lattice. Valence Bonds only form between neighboring sites. Chapter 2. Valence Bond state and Short-Range Valence Bond state 5 <>;9"--:9—4 i-----c—<? i n (a) 1 ! 2 (b) i Figure 2.3: (a) and (b) are two configurations of dimers which differ only in local arrangement of the dimers at the sites 1 , 2 , 3 and 4. Short-Range R V B states are linearly independent, they are not orthogonal. To see this, consider two dimer coverings as shown in Figure 2.3. Before calculating the overlap of these two states, one must pick a convention for the signs of the valence bond states. I will use the convention originally used by Rokhsar and Kivelson [5] for bipartite lattices. A bipartite lattice is con-sist of two inter-penetrating sublattices, call them the R(red) and B(black) sublattices. For nearest-neighbor valence bonds, a dimer always connect between a red and a black site. So one can fix the sign of a valence bond in the following way. | VB) = H 1t*- l6>-JtBl/l ) (2.3) pairs * In Figure 2.3, let 1 and 4 (2 and 3) belong to the red(black) sublattice. The overlap (a \ b) is equal to (b\a) = \ ((T1I2I —(Taiil) * «T43| -<T 3Lt|) * (IT1I3)- ITsli)) * (IT4I2)- IT2I4)) = \ «Til2T4l 3 | |Til2T4l3> + U1T2 .UT3IU1T2i.4T3>) 1 2 • More generally, overlaps between two arbitrary Short-Range V B states can be calculated through the transition graph. Figure 2.4 shows the transi-tion graph between dimer covering a and b from Figure 2.3. It is constructed Chapter 2. Valence Bond state and Short-Range Valence Bond state 6 Figure 2.4: The 'transition graph' of the dimer coverings (a) and (b) from Figure 3 is their superposition. The only non-trivial loop connects the sites 1, 2, 3 and 4 with 4 links(dimers). It represents the overlap of the non-orthogonal V B states | a) and | b) associated with the dimer covering in (a) and (b). by drawing configurations a and b on the same lattice. The transition graph consist of many closed loops u, each having an even number of links. Let 2L(M) be the number of links that make up for loop u. Then one can show that the overlap equals to: (a | b) = J j 2 X 2 < - L W . (2.4) U = 2p^bh^f (2.5) where P(a,b) is the number of loops in the transition graph while N is the (even) number of sites. Notice that in equation 2.4, contributions to '(a | b) only comes from non-trivial loops in the transition graph. One can make this point more explicit by considering site 5 and 6 in Figure 2.3. The valence bond expression for dimer covering a and b is, I a) = ^ ( I t s l e ) - I T e l s ) ) ® I-..) (2.6) I b) = ^ ( | T 5 | 6 > - | T 6 l 5 » ® l * * * > (2.7) i where | ...) and | ***) represents the singlet state for the other sites. Because r Chapter 2. Valence Bond state and Short-Range Valence Bond state 7 both a and b have a dimer between site 5 and 6. The contribution of this trivial loop is simply one. . <a | 6) = | « T 5 l 6 l T 5 i 6 > + <T6i5|T6i5»«— | * * * » (2.8) = (... I * * *) (2.9) This concludes the introduction to valence bond theory. In the next chapter, we will study the simplest Hamiltonian proposed by Rokhsar and Kivelson [5] on a square lattice. The Hilbert space is comprised of short-range valence bond states and this system is named the Quantum Dimer Model (QDM). Chapter 3 Effective Hamiltonian of the Quantum Dimer Models Rokhsar and Kivelson wished to study the spin system in terms of valence bond states and they came up with a simple effective Hamiltonian that consist of a kinetic and potential term. HQDM = Hkin + Hv (3.1) Hkin = - * £ (I H>H +&•<:.) (3.2) Plaq Hv = (I 11X11 I + HH) (3-3) Plaq The sigma sum go through all the plaquettes in the lattice and v,t > 0 .The kinetic term is an off-diagonal matrix that generates resonance by flipping parallel dimers on a plaquette as shown in Figure 3.1. The potential term is diagonal and counts the number of plaquettes having parallel dimers. The Hilbert space of the Q D M consists of all dimer covering that sat-isfy the hard core constraint, in which each lattice site can only belong to one dimer. On average, there are one electron per site(half-filling). When varying the value of t and v the ground state has, in general, a crystalline ground state. Before discussing the ground state in different limit of j. I will first classify the dimer configurations. n-n Figure 3.1: Resonance process. Flipping parallel dimers along vertical di-rection to horizontal direction and vice versa. Chapter 3. Effective Hamiltonian of the Quantum Dimer Models 9 Figure 3.2: (a) shows a columnar state and (b) shows a staggered state. They belong to different topological sectors, meaning that the staggered state cannot be obtained from the columnar state by resonance(flipping) movement. 3.1 Topological Sectors One can classify all the possible hard core dimer configurations in the Hilbert space into different topological sectors. A configuration within one topolog-ical sector cannot connect to a second configuration in another topological sector by the application of the Hamiltonians (by flipping parallel dimers). For instance, the columnar state and the staggered state shown in Figure 3.2 do not belong to the same sector. In the original paper written by Rokhsar and Kivelson [5], they label the topological sectors using winding numbers. For cylindrical boundary condition, there is one winding number and for a 2-torus, there are two winding numbers .(flx,Qy). Rokhsar and Kivelson [5] define the winding number in the following way. One must first pick a state as the reference state| C0). Then to find the winding number of the topological sector of an-other state(| C))> o n e must assign an direction to each dimer in the two state (| C0) and | C)). Recall that on a bipartite lattice, there are two sublattices which were labeled the Red(R) sublattice and the Black(B) sublattice. For nearest neighbor singlet bonds, each dimer is linked to a black site on one end and a red site on the other end. One can then assign a direction of going from red to black for the dimers in the reference state | C0) while assigning from black'to red for the state | C). I must note that the choice of direction here for the two states is irrelevant as long as they are assigned in the opposite way. The winding numbers are given by the net number Chapter 3. Effective Hamiltonian of the Quantum Dimer Models 10 • !>• • ®-«a • 0 - 4 • 0 ^ — ^ — « ««g # @-< • ® — • +-«t ® #-«9 • (a) (b) ••«< 0 ^ 4 — 9 * 4 — — @ - < a ©<at># -«a @-<9 (c) Figure 3.3: (a) shows the columnar reference state with dimers pointing from red to black, (b) shows the staggered state with dimers pointing from black to red. (c) shows the overlap of (a) and (b); In this 4X4 lattice, it has two loops encircling the 2-torus in the x direction, which gives a winding number of (2,0) of loops (clockwise minus counterclockwise) encircling the 2-torus in the x and y direction in the transition graph. For example, let pick the columnar state in Figure 3.2 as the reference state. The staggered state has winding numbers (Qx,Q.y) = ( ± y , 0 ) , which is illustrated in Figure 3.3. In addition, Figure 3.4 shows the winding numbers for the vertically aligned columnar state are (0,0). This means that the vertical columnar state belong to the same topological sector as the reference state. The can be easily observed as the vertical columnar state can be obtained from the reference columnar state by flipping parallel dimers ^j- times, where i V 2 is the system size. For triangular lattice there are only two topological sectors for cylinder boundary condition and 4 sectors for torus boundary condition. It is defined Chapter 3. Effective Hamiltonian of the Quantum Dimer Models 11 0 I • i • #««g -® • • • • • ®— T • ® A • 0<i ® • ? • • • T • (a) (b) •# i 1 j 0<c3 9 t>4 0 -9 © — • » # I A 1 f I ? • «Mg • (c) Figure 3.4: (a) shows the columnar reference state with dimers pointing from red to black, (b) shows another columnar state with dimers pointing from black to red. (c) shows the overlap of (a) and (b); There are no loop encircling the 2-torus in the x and y direction, which gives a winding number of (0,0) \ Chapter 3. Effective Hamiltonian of the Quantum Dimer Models 12 by the parity (even/odd) of the dimers along the reference lines in the x and y direction. More will be discussed in chapter 8. 3.2 Ground State in Various Limits In the limit j > 1, the system enters a staggered phase where the ground states are made up of a class of individual configurations which comprise of their own topologically distinct subspaces (one-man sectors). These config-urations contain no parallel nearest-neighbor dimers and are the zero-energy eigenstates of HQDM- Any other state with one flippable dimer would give an energy> 0 when applying HQDM- Figure 3.5 shows a few example of the ground states in this limit. For infinitely large system , there are infinitely many such states which tilt in different directions. At the RK point, j — 1, each sector has its own ground state, namely, the equal amplitude state (short-range RVB state) with zero eigen energy. \GS;(nx,Qy))= J2 I") ( 3- 4) where R(flx,fly) represents the set of all hard core dimer configurations with winding number (flx,Q,y). Therefore, the ground state at | = 1 is a dimer liquid. To show that this is indeed a eigenstate of the Hamiltonian, let consider a 2X3 lattice with no continuous boundary. Figure 3.6 shows all the possible states in the columnar topological sector along with the resulting state when applying the kinetic H \ - i n and potential H v part of the Hamiltonian. Notice that in this example (when v—t), HQDM \GS; (0,0)) = H k i n \ GS; (0,0)) + Hv \ GS; (0,0)) (3.5) = 0 (3.6) This is because the kinetic part and the potential part cancel each other term wise. In general, applying HV to a configuration would generate m identical terms, where m is the number of flippable plaquette. On the other hand, applying Hkin to the same configuration would generate m distinct terms, where each term is just one flip away from the original configuration. When summing over all dimer covering in the same topological sector (the equal amplitude state), the kinetic part and the potential part would cancel each other term wise. The number of degenerate ground state equals to the number of topological sectors. The one-man sector states discussed in the Chapter 3. Effective Hamiltonian of the Quantum Dimer Models 13 (c) (d) Figure 3.5: A few examples.of the ground states in the staggered phase for a 6X6 (N=6) lattice, (a) shows a staggered state with a winding number (0,y) and (b) has a winding number of (y,0) . (c) and (d) are staggered states that tilt in different directions with winding numbers of ( y , y ) and (—y,—y) respectively. Chapter 3, Effective Hamiltonian of the Quantum Dimer Models 14 IGS;(0,0)> = Hki„IGS;(0,0)> = -t( 1 1 + 1 1 + — . + — ) Hv IGS;(0,0)>= v ( — +—.+••+ 1 I ) Figure 3.6: Equal amplitude ground state for a 2X3 lattice. At the R K point(t = v), this is an eigenstate of the quantum dimer model Hamiltonian with zero eigen energy. When applying H^m and Hv to the ground state they generate terms that cancel each other term wise. limit f > 1 are also ground state here. For j <§; —1, the system is in a c o l u m n a r crystal phase and the ground states are four fold degenerate. Figure 3.7 shows the four columnar ground states which belong to the same sector with winding number (0,0). This can be seen by thinking about what happens when t = 0 and v —> —oo. It is now strongly favourable to have flippable plaquettes to lower the energy. The four columnar states in Fig. 3.7 are configurations that have the most flippable dimers. As j increases, the system undergo a first order transition to a p laquette phase [6]. In a plaquette state, every other plaquette has equal probability of being in || or in = and is independent of other plaquettes, as shown in Fig. 3.8 This phase is characterized by an order parameter that vanishes continuously at v — t, the R K point. The transition between all three phases are first order transitions [6]. The short-range R V B state is a dimer liquid while the V B crystals are dimer "solids. In the next chapter, I will discuss the relationship between Q D M and Ising gauge theory, which is a summary of the work done by R. Moessner, S. L . Sondhi, and E . Fradkin. Chapter 3. Effective Hamiltonian of the Quantum Dimer Models 15 I i I I ~ ~\ I i I I ~ Z 00 _ _ _ M l i i l : ~ : 1111 - — - T i l l (b) . (d) Figure 3.7: The four degenerate ground states in the limit | < —1. The boundary condition here is periodic in both x and y direction. A l l four states belong to the same topological sector with winding number (0,0) f m m i u i f f i i i i i u i i # • # i i i i i m i i # BH EST EQ 011111111110 Figure 3.8: In a plaquette state, every other plaquette has equal probability of being in || or in = and is independent of other plaquettes. 16 Chapter 4 Quantum Dimer Model and Ising Gauge Theory This chapter is based on the work done by R. Moessner, S. Shondhi, and E . Fradkin [1]. It describes the formulation of the quantum dimer model as an Ising gauge theory and of general Ising gauge theory as generalized dimer model. 4.1 The Quantum Dimer Model as an Ising Gauge Theory This section discuss the mapping from the Q D M to an Ising gauge the-ory(IGT). Recall that the naive Hilbert space, inclusive of any gauge equiv-alent states, of any I G T is defined by an Ising variable ax = ± 1 on each link of the lattice. The corresponding operator is <x(ri,r"2), where ( r i , ^ ) represents the link between two lattice points f{ and r j . One can identify the link variable with the presence or absence of a dimer in the following way. n(r1,r2) = - ^ (4,1) where n(f[, r"5) represents the number of dimer on link ( r i , r^) of the lattice. W i t h this relation a(ri,r"2) = —1 corresponds to no dimer(n(r"i,r"2) — 0) while cr(ri,r"2) = 1 corresponds to one dimer(n(r"i,r"5) = 1). The dimer number operators n(r~i,r 2) is related to the Ising variable operator in the same way (n(rl ,r2) = ^(1 + <7 x (r"i ,r 2 )). The hard core dimer constraint requires every site to have only one dimer connecting from it. In terms of IGT, this is expressed as - an operator, G, which leaves invariant when applying on the physical states | phys) that satisfy the hard core condition. Gj,a | phys) =| phys) (4.2) Chapter 4. Quantum Dimer Model and Ising Gauge Theory 17 where j is the lattice site index and for any phase variable a. One can construct G by first writing the constraint in terms of ox for each site j . ox | phys) = ( - n c + 2) | phys) (4.3) + where means summing over the nearest neighbor of a lattice site and nc is the coordination of that site. Coordination number is the number of nearest neighbor. For examples, nc = 4 for square lattice and nc = 6 for triangular lattice. Define Gj<a as: Gj<a = exp ia V (ax + 1 - — (4.4) The Hamiltonian at half-filling can be expressed in terms of the usual raising and lowering operators, = crv ± iaz. The commutation relation between and ax is [cr^o^] = ±2<r ± and the Hamiltonian is: Hi = -tf + vV (4.5) T = 52(&twt&I+h-c) (4-6) plaq V = i ^ ( ( l + ^ ) ( l + <Tf) + ( l + ^ ) ( l + ^ ) ) (4.7) plaq where S sums over all the plaquettes. I have suppressed the link indices by using 1,2,3, and 4 to label the link of a plaquette in a clockwise fashion as shown in figure 4.1a. It can be easily seen that equations (4.5), (4.6), and (4.7) correspond exactly to equations (3.1), (3.2) and (3.3). The raising operator <x+ creates a dimer while the lowering operator &~ destroys a dimer on the link p. The Ising Hamiltonian Hj satisfies = 0 (4.8) Eqn 4.8 is easy to prove, let consider E &X>A'] = - * E * * . T 1 + « E V] (4.9) + + H-Notice that the term J2+ <* x sums over all the nearest neighbor links of a lattice site while the Hamiltonian sum over all plaquette. [J2+ &x\ V] equals zero as V only contains .ax terms. For the kinetic commutation relation Chapter 4. Quantum Dimer Model and Ising Gauge Theory 18 Figure 4.1: (a) shows the numbering of the link indices 1,2,3, and 4 on a plaquette. (b) shows the two of the overlap links between summing over plaquette and the neighboring links of a site (+). E+ &X,T]> I w m show how two of the terms cancel each other and the rest of the terms cancel in a similar way. Please refer to Figure 4.1b for the calculation. E>*, f . ] = [^,a+a 2-a 3 +a 4-] + [ a f , a 1 + a 2 - a 3 + ( 7 4 - ] + . . . (4.10) + = Vl&iot °Z - d t °2°3 &4 +••• (4-12) = 0 (4.13) When ox, H] = 0 then [exp(icx J2+ & x), ^ ] = 0 and equation 4.8 follows. The physical states and the Hamiltonian have a larger local gauge symmetry that is of the form of a U ( l ) gauge theory. In fact, for a link p connected to site j, one can show that Gj^afGj^ = exp(±ia)a± (4.14) The above result tells us that since the number of valence bonds (dimers) is conserved, the effective Hamiltonians associated with these states must have a natural local conservation law, and consequently a local 17(1) sym-metry, instead of the natural Z2 symmetry of an Ising Hilbert space. To summarize, there exists a natural physical interpretation of the Hilbert space of R V B phases, and that its Ising character follows directly form the nature of the states themselves: short-ranged R V B states are naturally de-Chapter 4. Quantum Dimer Model and Ising Gauge Theory 19 scribed in terms of short range spin-singlets which are either present or absent. Thus from the point of view of the space of states, a description of the dimer Hilbert space should have a natural description in terms of Ising variables living on the links of the lattice. 4.2 Ising Gauge Theory as a Generalized Dimer Model The goal of this section is to show that the familiar three dimensional Isirig gauge theory can be viewed as a Generalized dimer model (GDM) which features dimers on links that obey a generalized dimer constraint. Two sets of constraints will be considered. We will first look at the simpler even I G T that only allows even number of links emanating from a lattice site. After establishing the Hamiltonian based on the even IGT, we wil l look at the odd IGT. It requires the addition of the Polyakov loop term to the action from the even IGT. Finally, we will study the Q D M limit of the odd I G T and examine some of the observations made by Senthil and Fisher [10] about IGTs in general. 4.2.1 Even Quantum Dimer Model Consider a classical 2+1 dimensional Ising model and place variables oz = ± 1 on each link. To see how it corresponds to the generalized quantum dimer model, we will derive the quantum Hamiltonian using the Transfer Matrix Formalism. The standard 2+1 I G T has the following action. — K \ ' rrz - nz rrz rrz s / J  ar,r+xar+x,r+x+yar+x+yf+y®r+y,r f where r represents any lattice site, f is the unit vector in the time direction and v is an unit vector in the spatial directions. The first term sum over plaquettes with two temporal links and two spatial links, and the second term sum over plaquettes with four spatial links. It is necessary to make the action anisotropic when taking the time continuum limit in the derivation of the Hamiltonian. It is convenient to work with a particular temporal gauge, az(r,r + T) = 1 Vf (4.17) (4.15) (4.16) Chapter 4. Quantum Dimer Model and Ising Gauge Theory 20 W i t h our choice of the tempora l gauge, the first te rm simplif ies to: Kr^oz(r,r + D)o-z(r + v + T,r + Q) (4.18) T h e act ion can be rewr i t ten in a more symmetr ic form by add ing an irrele-vant constant. ,' 5 = ^ L ( i + l , i ) " (4.19) t L(t+l,t) = ^KT (crz(m + P + T,m + f)-oz(m + 0,m))2 (4.20) rheM(t),0 -l-Ks + <yl+r°2+rVl+r°l+r) (4-21) m where M ( t ) is the set of latt ice sites at the tth level i n the t ime d i rect ion. Th i s means that the vector m is of the form m = n\x + niy + tf for any integers ri\ and n2. I have once again suppressed the l ink indices i n the second term. T h e next step is to interpret T(t + l,t) = e( - L(t + l,t)) as a mat r i x element between two different states located at the tn and tn+i level in the t ime direct ion. T(t + l,t) = ({crn}\T\{on+1}). E ach state {an} describes a l l the Ising variables on the x and y plane at t ime level tn. If we imagine that the t ime axis of our classical Ising mode l is the t ime axis of the gener-al ized quantum dimer mode l , the T carries in format ion f rom one t ime to a neighbor ing t ime. We can identify the transfer ma t r i x as the t ime evolut ion operator for the quan tum system. Hence, the fol lowing re lat ionship holds: f = e~T" KI-TH (4.22) in the l imi t r —* 0 which is the t ime cont inuum l imi t . Th i s is not true in general for the Ising system, but there exists a l imi t for the parameters KT and Ks such that it is consistent w i t h the requirement r —> 0. We do not need to solve for T expl ic i t ly , instead one could look at the ind i v idua l ma t r i x elements to come up w i th the scal ing l imi t for KT and Ks and the quan tum Hami l ton i an . For the diagonal element (0 flips), cr^+- = for a l l m, and equation Chapter 4. Quantum Dimer Model and Ising Gauge Theory 21 4.20 reduces to • • L(0 flips) = -Ka52<rl(T}<TZ<Tl ' (4.23) in If there is one spin nipped between the (£+ l)th level and the tth level, then L(l flip) = 2KT - \KS ]TK<72Vf<T| + <r*1+fofaal+iaf+f) (4.24) Consider the various matrix elements of T f(0 flips) = exp(KsY,o\ol<jl<j\) (4.25) rh « i - r H \ o f U p s . (4.26) f ( l /Zip) = e _ 2 i < : T e 5 i f s ^ m K f f ! f f 3 f f I + f f i + f f f 2 + f < 7 l + T < T l + f ) (4.27) « - T f f | i / K P (4.28) The above equations will determine the r dependence of the coupling con-stants KT and KS. From equation 4.26, we learn that KS oc r and from equation 4.28, e _ 2^T a r. Therefore -FC, and e~2Kr must be proportional. Define the proportionality constant K as KS = K e - 2 ^ (4.29) We can pick — TT = e~2Kr so that the coupling between nearest-neighbor spins is KS = KT. Using this relation, we can identify the Hamiltonian H in the limit as KT —* 00 and fixing KS —* ne~2Kr. H = T^ax{m,m + v) - K^a{dlalol (4.30) m,0 rh The Hamiltonian in Eqn 4.30 still have a gauge invariant upon flipping all the spin emanating from one site. This transformation corresponds to the condition that GJCT = tl+cfx = 1 at every site. The meaning of these constraints become transparent if we consider the system in the o~x basis. In order for GiGr\phys) — \phys), there must be an even number of link with o~x = 1. Identifying the presence (absence) of a dimer with cr1 = ± 1 , we see that the constraint GIGT = 1 implies the presence of an even number of dimers emanating from each site. Chapter 4. Quantum Dimer Model and Ising Gauge Theory 22 4.2.2 Odd Ising gauge theory and the Polyakov loop term T h e o d d I G T , wh ich only allows o d d number of dimers at each site, requires the addi t ion of the Polyakov loop te rm SP to the act ion 4.15. e S ' = U < r z t (4 .31) . t where the product runs over a l l t empora l l inks . It can be shown (see A p -pend ix B of [1]) that this is equivalent, i n the t ime con t inuum l imi t , to choosing GICT — — 1- For the square lattice, the inc lus ion of SP repre-sents a mix tu re of dimers (one l ink occupied) and tetramers (three). W h i l e dimers have a phys ica l interpretat ion of valence bonds, there is no s imi lar interpretat ion for polymers . 4.2.3 Quantum dimer model limit of odd Ising gauge theory T h e quan tum dimer model have the ha rd core dimer constraint . One can retrieve this constraint from the o d d I G T by hav ing a very large coupl ing constant T. In this l imi t (1? —» oo), the term —K-Ylpiaq®*becomes exact ly equivalent to the kinet ic t e rm of the quan tum dimer model . I must note that the potent ia l te rm of Q D M is not presence i n E q n . 4.30. For a square lattice, the known result of the o d d I G T is that i t has a confinement t rans i t ion accompanied by t ransla t ional symmet ry breaking as the Q D M l imi t is approached. T h i s result follows from analyzes of the dua l transverse field Ising mode l [11]. Consequently, the purely kinet ic Q D M on the square lat t ice gives rise to a valence bond crysta l w i t h confined spinons. T h i s concludes the chapter on I G T . In chapter 5, we w i l l look at the mapp ing of Q D M into Polyakov 's compact Q E D . Chapter 5 23 Quantum Dimer Model and Polyakov's Compact QED This chapter is based on E . Fradkin and S. Kivelson's work which discuss short range resonating valence bond theories and superconductivity [3]. The central idea is the realization that dimer models are exactly equivalent to Polyakov's compact electrodynamic in 2+1 dimensions [7]. However, upon mapping to the lattice gauge theory, the system is no longer in the familiar vacuum sector, but rather in the case in which there are sources of "elec-tric" field on the lattice. Despite this fact, one can still draw important conclusions based on this mapping. Let first look at the mapping. Consider an enlarged Hilbert space defined on the links of the lattice. Let {lj(x)} be a set of integer valued variables defined on the links(x, x + ij) of the lat-tice. The states \{lj(x)}) span the unrestricted Hilbert space. Let Lj(x) be the angular-momentum operator with the integers lj(x) as the eigenvalues and |{Zj(x)}) as their eigenstates. One can restrict this enlarged Hilbert space into a subspace in which lj = 0,1 by assigning an infinite energy to all unwanted states. Restricting each link to have an eigenvalue of 0 or 1 would corresponds to having 0 or 1 dimer on each link. This can be done by introducing the constraint term in the following form: Figure 5.1 below shows the dimer energy of a single link at different value of lj(x). In the limit as k —> 0, the energy to any unwanted state would be infinite which can effectively impose the condition that each site can have either zero or one dimer. To map the kinetic and potential term of the Quantum Dimer Model Hamiltonian into compact Q E D , let first define the canonically conjugate(to Lj(x)) phase operator a.j(x) satisfying: (5.1) [aj(x),Lji(x*)] = iSjjSz^ (5.2) Chapter 5. Quantum Dimer Model and Polyakov's Compact QED 24 Figure 5.1: The dimer (Hfiimer) energy of a single link (x,x + ej) as a function of lj(x). For any value of the coupling constant k, configurations with lj = 0,1 have exactly zero energy while any other state will have energies growing like \ . Chapter 5. Quantum Dimer Model and Polyakov's Compact QED 25 While the eigenvalue of Lj(x). are discrete integer, the eigenvalue of G\J(X) should.be continuous and fall in the range of 0 < dj(x) < 2TT, and the corresponding Hilbert space is the space of the periodic functions aj(x) with period 27r, independently at each link. It is easy to show that e ± m o ^ x ) is the ladder operator on eigenstate of L with step size ±m. Proof: & e < m a l = E ^ p L M " ] (5-3) = e i m a m (5.5) Therefore, applying elma to an eigenstate of L would raise the eigenvalue by m. steps. Leima\l) = eimh(e-imhLeimh)\l) (5.6) = e-imd-(l+m)\l) (5.7) = (I + m)eimh\l) (5.8) We can identify elma\l) = \l + m), which is what I set out to show. In the context of Q D M , e'^^/e"™^ 1 ' would be the dimmer creation/annihilation operator on link (x, x + ij). The kinetic term removes two parallel dimers and replace by a perpendicular set of parallel dimers. It can be expressed using creation/annihilation operators: HKIN = ~tJ2(\ H > H +h-C') ( 5- 9) Plaq _ _ i ^ ( e i [ a i ( x ) + a i ( x + e 2 ) - a 2 ( x ) - a 2 ( x + e 1 ) ] _|_ ^  c_-j (5.10) _ ; x = - 2 t E C 0 S ( a i ( ^ ) + a i (z + 62) - a2(x) - a2(x + ei)) (5.11) X The diagonal term are HV = «E(MIKHI + I = ) H J ' (5-12) Plaq = vY^iLi^L^x + e2) + L2(x)L2(x + ei)) (5.13) and the constraint that every lattice site can only form one valence bond Chapter 5. Quantum Dimer Model and Polyakov's Compact QED 26 can be expressed by Q(x) = Li(x) + Li(x - e i ) + L2(x) + L2(x - e 2) = 1 (5.14) The allowed state, denoted by | Phys), must satisfy. Q(f) | Phys) = | P/iys> (5-15) In the next section, I will highlight the correspondence of Q D M in the con-text of compact Q E D . One can show that this model is exactly equivalent to Polyakov's lattice Q E D in 2+1 dimensions with two main differences. 5.1 Quantum Dimer Model in the context of Quantum Electrodynamics The Hamiltonian formula in the last section can be written in a much more transparent and familiar way, by staggering the configuration on the two fields {a.j(x)} and {Lj(x)}. Define the staggered lattice gauge field A(x) and lattice "electric field" Ej(x) by Aj{x) = e^ajix) = {-l)x+vaj(x) (5.16) Ejix) = e^"-sLj(x) = (-l)x+vLj(x) (5.17) with x — (x,y) and Q0 — (TT,TT). This can only be done consistently for a bipartite lattice. W i t h these definition, we can write the constraint of Eq. 5.14 in the form [AjEj(x) - p(x)] | Phys) = 0 (5.18) where Aj is the lattice divergence AjEj(x) = Ei(x) - Ex(x - ei) + E2(x) - E2(x - e 2) (5.19) The staggered density p(x) is defined to be p{x) = ei(3°-x = (5.20) The local condition that each site only links to one dimer now has the standard from of Gauss's law. p(x) represents a background charge density that equals to +1(-1) on red(black) lattice sites. Figure 5.2 shows how a allowed state satisfy Gauss's law under the mapping described. In the Chapter 5. Quantum Dimer Model and Polyakov's Compact QED 27 I I • red • black Figure 5.2: The above figure shows how an allowed state would satisfy Gauss's law described in Eqn. 5.18. Note that there is a background charge density that equals to +1(-1) on red(black) lattice sites. lattice, we assume el®°'x = +1 for red site and -1 for black site. In Fig 5.2, the divergence of the lattice 'Electric' field for site x0 and x\ would be AjEjixo) = El{x0)-El{x0-el) + E2{x0)-E2{x0-e2) (5.21) = 0 . - 1 + 0 - 0 = - 1 (5.22) AjEjixi) = E^xi) - E^Si - + E2(2i) - Eh(xi - e2) (5.23) = 0 - ( - l ) + 0 - 0 = 1 (5.24) In this representation, the Hamiltonian takes the following form. H=^Y,^J(S)-a]{x)f-a}{x)) (5.25) x,j - 2 i ^ c o s ( A 2 i i O ? ) - A i i 2 ( £ ) ) (5.26) X - v Y ( ^ ) E 1 ( x + e2)E2(x)E2(x + e1)) (5.27) where CXJ(X) = ^el<^°'x. By expanding the square in the first term,'we can rewrite the Hamiltonian as H = ^ fe^*)- f ) - f c E c o s C A a i ^ - A i ^ a O X ^ S ) \ x,j ) X Chapter 5. Quantum Dimer Model and Polyakov's Compact QED 28 + ^ E ( ( A i ^ ( ^ ) ) 2 + ( A i ^ ( x ) ) 2 ) - (5.29) X where L is the linear size of the system. Notice that the cosine argument can be written as a curl (V)* x A = A2Ai — AiA2, which we can label as the Magnetic field. When the gauge field A is slow varying, the Hamiltonian becomes H = E + 2I^V)) + \ E ((Ai£ 2(£)) 2 + (AlE2(x))2)+const X X (5.30) where E = (E\,E2,<S) and B = (0,0, Bz) It should be stressed that this field do not represent the standard electromagnetic fields as the Hamiltonian is not the standard E M theory. This first two term of the Hamiltonian can be recognized as the Hamilto-nian for Polyakov's lattice Q E D in 2+1 dimensions for the gauge field Aj(x). The main difference is the constraint selects a space of states that is not the usual vacuum (p = 0) but has an array of sources, p(x) = ±1. In QED, the elementary excitation (photon) should have Aj to be small, slowly varying and gapless. Because we are working with staggered variables, this photon should has wavevectors close to Q0 = (TT, 7r ) . In the context of Q D M , this corresponds to the excitation (resonon) about the R K point (t = v) that Kivelson and Rokhsar argued. When the system is away from the R K point, this mapping tells us that resonon does not exist. Polyakov has showed that Q E D is a confining theory which tells us that (1) The ground state is a sin-glet state that is gapped, and (ii) holons/spinons are virtual processes. In conclusion, under this mapping we have learned that resonon is the ground state at the R K point and that it doesn't exist when the system is away from R K point. One can study furthur by making a duality transform of the com-pact Q E D mapping. Before going into that, I will first present the work by Christopher Henley in 1997. He mapped the Quantum Dimer Model into Height Model at the R K point. 29 Chapter 6 Q u a n t u m D i m e r M o d e l w i t h H e i g h t R e p r e s e n t a t i o n This chapter discusses the results in Henley's paper [2] by mapping the Quantum Dimer Model into height model at the R K point. It is known that for two dimensional square lattice, the system has equal, amplitude ground state at the R K point. Since each dimer covering has equal weight, one can study the system with stochastic dynamics. For the remaining of the section, the system is taken to be a L x L square lattice with periodic boundary condition. I will discuss the mapping of the dimer packing (con-figuration by configuration) to configuration of heights {z(r)} lining on the dual lattice sites. {z(f)} form a rough interface in an 3 dimensional abstract space. In section 6.2, I will define the dynamics on the rough interface that corresponds to a dimer flip in the quantum dimer model. In section 6.3, a continuum limit of the height representation is taken, which is. defined by the standard Langevin dynamics. This approximate coarse-graining of the microscopic describes most of the slow eigen modes. Finally, in section 6.4, another branch of excitation called the Height-shift mode is studied. 6.1 Height Representation To map the Quantum Dimer Model into Height Representation, we first subdivide the square lattice into two sublattices (red and black). Each red site is dressed with an clockwise orientation. As shown in Fig. 6.1, x represents the position of the lattice site while f represents the position of the dual lattice site. The rules for constructing a height pattern {z(f)}, given a snapshots of the dimer, are as follow: 1. When going along the direction of the arrow about a red lattice site, the height decreases by 1 if there is no dimer in between. 2. When going along the direction of the arrow about a red lattice site, the height increase by 3 if there is one dimer in between. Chapter 6. Quantum Dimer Model with Height Representation 30 ^ 3 i r Figure 6.1: The above figure shows the clockwise orientation assigned to each red site, x represents the position of the lattice site while r represents the position of the dual lattice site. In term of equations, let r2 be a step clockwise from f\ about a red site, then: 1. z(r2) — z(ri) = —1 if there is no dimer in between. 2. z(f2) — z(f\) = 3 if there is one dimer in between. Figure 6.2 illustrates the height pattern for a dimer covering. Notice that if the dimer covering satisfy the hard core dimer constraint, the net height difference is zero for the path around one plaquette. However, adding the same constant to each z(f) makes an equally valid height representation of the same dimer configuration. Periodic boundary conditions for z(f) are needed in order to define the Fourier transform when taking the continuous limit. However, with the above mapping, the rough interface at the boundary only satisfy: z(L,y) z(x, L) Z(0,y)=w: Z{x,0) (6.1) (6.2) where wx and wy are multiples of 4. Fig.6.3 shows two dimer coverings with (wx, wy) — (0,0) and (0,12). The columnar state has a height difference of (wx,wy) = (0.0) at the boundary while the staggered state has (wx,wy) = (0,12). There is a one to one correspondence between (wx,wy) and the winding numbers (ttx, Q,y) defined by Rokhsar and Kivelson earlier. In facts, Chapter 6. Quantum Dimer Model with Height Representation 31 Figure 6.2: The above figure shows the height pattern {z(r}} for a dimer covering. The net height difference for the path around one plaquette is zero. Also, adding the same constant to each z(r) makes an equally valid height representation. z(0,y) (a) z(Uy) 11 12 11 12 11 12 11 10 9 7 8 6 5 • i 3 4 1 2 • i -/ 0 (b) 10 9 10 9 10 7 _ 8 _ 7 8 7 6 5 6 3 4 3 1 2 1 5 _ 6 4 3 2 1 - 1 0 - 1 0 -1 Figure 6.3: The above figures show two dimer coverings of a 6x6 system with periodic boundary condition, (a) The columnar state has a height difference of (wx,wy) = (0,0) at the boundary; (b) The staggered state has (wx,wy) = (0,12). Chapter 6. Quantum Dimer Model with Height Representation 32 they are related by (wx,wy) = (4Q,x,4fly) (6.3) With this relation, it is easy to see that local updates rule (to be defined later) conserve the winding numbers (wx,wy). To obtain periodic boundary conditions, one could define the subtracted heights z'(x, y) = z{x, y) - ^j-x - ^-y (6.4) for each topological sectors. The tilt is defined to be the gradient in the height field. For a dimer covering, the mean tilt is (^ , ^-). In the limit of small tilts, z'(f) would obey the same continuum dynamics. However, if tilts are non vanishing as L —> oo, the free-energy functional must be gen-eralized to have different stiffnesses for components of the gradient parallel and transverse to the tilt direction; The same power laws for the dynamics would be deduced. For the remaining of the chapter, we will work with configurations in the columnar sector (wx,wy) = (0,0). 6.2 Dynamics The dynamic of the quantum dimer model is defined by the kinetic term in the Hamiltonian. Because |||)(=| runs over all plaquette, the Hkin term can be rewritten as a sum over \P){ct\ for every possible pair of configurations (p, a), such that /? differs from a only by one dimer flip. Hkm = -*E(III>H + I=>(III) (6-5) Plaq = - * I » M (6-6) This model is endowed with stochastic (Monte Carlo) dynamics in con-tinuous time as follow: 1. Select plaquette at random, at a rate of N plaquettes per unit time, where N = L2 is the number of sites. 2. Flip the plaquette if it has two parallel dimers, otherwise do nothing. At the RK point of the quantum dimer model, the exact ground state Chapter 6. Quantum Dimer Model with Height Representation 33 wavefunction is an equal-weighted superposition of all dimer coverings. V J v s a where the sum is over all valid dimer configurations within a given topo-logical sector. This ground state corresponds to the probability weight of pa = jt~, where N„ is the number of configuration, just as in the classical ensemble. Now let {pa(t)} be the probability of being in microstate a at time t. Then the discrete classical model's dynamics is described by the master equation. £ (Pa(*)-P/?(*)) (6-8) <a-*0> P0 ( 6- 9) where < a —> /3 > means summing over configurations /? that are one dimer flip away from a. The matrix W is an NsxNs matrix defined as follow W Q / 3 = Fa5a(5 - Aap . (6.10) where A is the adjacency matrix. It equals to unity if a and (3 are related by one dimer flip and zero otherwise. Fa = YlpAaf3 is the number of flippable plaquettes in configuration a . Since W is a stochastic matrix, we can apply the Perron-Frobenius theorem to assert that W has nonnegative eigenvalues. In this case, the lowest eigenvalue is zero and it corresponds to the eigenvector pa = ^- which is just the weight of the (equilibrium) steady state. Note that this steady state is unique since dimer flips connect all the microstates. In addition, any time evolution of the system can be decomposed into eigenmodes of the W matrix, in the form £ c A e - A t ( 6 . n ) A In the quantum dimer model, the Hamiltonian is taken to have the matrix elements H = E ( - * ( l l l > H + f t - c ) + «(lll><IIH-l=)H)> (6-12) Plaq = £ (-* | 0)(a | +v | a)(a |) (6.13) {0,*} dPajt) dt Chapter 6. Quantum Dimer Model with Height Representation 34 i i i i i 1 i i 1 i i i i I 1 I. I 1 I i I i i ! Figure 6.4: The above figure shows the change in the height variable when a dimer is flipped. The corresponding height variable is raised/lowered by 4. Ha0 = -tAap + vFa6aP (6.14) At the R K point when v = t, the quantum Hamiltonian matrix is propor-tional to the classical W matrix: Ha0 = tvVap (6.15) This implies that all the eigenvectors of the quantum matrix are the same as those of the classical matrix. There is a one-to-one correspondence of all the eigenstates of the quantum dimer model at R K point to the normal modes of the master equation. Furthermore, the eigenenergies of H are equivalent to the eigenvalues of W: Ex = t\ (6.16) This is the key result Henley has observed. In the next section, we wil l look into the continuous (coarse :grained) version of the dynamic. Let first define the effect on the discrete height field {z(r)} as a result of the dynamic. To eliminate the arbitrariness in defining z(r) and to ensure that the coarse-grained dynamics is continuous in time, we relate z(r, t) at different time, by specifying that a dimer flip on a plaquette changes only the z(r) value in that plaquette's center. As a result, the elementary flipping | ||)(= |of a plaquette' corresponds to raising/lowering the height variable of that plaquette by 4, as shown in Fig. 6.4 6.3 Continuous Theory of Dynamics In the last section, we have showed that the quantum dimer model is equiva-lent to a classical system at the R K point. We have also defined the dynamic on the discrete height field {z(r)}. This section develops the spatially contin-Chapter 6. Quantum Dimer Model with Height Representation 35 uum (coarse-grained) version of the dynamics. In this limit, the model has an easily visualized physical meaning and is solvable by standard techniques. It will help us understand the critical behavior of the quantum dimer model. It should be noted that this theory is general to all rough height models. The only information specific to the dimer model is the value of the elastic constant K and of the height space lattice constant a^. 6.3.1 Continuum Equations and Fourier Modes Let h(x, y) be the spatially continuous height field defined in two dimensional plane. Recall that in our mapping to the height representation, adding the same constant to each z(r) makes an equally valid height representation of the same dimer configuration. Therefore, the free energy F of this height field should be invariant with respect to uniform displacements of h(x, y) —> h(x, y) + c. Furthermore, the free energy should be translationally invariant. The simplest form for F consistent with these requirement is 1 L d2r~ \Vh(r)\2 (6.17) [0,L] 2 where h(r) represents a smoothed version of z[f) and K is the stiffness con-stant controlling the fluctuations of the interface. The dynamic associated with this field theory is formulated as a Langevin equation: where V is the kinetic (damping) constant measuring the linear-response to the force Jj^/j, and £(f, t) is a random source of Gaussian noise. The random noise is not correlated in space or in time. < C(r, tXif, t') >= 2T5(f - f)S(t - t') (6.19) As described in Fig 6.4, an elementary dimer flip changes z(f) on just the plaquette involved by Az = ±4. Thus, the net height is not conserved, and the change is local. In addition, the location of the dimer flip is random. A l l these are modeled by the identical properties of the Gaussian noise in Eqn. 6.19 which is uncorrelated in space and time. Another remark is that the hard core dimer constraint is automatically satisfied when taking the smooth continuum limit. Notice that if the micro-scopic dimer covering satisfy the hard core dimer constraint, the net height Chapter 6. Quantum Dimer Model with Height Representation 36 3 4 3 4 • 11 • 2 5 • 2? 6? (b) 5 (a) Figure 6.5: In the above figure, (a) shows the change in height about a lattice site satisfying the hard core dimer constraint, (b) A discrete height field for a monomer. A monomer is a site with either zero or two electrons. difference is zero for the path around one plaquette. Fig. 6.5(b) shows a discrete height field for a monomer. A monomer is a site with either zero or two electrons. In this case, the continuous height field h(r) becomes a vortex because h(r) represents a smoothed version of z(r) and is continuous. The Langevin equation in Eqn. 6.18 can be solved by standard tech-niques. Let first define the Fourier transform of h(f,t), denoted as hq(t): Since h(f,t) has periodic boundary condition, one can take the Fourier transform of Eqn. 6.18 and 6.19. The short distance cut off for this contin-uum field theory should be within the first Brillouin zone; In other words, the only allowed q value is [—TT,TT] 2 . Eqn. 6.18, after Fourier transform, becomes (6.20) and the inverse Fourier transform is defined by: (6.21) dh?(t) dt = -TK\q\%(t) + Cq~(t) (6.22) and the Gaussian noise < (q<t) * (*') >= 2 r ^ i ? ( j ( t -1') (6.23) Note that 5^ is the discrete delta function . The different Fourier Chapter 6. Quantum Dimer Model with Height Representation 37 components are decoupled in Eqn.6.22 for each wavevectors except for q = 0. Eqn.6.22 is the one-dimensional Langevin equation with a restoring force. 6.3.2 Dynamic Scaling From the Langevin equation, it is easy to show that the correlation function of the height field hq~(t) is exponential. where \h{o) is the relaxation rate \h(q) = TK\q\2 (6.25) The smallest nonzero wavevector is qmin = (x>0) a n d the smallest relax-ation rate is therefore: , ^ , 4w2TK .„ „. K{qmin) - —j2— (6-26) In the next section, we will show that \h{q) is the eigenvalue of the excitation spectrum of the Fourier mode. 6.3.3 Fokker-Planck Mode Spectrum Similar to the master equation (Eqn. 6.8) in the discrete stochastic dynam-ics, the Fokker-Planck equation describes the continuum stochastic dynamics for the evolution of the probability density P({h$(t)}). jP{{h&)}) = wh{P({h?(t)})} (6.27) = r S ^ t ^ + m^Pahfit)}) (6.28) The Fokker-Planck equation can effectively describe the slowest Fourier mode at each wavevector. However, there are many more eigenmodes be-yond the ground state. These modes are worth studying because they cor-respond to excited states in the quantum dimer model at its critical point (RK point). Recall that in the Q D M , the ground state at the R K point is the equal amplitude superposition of all the states, which is equivalent to having equal probability for all the possible height configuration. Upon taking Fourier transform, this unique zero energy groundstate corresponds Chapter 6. Quantum Dimer Model with Height Representation 38 to the Boltzmann distribution which is Gaussian: P0({hqit)}) = exp ^ - £ \m%h_^j (6.29) To construct all the other eigenfunction, it is convenient to write $>({hq~}) = _ £ ( M 1 L so that the time evolution operator of ^ /({hg}) is Hermitian: vhere % s ^ S ^ - ^ ^ f e ^ ™ ^ ) ( 6 - 3 1 ) = £rif|,fe]e,- (6.32) 9 9 ^ 0 where the annihilation operator is C^[\K\^y l (-±-+1-K\g\%) (6.33) - 9 and the corresponding creation operator is 4= ( W )"M--^ + ^ k 1 2 ^ ) (6.34) For all <? q, C~ commutes with C^. The Hamiltonian shown in Eqn. 6.31 is mathematically identical to the quantum Hamiltonian for a discrete set of harmonic oscillators. The frequency for each wavevector q is A/J(<J) = r i ^ g ] 2 . 1/2 The ground state wavefunction is simply \I/0 = P0 • The eigenfunction for the excited state in this mode can be expressed in terms of the annihilation and creation operator: *({M; W f l } ) = ( n ^ 0 ( c t ) " w ) (6.35) the set of nonnegative integers {n(q}} labels the occupation of oscillators at different wavevectors. The total energy of the oscillators with occupation Chapter 6. Quantum Dimer Model with Height Representation 39 {n(q)} is: hot({n{q)}) = 2~Z n(Q)*h(q) (6-36) In terms of the probability P({hq{t)}) = ^({/ig-(i)})P0({/ig-(i)})1/2, each eigenstate is a product of P0({h^(t)}) times polynomials in {(^rKlq}2)?^}. For elementary excitation, in which n(q) = 1 for one wavevector and zero for all the others *ee((A-}) = 4o*»(^» (6-37) = h 1 (-^ + U\q0\2h_qJ*0({hq-}) (6.38) = 2^« l 2 fc - - , „*o({M) (6.39) From Eqn. 6.36, degeneracies of the excitation come from many ways: 1. Global Symmetries - q —> —9 2. less trivial degeneracies: n(<f) = n(—q) = 1 and n(q) = 2, n(g) = 0 In summary, the mapping into height model tells us that at the R K point, the excitation has a photon like dispersion. This corresponds exactly to the resonon discussed by Rokhsar and Kivelson. Similar conclusion are discussed when Fradkin map the quantum dimer model into Polyakov's compact QED. However, Henley noticed that in addition to Fourier mode discussed in this section, there is another branch of excitation that arises from the additional symmetry of translation in 'height space'. This is referred as the height shift mode and will be discussed in the next section. 6.4 Height-Shift Mode In the last section, we used the periodic boundary condition for the height function to obtain excitations pertained to the Fourier mode. In addition to translation in real space, there is a translational symmetry in the height space as well. The mapping from the dimer covering to height representation is one-to-many. For example, a global shift of z(r) —> z(r) + 4 describes exactly the same dimer configuration. With this property, one would expect Chapter 6. Quantum Dimer Model with Height Representation 40 the probability of having a height configuration {z(r)} be the same as the probability of obtaining the height configuration {z'(r}} where z'(f) = z(r) + 4 for all r. P({z(r)})=P({z'(r)}) (6.40) A corollary in [8] states that if we write the dimer or spin configuration as a function of height, then this function must have a period of 4. Therefore, Eqn 6.40 define the translational symmetry in height space. This symmetry allows us to obtain another kind of slow mode (in a finite system), corre-sponding to random walk of the mean height, Consider the average height, ^ - I j f r d2rHr,t) (6.41) Note that h(t) = -^h0(t), which is just the q = 0 mode that was excluded in the Fourier mode. The Langevin equation (Eqn.6.22) with \q\ = 0 tells us that j0(t) cc j(t) simply executes Gaussian random walk. ' ^ = C o ( i ) (6-42) where C_,(i) is a random source of Gaussian noise that is not correlated in space and in time. The random walk behavior is: (\h(t) -h{0)\2) =D(N)t (6.43) with a diffusion constant 2F D(N) = — (6.44) The diffusion constant is inversely proportional to the system size, which means that this kind of slow mode is only significant in finite system. When h(f) describes a genuine interface, states with different h are all distinct, and the distribution of h simply spreads diffusively without ever reaching a steady state. However, the image space of h(r) should be considered a circle of diameter 4. Thus the distribution function, which begins sharply peaked at a particular value of h, will evolve to a uniform distribution at some rate. In the next section, we will study the Fokker-Planck Modes of the mean height h(t) and present the complete set of eigenfunctions and eigenvalues of Q D M at the R K point. Chapter 6. Quantum Dimer Model with Height Representation 41 6.4.1 Fokker-Planck Modes of the Mean Height h(t) The Fokker-Planck equation for h(t) is simply the diffusion equation (Eqn 6.27 with 9 = 0); Its eigenfunctions are simply plane waves as a function of h, $Q(h) = eiQ~h (6.46) and the corresponding eigenvalue is h(Q) = JjQ2 (6-47) = \D{N)Q2 (6.48) where D(N) = Note that Q is one-dimensional and is the wavevector conjugate to the mean height h. The translational symmetry in the height space require that ^f(h) is periodic under h —> h + 4. $(7i) = $(h + 4) (6.49) This condition determines the possible values for Q. e ^ 4 = 1 =$> 4Q = 2n 07r Q = no-where n0 is an integer. The smallest of such eigenvalue is given by no = ±1 X r = X-h{±\) = ^ (6.50) The complete set of eigen functions are the product of Eqn. 6.35 and 6.46 with the excitation labelled by the set of integers {n0, {n(q)}}. * ( { M ; W f l . 9 V o } ) * Q ( / i ) (6.5i) and the corresponding eigenvalue is ^tot({n(q)},q^0)+n20Xfn (6.52)-Chapter 6. Quantum Dimer Model with Height Representation 42 By mapping to the height model, Henley has presented two modes (height-shift and Fourier modes) that describe the low energy excitation of the quan-tum dimer model at the R K point. The relationship of the height-shift mode and the Fourier modes is analogous to the relationship between two kinds of low energy excitation in a quantum spin system. The height-shift mode is the analog of the change in total spin number while the Fourier mode is the analog of a spin-wave mode. This concludes the study of quantum dimer model with height-representation. In the next chapter, I wil l discuss the duality between the height model and the Polyakov's compact QED. 43 Chapter 7 Duality between the Height-Representation and Polyakov's Compact QED This section discusses the duality relationship between the mapping of quan-tum dimer model in height representation (Chapter 6) and in Polyakov's Compact QED (Chapter 5). I will first define the duality transform from the compact Q E D representation and then show that the resulting Hamil-tonian is exactly equivalent to the Henley's height representation within the appropriate limit. Recall that Henley's height representation focusses on the R K point of the model which showed the excitation above the equal amplitude ground state. 7.1 Duality Transformation of Polyakov's Compact QED Let x labels the sites of the direct lattice while r labels the site of the dual lattice. The site of the dual lattice resides on the center of each plaquette in the direct lattice as shown in Figure 7.1. Let S(r) be an integer valued operator with eigenvalue S(r) and it acts on the dual lattice site f. Let B(r,r + ej) EE Bj(f) be a classical background real valued field which resides on the link of the dual lattice. The duality transformation is defined to be: Ej(x) = ejk{AkS{r) + Bk(f)) • (7.1) where ejk is the Levi -Civita tensor. if j = i,fc = 2. if j = 2, k = 1 otherwise Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 44 • 4 o » 4 0---1 4 > o - - o Ii Ik ft n 6— ' " " t r " P IF - - Q 6—-O 4 c » 4 9 W 0 » • Figure 7.1: The sites of the direct lattice are labelled by x while the sites of the dual lattice are labelled by r and Afc is the lattice derivative AkS(r) = S(r) -S(f-ek). Note that Eqn 7.1 can be written in a more suggestive form if we treat S(r) as the z-component of a vector field with zero x and y components(ie. S(r) = (0,0, S(f)). In this case, the electric field is the curl of the integer vector field S(r) plus the background term. E{x) ~ V x S(r) (7.2) In term of the new dual operator 5(f) , the electrostatic-like constraint (i.e. Gauss's law) has become: V-E = AjEj(x) = ejk(AjAkS(r)-^AjBk{ff)) (7.3) = ejkAjB^r) (7.4) = P(2) (7.5)• where e i f c A j A f c 5 ( r ) = e 1 2 A 1 A 2 5 ( r ) + e 2 1 A 2 A 1 ( S ( f ) (7.6) = AxA2S(r) - • A 2 A 1 5 ( f ) . (7.7) = 0 (7.8) Notice that the electrostatic-like constraint has become the magneto-static constraint if we treat the Bk(r) as having only x and y direction, (ie. B(f) = (Bi(f),B2(r),0). This is the usual electric, magnetic duality. V x B(r) = ejkAjB^r) = p(x) = ( -1) X l + X 2 (7.9) Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 45 The set of possible solutions in Eqn. 7.9 is in one-to-one correspondence with the set of classical dimer configuration since Eqn. 7.9 is just the dual version of the constraint. Because of this correspondence, there must be a way of classifying these configurations. The next session wil l discuss the winding numbers in this representation with a familiar notion - Gauge Transforma-tion. 7.2 G a u g e T r a n s f o r m a t i o n a n d G r o u n d S ta te Degene r a c y If we treat Bk(f) as a two dimensional vector potential and S(r) as a'scalar potential, then we can introduce the idea of gauge transformation similar to that in electromagnetic. where , without the loss of generality, r(f) is an integer-valued function on the dual lattice. A local change in Bk(r) —* B'k{r) and S(r) —> S'(r) leaves the constraint invariance. A ^ ( f ) = ejk{A3AkS'{f) + AjB'k(r)) = ejk(AjAkS(f) - A , A f c r ( r ) + A3Bk{r)) + A,A*r(f)) = ejk(AjAkS(r) + A3Bk{r)) These kinds of transformation are defined as the local gauge transfor-mation, which leave the constraint unchanged (gauge invariance). However, there is a set of transformations Bk(r) = B'k(f) — Bk(r) = AkT(f) that do not have gauge invariance. They are defined as global gauge transformations1. There are explicit conditions that distinguish between the local an global gauge transformation. Let first define the line integrals of Bk(r) along two non-contractible loops (71 ( f ) and 72 ( f ) around the torus as shown in Fig. B'k(f) = Bk(f) + AkT(r) S'(f) = 5 ( f ) - r ( f ) (7.10) (7.11) 7.2 1 T h e term global gauge transformation is counter-intuitive as gauge transformations are often referred to transformations that are gauge invariant. Nevertheless, I w i l l use this terminology. Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 46 Figure 7.2: A square lattice with periodic boundary condition is isomorphic to a torus. The two non-contractible loops must wrap around the boundary of the torus. 71 ( f ) is along the x\ direction while 72 ( f ) is along the x2 direction and both loops go through the dual lattice site f. L £ B^f+nxh) (7.12) 7 1 1 = 1 L £ £ 2 ( f + n 2 e 2 ) (7.13)-ri2=l Non-contractible loops must wrap around the boundary of the torus. In Fig. 7.2, 71 ( f ) is along the x\ direction while 72 ( f ) is along the x2 direc-tion and both 71 ( f ) and 72 ( f ) go through the dual site f. The line inte-grals Iyi^[B] and Il2^[B] are invariant under local gauge transformation. However, global gauge transformations, which do not respect the periodic boundary condition, change the values of I71^[B] and J 7 2 ( ^ [JB] . The table below summarizes the two kinds of gauge transformations. Gauge Transform Does Bk = (B'k - Bk) has Periodic Boundary? Condition Local Yes 7 7 l ( f 0 [S ] = / 7 l ( r - 3 [ B ' - B ] = O J 7 2 ( f ) [ S ] = 0 Global No 71(f) 7 2 M Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 47 7.2.1 The Meaning of Local and Global Gauge Transformation Recall that the electric field Ek(x) is related to the integer valued operator S(f) and the classical background field Bkif) in the following way: Ej{£) = ejk(AkS(r) + Bk(r)) (7.14) If the operator S(r) is treated as the quantum fluctuations and is set to zero. Then the classical background fields Bk(r) describe classical configu-rations that are the parent states for the quantum evolution of the system. As a result, the line integrals from Eqn. 7.12 become: IJi(rl[B} = £ ^ ( r ) (7.15) 1i(r) = Yl^iEjiS) (7.16) 7*(£) = e j i 2 Z ( - ^ X l + X 2 + n i L j ^ + ^i) (7-17) Ui — l which is simply the sum of the differences in the number of dimers occupying neighboring parallel links. For example, the configuration shown in Fig. 7.3(a) would have the value of Iyi^[B] = 1 and Il2^[B] = 2. It is easy to see that flipping parallel sets of dimers conserve the value of the line integrals. In Fig. 7.3(b), Iyi^f)[B'] and I72^[B'] is the same as 7 7 l(^[B] and I^2^[B] after having two plaquettes flipped. Local gauge transformations are equivalent to transforming from a classical dimer con-figurations by flipping a set of plaquettes with parallel dimers. On the other hand, global gauge transformations correspond to processes which bring configurations from one topological sector into another one. Thus, there is an one to one correspondence between the line integrals and the winding numbers introduced by Kivelson and Rokhsar. (iJl(rl[B],iMS)[B}) = (nx,ny) (7.18) Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 48 (a) (b) Figure 7.3: In (a), the value of the line integrals in Eqn. 7.15 are I71^[B] = 1 and Iy2(fj[B] = 2.The boundary are continuous. In (b), two plaquettes are flipped as indicated by grey circles but line integrals are still unchanged. / ^ [ J B ^ l a n d J 7 a ( f ) [ B ' ] = 2 . 7.2.2 Ground State Degeneracy Recall the constraint equation for the vector field Bk{r). tjkAjBkif) = p(x) = (7.19) The number of distinct solution satisfying 7.19 for a topological sector with a given winding number is equal to the degeneracy of the ground state in that sector. This is because the line integrals are invariant under the flipping dynamics and the Bk corresponds to all the possible classical configurations. By keeping track of the degeneracy for each winding number, one could determine the degeneracy of the full quantum theory unless there are extra degeneracies such as gapless modes. Consider the columnar sector with winding number (0,0) as an example. I^f)[B]=^Bi(f)=0 (7.20) 7i(f) In the gauge B\{r) = 0, there are two possible solutions satisfying Eqn. 7.19 B^r) = 0 (7.21) B2(r) = + ( 1 ± ( 2 ± 1 ) n ) ( - i r (7.22) Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 49 Similarly, within the gauge B2(r) = 0, there are also two solutions. These four states correspond to the four-fold degenerate states in the columnar sector (for an infinite system). 7.3 Dual Form of the Hamiltonian The goal of this section is to write the Hamiltonian from the compact Q E D representation in terms of the dual variables defined in 7.1. Later on, I will show that this corresponds exactly with the height Hamiltonian. Let first assume we are working in a topological sector with winding number (Iyi(f)[B}, Il2(f)[B]). Let P{r) be canonically conjugate to the integer valued field S(f) . •[P(ff),S(P)]=iSpp (7.23) While the eigenvalue of S(r) are discrete integer, the eigenvalue for P(f) should be continuous and fall in the range of 0 < P(r) < 2ir, and the corresponding Hilbert space is the space of the periodic functions P{r) with period 2-7T. Recall the relation in Eqn. 7.2 that E(x) ~ V x S(r). From the dual nature of the transformation, it is easy to see that P(r) = A 2 i 1 ( x ) - A 1 i 2 ( x ) (7.24) ~ ( V ) 2 x i ( £ ) (7.25) where A = (A\, A2,0). Let verify 7.24 be showing that the following com-mutation relations are equivalent \E3{x),V x A(x')} = lE3(x),P(f)] (7.26) The left hand side of the equation make use of the conjugate relation [Ej(x), Aji(x')} — —iSjjiS.. -,. [Ej{x),V x A] = [Ej(£),A2A1(x')-A1A2(x>)] = [£,•(£),iiO?) - Ax(x' - e2) - A2{x') + A2(x' - ei)] = — i \5n5-. — 8i\8^ -, - — SioS- -, + 8i2$- */ - ) \ J1 x,x' J*- x,x' — e2  J x,x' x,x'—eij i(5-. . — (5- -*,), if 7 = 1 x x,x'—e<i x,x  n J i(5. -, - 5- - . ), if j = 2 v x,x x,x — e\''  J Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 50 The right hand side make use of the conjugate relation [S(f), P(r')] = r , r ' &•(£), P{?)\ = [ejkAkS(f).P(?)} [S(f)-S(f-e2),P(f)}, if j = 1 - [ 5 ( f = ) - 5 ( f - e i ) , P ( T 0 ] , if J" = 2 i(<5--,, . -<5--) , if j = 1 ^^r* ,r ' ^f^r'+ei^' i^ J 2 Comparing the two commutation relations, we can see that the relation in Eqn. 7.24 is true 2 The Hamiltonian dual to that of the Polyakov's compact Q E D Hamiltonian (Eqn. 5.30) is Hduai = ± ( p A f c S ( r ) + B f c ( f ) ) 2 - Y ) -2*£cos(P(r))(7.27) \ r,k ) r + ^ £ ( ( A 1 ( A 1 5 ( r ) + B 1(f))) 2 (7.28) r + ( A 2 ( A 2 5 ( r ) + J B 2 ( r ) ) ) 2 ) . (7.29) Next, I want to show that the dual transformed Hamiltonian in Eqn. 7.27is equivalent to Henley's Hamiltonian in height representation at the R K point. If we approximate the cosine term with the quadratic term cos(P(f)) « 1 - (7.30) then the Hamiltonian would become Hduai = t£P 2 ( r - ) + | £ ( A 2 5 ( r ) + A 2 5 ( r ) ) 2 (7.31) r r = *E^ 2w + iE( v 2 ^) 2 (7-32) f f where only the kinetic and potential term are considered. I also fix the 2 There is a sign different in the delta function between the right and left hand side of Eqn.7.26 (5- - instead of <5. -,. . ) This is because the lattice derivative Ak is not centered in its definition. Chapter 7. Duality between the Height-Representation and Polyakov's Compact QED 51 topological sector to be in the columnar sector so Bj.(r) takes one of the form described in Eqn. 7.21. Recall that at the R K point, the Hamiltonian in terms of height representation takes the following form. Hheight = r £ (p\ + -K2|9f h}) (7.33) where p$ is canonically conjugate to hg-. Comparing Eqn. 7.31 and 7.33, we can see that Height is just the Fourier transform of Hduai- This relation tells us that the two studies done by E. Fradkin (in chapter 5) and C. Henley (in chapter 6) are interconnected.' Chapter 8 52 Topologically Protected Qubit in Triangular Quantum Dimer Model In the past decade, many remarkable efforts have been made towards the development of a quantum computer capable of performing calculations that are impossible with classical devices. A well-known example, proposed by Peter W. Shor in 1994, is factoring large numbers exponentially faster that the best classical algorithm, thus rendering most modern cryptographic sys-tem potentially obsolete. However, when transmitting signals, one must consider the effect of noise, which might alter the original code. A n ap-proach that is similar in spirit of that used in classical computers is the quantum error-correction scheme. It involves redundant multi-qubit encod-ing of the quantum data combined with error-detection and recovery steps. Such a scheme is generic but requires repeated active interference during run-time. This chapter will discuss an alternate solution to this problem-the topologically protected qubit. The basic idea of topologically protected qubit is to make use of a many-body quantum system whose Hilbert space is comprised of mutually or-thogonal sectors. If the system has a gapped excitation spectrum, then representing the two qubit states (0 and 1) using these topological sectors can protect unwanted mixing through noise. To encode one-bit information onto a superposition of the two qubit states, the system must have global operators that allow such manipulation. A promising candidate that fulfills the above requirement is the triangular quantum dimer system. Ioffe and his collaborators [9] proposed a model of building the topologically protected qubit using triangular dimer system. In the next section, I wil l highlight the various properties of the triangular dimer system that suit the requirement for a topologically protected qubit. Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 53 Figure 8.1: One allowed configuration satisfying the constraint that every vertex must belong to only one dimer in a triangular lattice. H = H 0+ H 0+ H 0= 2-[-((l"X--| + l--X"l)+y(i--X--| + l''X<'l)] HQ= | H ( h ' X - l + l-X»l) + v(l-X-l + l»X"l)] H 0 = | [-t (KXsN + W ' l ) + v{KX''\ + ksXol)] Figure 8.2: The Hamiltonian for triangular dimer system. The kinetic term rotates parallel dimers while the potential term allows frustration of parallel dimers. The Hamiltonian is a sum over three different kinds of plaquettes 8.1 Triangular Dimer System In the triangular quantum dimer model, the allowed configurations must satisfy the constraint that every vertex belong to only one dimer as shown in Fig. 8.L The simplest Hamiltonian is a sum over three different kinds of plaquettes as shown in Fig. 8.2 The kinetic term rotates parallel dimers on appropriate plaquettes. When t > 0, the kinetic term favours the 'columnar' phases (Fig. 8.3(a) which have the maximum number of flippable plaquette. The potential them allows frus-tration of parallel dimers, and favours the 'staggered' phase as shown in Fig. 8.3(b). The combined effect of the two term produce a dimer liquid phase at the appropriate ratio of | . e will look at the two particular features of this model in the context of quantum computing : (1) the topological structure of the Hilbert space and (2) the presence of a liquid ground state with its robustness to disorder • Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 54 Figure 8.3: (a) is a columnar state with the maximum number of flippable plaquette. (b) is a staggered state with the no flippable plaquette. 8.1.1 The topological structure of the Hilbert space In the square lattice, the topological sectors are classified by the winding numbers (£lx,Qy) that grow with system size - § < n * , f i y < § (8.i) where L2 is the number of vertices in a square lattice. Consider a triangular lattice with cylindrical boundary condition (continuous in the x-direction and free in the y-direction), the parity of the dimer count along a reference line 7 parallel with the y-axis defines a topological order parameter. As shown in Fig. 8.4, if there are even(odd) number of dimers cutting across the reference line 7 , it will stay even(odd) upon rotating any set of parallel dimers. Therefore, the action of the Hamiltonian leaves the parity invari-ant and the Hilbert space splits into two topological sectors Tte and Tt0 characterized by even and odd dimer counts. 8.1.2 Liquid ground state in triangular lattice dimer model The liquid phase of the square lattice dimer model is only presence at the R K point v = t with its ground state characterized by the equal amplitude superposition of all the state within a topological sector. However, it is difficult to maintain a system at a particular point (v = t) in order to obtain a liquid ground state. This problem is overcome in the triangular lattice where there are numerical evidence which shows that the liquid phase is presence over a finite region for the parameter j. Recent Monte Carlo simulations [12] on large systems with Lx<y = 36 exhibit short-range dimer-Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 55 Figure 8.4: The even and odd dimer count along the reference line 7 is invariant under the action of the Hamiltonian. dimer correlations within a parameter region of ! < H < 1 (8.2) , indicative of a liquid state. Ioffe and his collaborator [9] has studied its robustness to disorder by numerical diagonalization studies with system size going up to Lx<y = 6. They find that: 1. The triangular dimer model develops an isolated twofold degenerate dimer liquid ground state for a cylinder within parameter region 0.8 < j < 1. The gap A separating the ground state from excited states is of order O.li in this region. Note that the degeneracy is only exact in the thermodynamic limit (infinite system size). For finite system, the twofold degenerate ground state is separated by a small gap A<2 which is significant in determining its robustness to disorder. Putting this issue aside, the mixing of the two ground states involves creation of topological defects (real and virtual) which violate the dimer constraint and is- exponentially suppressed in the system size Lx. 2. When perturbed by a quenched disorder potential of strength d, the ground state degeneracy is robust to within a factor of 10~ 3 to 10~ 2 of d. This robustness is expected to increase exponentially in the system size L,,. Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 56 8.2 Basic operations of a qubit There are two kinds of manipulation in encoding an qubit, which is expressed in the qubit Hamiltonian: Hqubit = hxcrx + hzoz (8.3) where the two qubit states are now represented by spin up (| "[")) and spin down (| i)). o~x and oz are Pauli matrices and hx, hz are the (manipulable) parameters producing the amplitude (a) and phase (x) mixing in the qubit state vl+. cr The implementation of hx and hz requires a controlled mixing of the pro-tected dimer states, implying a reduction of the ground state's topological protection. The details of implementing the amplitude and phase mixing operators are specific to how the triangular dimer system is constructed. In the next section, I wil l discuss a construction using Josephson junction array, which was proposed by L. B. Ioffe in [9]. 8.3 Implementing the topologically protected qubit using Josephson junction array The array is made up of Josephson junctions to emulate the triangular dimer system. I will first explain how the design naturally pick classical dimer configurations as ground states. Next, We will look at the energy cost for dimer flipping and the coulomb potential between parallel dimers. Finally, basic qubit operations and the construction of a register of qubits wil l be discussed. Fig.8.5 shows the basic construction of the array. Each vertex or lattice site is made up of six Y-shaped superconducting islands with two ends join-ing the hexagonal vertex and the third end linking the neighboring hexagon. Each link is made up of a Josephson junction with capacitance C;, charging energy Ef = ^~r, Josephson current /(, and Josephson energy Ef = ®° = te denotes the flux quantum with h, c, and e denoting Planck's con-stant, the velocity of light, and the elementary charge respectively. Each hexagonal vertex contains six Josephson junctions with the same capaci-tance and energy relation, which I will denote with Ch, Ih, E%, and E£. Finally, Cy is the capacitance between the array and the ground-plate. \ Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 57 Figure 8.5: The array is made of Josephson junctions to'emulate the trian-gular dimer system. To understand the physics of this construction, we first look at the Quan-tum phase model, which is often used to describe the Josephson junction array. Eqn. 8.5 below shows the Hamiltonian for a uniform two dimen-sional square lattice. Although it is not triangular, we can still gain some qualitative understanding of the behavior. where 2eqi is the net charge on the ith island and (ij) is a notation for nearest neighbor. A n external gate voltage Vx contributes to the energy via the induced charge qx = Ylij Ci^x • EJ is the Josephson energy which describes the Cooper pair tunneling. Uij is the electrostatic energy between junction i and junction j and it is determined by the capacitance matrix C y . The two contributions in the Hamiltonian (Eqn.8.5) favour different types of ground states. When the Josephson energy is large (EJ » Uij), the system tends to establish phase coherence or in the liquid/superconducting phase. On the other hand, large charging energy (Uij » EJ) favours charge localization on each island or in the Mott insulating phase. Wi th this qualitative relations in mind, we can proceeds to the construction of the qubit. We choose a large capacitance Ch to join the islands electrically into one hexagonal vertex. A small capacitance Cy is needed to define the large Ec charging energy Ehex w -g- of the hexagonal array. A large charging energy protects unwanted charging or discharging of the entire array. Next is to Qx)Uij(qj - qx) - EJ Y C0S(<Pi ~ <t>j) <ij> (8.5) Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 58 i—^ i—\ i—\ i—\ < <^ > < ) ( > "K 6^P1 /^ "\ Step 2 / " \ ' \ i \—i \ i \ i i—^ i—\ < ) —< > \ / \ _ _ / Figure 8.6: The series of steps in a basic dimer flip process. The overall Ej2 hopping amplitude is of order t ~ apply an external gate voltage (Vx in the Quantum phase model) such that the vertices have equal energy (to an accuracy better than Ef) for two states differing by one Cooper pair. This wil l restrict each link junction to have either zero or one cooper pair because the large charging energy E^ex lifts other charge states to high energies. Wi th only one electron per hexagonal vertex, each vertex is involved in the formation of a cooper pair with one and only one other vertex through the link junction. This defines a classical dimer configuration by regarding the cooper pair between two vertices as a valence bond. The corresponding Hilbert space of dimer states is protected by the energy scale Ef. Fig.8.6 describes a basic dimer flip which first involves localization of a dimer to one vertex with an energy cost of Ef. Then the localized dimer hops over junctions with energy E f to the new vertex islands. This would force the originally parallel dimer touching the new island to be localized. The second dimer will also hops over junctions with energy E f to a new vertex. Overall, the hopping amplitude is of order FJ2 The electrostatic interaction between parallel dimers depends on the choice of the capacitance matrix and its dependence on the dimer configu-Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model 59 Amplitude Shifter Figure 8.7: A possible construction of a qubit with cylindrical boundary condition using Josephson junction array. The phase of a qubit state is controlled by a gated superconducting strip (GSS) placed along the reference line 7. The amplitude mixing is controlled by a tunable Josephson junction placed at the inner qubit boundary. ration. This can be done by comparing the charging energy of a staggered configuration in which there are no parallel dimers to the charging energy of a columnar configuration in which there are a maximum number of par-allel dimers. The interaction energy v between parallel dimers; in the limit Ci«CY,Ck is v = tf& f 8 7) C f c[(l + & ) ( 1 + 3&)P The electrostatic energies of the liquid phase dimer configurations indeed scale(to within ~ 5% accuracy) with the number of parallel dimer pairs. The condition C; << Cy guarantees a short range interaction between dimers. Fig. 8.7 shows the possible construction of a qubit with cylindrical boundary condition. We label the periodic direction as x with length and the free direction as y. The entire qubit is hexagonal in shape with dimensions Lx and Ly. As discussed in the last section, basic qubit operation involves the am-plitude (a) and phase (x) mixing in the qubit state M = M ± £ f M . ( 8 . 8 ) V I + o r Chapter 8. Topologically Protected Qubit in Triangular Quantum Dimer Model ,60 In our Josephson junction array, amplitude mixing can be achieved in the following steps: 1. Localizing a dimer near the inner boundary where one Cooper pair (particle) retreats to one hexagon, leaving the partner hexagon empty (hole). This virtual particle-hole excitation costs an energy of Ef. 2. While the particle remains pinned down at the weak junction, the hole is taken around the inner boundary through appropriate dimer flips and is subsequently recombined with the particle. / E J \ M 3. This process results in a mixing amplitude hx ~ Ef ( > where M is the number of links on the inner boundary. The phase shifter can be implemented through a gated superconducting strip placed along the reference line 7 as shown in Fig.8.7. It is capacitively coupled to the array and can shift the energy of the two ground states with respect to each other upon biasing the strip. In terms of Quantum phase model, this is equivalent to applying external gate voltage Vx along each link on the reference line. This will result in the qubit's phase x being modified. To avoid excitations within the dimer liquid, one must carefully control the biasing amplitude u (the energy acting on one dimer) and the duration r of the manipulation, u and r must satisfy the constraints: u ~ t << Ef and r > 2j; A is the energy gap between the ground state and the first excited state. Furthermore, the strip must be electrically disconnected during idle time because a fully connected strip could decohere, the system by electric fluctuations fed to it. Therefore, we should construct the strip from an isolated superconducting island which is to be connected when the phase shifter is needed. The two qubit operations requires a reduction on the system's topological protection; While the amplitude shifter takes the system out of the protected Hilbert space through virtual breaking of a dimer, the phase shifter requires introduction of a global operator. In turn, these processes are strongly inhibited during idle time and give the qubit its robustness. In addition, the level of protection grows with the qubit size. While amplitude mixing is exponentially small in Lx, the phase drifting is exponentially small in Ly. 61 Chapter 9 Conclusion We have reviewed the previous studies done on the quantum dimer model with emphasis in the context of topologically protected qubit. While Ioffe has proposed a physical implementation using Josephson junction array, we believe this idea can be applied to optical lattices with more control over the many-body system. This thesis is an introduction of the quantum dimer model and its application in quantum information. One extension to this project is to study the physical implementation needed to emulate the triangular dimer system using optical lattices. 62 Bibliography [1] R. Moessner, S.. L. Sondhi and E. Fradkin Phys. Rev. B 65, 024504 (2002) [2] C. L. Henley, J. Stat. Phys 89, 483 (1997) [3] E. Fradkin and S. A. Kivelson, Mod. Phys. Lett.B4, 225 (1990) [4] P. W. Anderson, Science 235, 1196 (1973) [5] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2379 (1988) [6] E. Fradkin, D. Huse, R. Moessner, V . Oganesyan and S. Sondhi, cond-mat/0311353v4, (2004) [7] A. M. Polyakov, Nucl. Phys. B120, 429 (1977) [8] C. Zeng and C. L. Henley, Phys. Rev. B 55, 14935-947, (1997) [9] L. B. Ioffe, M. V . Feigel'man, A. Ioselevich, D. Ivanov, M. Troyer and G. Blatter Nature 415, 503 (2002) [10] T. Senthil and M. P. A. Fisher, Phys. Rev. B 62, 7850 (2000) [11] R. Moessner, S. L. Sondhi and P. Chandra, Phys. Rev. B 41, 11693 (1990) [12] R. Moessner, S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001) 

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