Loculizution oz u durticle due toDissipution in 1 und 2 DimensionulLutticesbyMatthew HasselfieldB.Sc., The University of Manitoba, 2003A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinThe Faculty of Graduate Studies(Physics)The University Of British ColumbiaOctober 2006c© 2006, Matthew HasselfieldAvstruwtWe study two aspects of the problem of a particle moving on a lattice while subject to dissipa-tion, often called the “Schmid model.” First, a correspondence between the Schmid model andboundary sine-Gordon field theory is explored, and a new method is applied to the calculationof the partition function for the theory. Second, a traditional condensed matter formulation ofthe problem in one spatial dimension is extended to the case of an arbitrary two-dimensionalBravais lattice.A well-known mathematical analogy between one-dimensional dissipative quantum mechan-ics and string theory provides an equivalence between the Schmid model at the critical pointand boundary sine-Gordon theory, which describes a free bosonic field subject to periodic in-teraction on the boundaries. Using the tools of conformal field theory, the partition functionis calculated as a function of the temperature and the renormalized coupling constants of theboundary interaction. The method pursues an established technique of introducing an auxiliaryfree boson, fermionizing the system, and constructing the boundary state in fermion variables.However, a different way of obtaining the fermionic boundary conditions from the bosonic the-ory leads to an alternative renormalization for the coupling constants that occurs at a morenatural level than in the established approach.Recent renormalization group analyses of the extension of the Schmid model to a two-dimensional periodic potential have yielded interesting new structure in the phase diagram forthe mobility. We extend a classic one-dimensional, finite temperature calculation to the caseof an arbitrary two-dimensional Bravais lattice. The duality between weak-potential and tight-binding lattice limits is reproduced in the two-dimensional case, and a perturbation expansionin the potential strength used to verify the change in the critical dependence of the mobilityon the strength of the dissipation. With a triangular lattice the possibility of third ordercontributions arises, and we obtain some preliminary expressions for their contributions to themobility.iihuvly of ContyntsAvstruwt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iihuvly of Contynts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAwknowlydgymynts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 Introduwtion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Caldeira-Leggett Dissipative Quantum Mechanics . . . . . . . . . . . . . . . . . 51.3 The Schmid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 The String Theory Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 14F Fryy Bosoniw Fiyld hhyory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Action and Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Normal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Conformal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.5 Primary Fields and Conformal Dimension . . . . . . . . . . . . . . . . . . . . . 252.6 Radial Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.7 Mode Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.8 Free Fermion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29G Boundury ginyAGordon aodyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Boundary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Auxiliary Boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Fermionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Boson momenta and fermion numbers . . . . . . . . . . . . . . . . . . . . . . . 413.6 Gluing relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 Fermion boundary state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.8 Evaluation of the Partition Function . . . . . . . . . . . . . . . . . . . . . . . . 493.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53iii4 aovility ut nity tympyrutury . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Non-zero temperature approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Yi-Kane Generalization to 2d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61I durtiwly aovility on u Bruvuis Luttiwy . . . . . . . . . . . . . . . . . . . . . . . 655.1 Reformulation in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Tight-binding duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Second order contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4 Third order contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.5 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Bivliogruphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76ivAwknowlydgymyntsI thank my supervisors. I would especially like to acknowledge Philip Stamp for sharing hisbroad interests and holistic view of the diverse domains of theoretical physics, and GordonSemenoff for inspiring me with his work ethic and enthusiasm for his science.Ian Affleck generously agreed to be my second reader, for which I am very grateful. Manythanks as well to the students and professors in the condensed matter theory group, especiallyLara Thompson and Dominic Marchand.Moral support for this work was provided by my friends and colleagues in the mezzaninehallway, and my loving family. I especially acknowledge Dave Wiens, that layman who neverstops asking interesting questions and reminds me of how important science really is.v1B IntroduwtionEver since Caldeira and Leggett proposed their model of dissipative quantum mechanics [1],attempts have been made to apply the formalism to simple situations and determine the effectsof dissipative forces in the quantum regime. A particularly important and general problem isthe question of the mobility of a particle in a periodic potential in the presence of friction.The application of the Caldeira-Leggett theory to the particle mobility problem at zerotemperature was first studied by Schmid [14]. His results suggested a sudden transition inthe dimensionless mobility from 1 (diffusive hopping) to 0 (localization) as the magnitude ofthe friction passes through a critical value. Subsequently, Fisher and Zwerger [7] generalizedSchmid’s model to non-zero temperature, obtaining general expressions for the mobility atarbitrary temperature. These results apply to the case of a one-dimensional periodic potential,or to hyper-cubic lattices where the behaviour in the different dimensions is decoupled.Following the discovery of a mathematical analogy between dissipative quantum mechanicsand open string theory [4], the problem was opened up to attacks and extensions from a stringtheory perspective. The similarity arises in the context of open strings with end-points tied tosome world-sheet boundary [3]. When the closed string modes in the bulk of the world-sheetare integrated away, the resulting effective action term for the open strings on the boundary isidentical to the non-local interaction found in dissipative quantum mechanics upon eliminationof the oscillator bath representing the environment.More recently, Yi and Kane attacked the two-dimensional particle mobility problem for theadditional cases of equilateral triangular and hexagonal lattices [15]. Their interpretation of1the renormalization group results of Schmid led them to propose an intermediate fixed point forthe non-square geometries that was unstable in the triangular case but stable in the hexagonalcase. They conclude that in a certain transition regime, the fixed point for the mobility in thehexagonal lattice varies continuously from 0 to 1.In chapters 2 and 3 of this work, we look at the string theoretical formulation of the Schmidmodel at the critical point in terms of a free bosonic field theory with interactions on a boundary.We demonstrate a new calculation of the partition function using fermionization. The solutionis given in a form such that the renormalized parameters may be more closely related to the bareparameters of a given condensed matter problem. While acknowledging certain weaknesses ofthe model’s applicability to real systems, there is some hope that attempts at analyzing similarsystems might benefit from the alternative approach outlined here.In chapters 4 and 5, we attempt to clarify the origins and consequences of the behaviourYi and Kane suggest. The analysis of Fisher and Zwerger is generalized to non-rectangulartwo-dimensional Bravais lattices, and traces of the behaviour predicted by Yi and Kane areinvestigated.Each of the two parts of this thesis contains an introductory chapter that provides a basicexplanation of the origins and machinery of the problem at hand. In the remainder of thischapter, we will introduce Caldeira-Leggett dissipative quantum mechanics and the Schmidmodel, and then outline the nature of the connection between the Schmid model and openstring theory.EBE cvyrviyw oz drovlymThe models studied here consider a particle of mass b moving in a periodic potential k (x), andsubject to some constant applied force F . Dissipation is introduced by Caldeira and Leggett’sprescription of coupling the particle to a bath of oscillators with a particular spectrum. Thebath variables are eliminated, leaving an effective action for the particle.2In the absence of dissipation, the effect of the periodic potential is to create Bloch bandsin the energy spectrum. The Hamiltonian eigenfunctions are also eigenfunctions of the crystalmomentum, and these states are extended. A particle cannot be restricted to any small region;it will quickly disperse to fill the entire volume.In the presence of dissipative forces, on the other hand, we might expect some restrictionof the particle’s mobility. Classically, “friction” impedes the free propagation of the parti-cle. Quantum mechanically, long-distance correlations in the wavefunction may be destroyed,leading to the localization of the particle.Classically, friction is added to a system by including a force that is proportional to theparticle velocity. The constant of proportionality is the friction coefficient , and the equationof motion for the system is thenb q¨ + q˙ + rk (q) = F : (1.1)The mobility indicates the ratio of the terminal velocity v to the applied force. In onedimension we have =vF(1.2)while in higher dimensions the more general mobility tensor ij describes the particle’s responsein the i direction to an applied force in the j direction:vi = ijFj : (1.3)The tensor may in general depend on F and v. For vanishingly small force we might only takethe constant part of ij ; this is the linear mobility.In the absence of the potential k , the classical mobility is found from the steady state3solution to (1.1):ij0 =1ij : (1.4)In quantum mechanics, where a particle’s position is not well defined, we approach themobility in a way that generalizes to the classical result in the high temperature limit. Usingthe expectation value of position as a function of time in response to an applied force, the linearmobility is thenij =Fjlimt→∞mi(t)−m i(0)t∣∣∣∣F=0(1.5)with the particle’s mean position m obtained from the reduced density matrix /(qP q ′; t), whereq and q′ represent spatial coordinates, which takes into account the effects of dissipation:X(t) = 〈q(t)〉 =∫yq /(qP q; t) q: (1.6)In the absence of an applied force, the mobility may be studied by analyzing the naturaltendency of a localized state to spread out in space with time. Such behaviour is encoded inthe two-point correlation function〈qi(t)qj(0)〉. An alternative definition of mobility is thento take the (suitably normalized) coefficient of the logarithmic dependence of〈xi(t)xj(0)〉 −〈xi(0)xj(0)〉. This quantity can be extracted from the two-point function’s Fourier transform:ij = lim!→0|!|∫yt zi!t〈xi(t)xj(0)〉: (1.7)As we will see, Schmid’s original analysis was made using this form of the mobility, whileFisher and Zwerger took the “terminal velocity” approach and used the definition (1.5).4EBF CulxyiruALyggytt Dissiputivy euuntum aywhuniwsIn Caldeira and Leggett’s model of dissipative quantum mechanics (DQM) [1, 2], a heat bath isrepresented as a set of oscillators, indexed by , with coordinates x, mass m, and frequencies!. The oscillators are coupled linearly to the particle, with coupling constants X.The action for the entire system is thenh[qP {x}] = h0[q] + hwath[{x}] + hint[qP {x}] (1.8)whereh0[q] =∫ t0y(12b q˙()2 − k (q)) (1.9)hwath[{x}] =∑∫ t0y(12mx˙2 − 12m!2x2)(1.10)hint[qP {x}] =∑∫ t0yXx · q (1.11)The density matrix / in position space, as a function of time t, is obtained by propagationof the initial configuration:/(qP q′; {x}P {x′}; t) =∫yf∫yf′∏∫ym∫ym ′ K(qP q′P {x}P {x′};fPf′P {m}P {m ′}; t)× /(fPf′; {m}P {m ′}; 0) (1.12)with the propagator K given byK =∫ qfDq˜∫ q′f′Dq˜′∫ xmDx˜∫ x′m′yx˜′ exp(i~h[q˜P {x˜}]− i~h[q˜′P {x˜′}])(1.13)5For our purposes it is reasonable to assume that the initial density matrix factors cleanlyinto some initial distribution for the particle and a bath distribution in thermal equilibrium attemperature i = 1Rk:/(qP q′P {x}P {x′}; 0) = /(qP q′)× /({x}P {x′}): (1.14)Since we are only concerned with the particle’s properties at time t, we can trace out thefinal state of the bath from the complete density matrix to leave the part relevant to particleexpectation values:/(qP q′; t) =∏∫yx /(qP q′; {x}P {x}; t): (1.15)Caldeira and Leggett showed that these integrations can be done for a general h0, and thatthis leaves a simplified expression for the particle density matrix that incorporates the baththrough an influence phase iΦ:/(qP q′; t) =∫yf∫yf′/(fPf′; 0)∫ qfDq˜∫ q′f′Dq˜′ z i~h0[equ− i~h0[eq′u+i[eq;eq′u (1.16)The influence phase Φ[qP q′] couples the “forward” and “backward” paths q and q ′,iΦ[qP q′] = −2i~∫ t0yt′∫ tt′ys y(s)I(s− t′)x(t′)− h2[y]P (1.17)with x and y the centre of mass and difference coordinatesx = 12(q + q′) y = q − q′P (1.18)6and h2 a temperature-dependent term coupling difference paths only,h2[y] =1~∫ t0yt′∫ t′0ys y(s)g(s− t′)y(t′): (1.19)The functions I and g determined by the oscillator spectrum and the temperature:I(s) =∫ ∞0y!J(!) sin!s (1.20)g(s) =∫ ∞0y!J(!) cos!s coth(12~!)(1.21)J(!) =∑Xm!2(! − !): (1.22)The most interesting behaviour results from taking an “ohmic” spectrum for the oscillators,where J(!) = !. In the high temperature limit, this choice of spectrum yields the classicalfrictional force −q˙. It is thus the spectrum most appropriate for systems where such frictionis observed at high temperatures.EBG hhy gwhmix aoxylSchmid studied the general problem of a particle in one dimension moving in a potentialk (q) = −k0 cos(2.qRv0): (1.23)In order to extract correlation functions from the system, he worked with a generatingfunctional in imaginary time:Z[F ] =∫Dq z−heff [qu−Ryt q(t)F (t) (1.24)where F (t) is a time-dependent source term and hzff [q] is obtained from (1.8) by tracing outthe bath modes with ohmic dissipation (this is a simplified version of the approach described7in the last section):hzff [q] =∫ ∞−∞yt(b2q˙(t)2 + k0 cos q(t))+4.∫ ∞−∞yt∫ ∞−∞yt′(q(t)− q(t′))2(t− t′)2 : (1.25)Both the mass term and the interesting non-local influence functional term can be Fouriertransformed (with the convention f˜(!) =∫yt zi!tf(t)) to givehzff [q] =12∫ ∞−∞y!2.(b!2 + |!|) q(!)q(−!) + k0 ∫ ∞−∞yt cos(q(t)): (1.26)With the friction term associated with a linear term in !, the mass term b!2 serves onlyas an high energy cut-off. In renormalization group language, the terms in ! are marginal whilethe !2 terms are irrelevant. We may then often treat b as zero in what follows, while pickingup a characteristic frequency =b(1.27)that acts as an effective ultraviolet cut-off for the low-energy theory.Dealing with the cosinusoidal potential in this framework is achieved by a “Coulomb gas ex-pansion.” This is essentially perturbative in k0. First the exponential of hk [q] = k0∫yt cos q(t)is expanded:z−hV [qu =∞∑n=0(−k0)nn!(12∑=±1∫ ∞−∞yt z2iq(t)Ra0)n: (1.28)Now the exponentials from the cosine interaction provide effective terms for the particle action:z−hV [qu =∞∑n=0(−k02)n ∑{i=±1}∫yt1∫yt2 : : :∫ytn exp(2.iv0n∑i=1iq(ti)): (1.29)8The exponent is rewritten as an integral over a charge density /(t′),2.iv0n∑i=1iq(ti) = − i~∫yt′ q(t′)/(t′) (1.30)where/(t′) = −2.~v0n∑i=0i(ti − t′): (1.31)Using this expansion we may write the generating functional (1.24) for the correlationfunctions asZ[F (t)] =∫Dq z− 12Rd!2eY−1(!)|q(!)|2×∑n∫yt1 : : : yt2n((−k0R2)nn!)2z−i~Ryt′(iF (t′)+/(t′))q(t′) (1.32)where (anticipating the imminent functional integration) the propagator isY˜(!) =(b!2 + |!|)−1 (1.33)and we have used that only neutral charge distributions (i.e. with i = (−1)i for i = 1P : : : P 2n)contribute finitely to the path integral. Note that for ! , the dominant term in Y˜(!) is1R!.The path integral is gaussian and givesZ =1√detY∑n∫yt1 : : : yt2n((−k0R2)nn!)2× exp(−12∫ys ys′ (iF (s) + /(s))Y(s− s′) (iF (s′) + /(s′))) : (1.34)9with Y(t) the inverse Fourier transform of Y˜(!),Y(t) =∫ ∞−∞y!2.z−i!tY˜(!) = −12|t| P |t| 1R− 1 ln |t| P |t| 1R: (1.35)From this we may pull down the correlation function〈q(t)q(0)〉 = 1ZF (t)F (0)Z∣∣∣∣F=0= Y(t)−∫ys∫ys′ Y(t− s) 〈/(s)/(s′)〉Y(s′) (1.36)with charge density correlation function given by〈/(t′)/(0)〉=1Z[0]1√detY∑n∫yt1 : : : yt2n((−k0R2)nn!)2/(t′)/(0)× exp(−12∫ys∫ys′ /(s)Y(s− s′)/(s′)): (1.37)We see from this expression that the “charges” in / interact via the essentially logarithmicpotential Y(t′). This analogy to electric charges moving in one dimension is the origin of theterm Coulomv gus yfipunsion. Using the expression (1.31) we can rewrite the exponential in(1.37) asexp(−12∫ys∫ys′ /(s)Y(s− s′)/(s′))= exp−12 ∑j;k(2.~v0)2jkY(tj − tk)= exp−12 2.~ ∑j;kjkY(tj − tk) : (1.38)where we have introduced the very important dimensionless dissipation parameter =v202.~: (1.39)10From (1.36) the mobility (1.7) is then = lim!→0|!| (Y(!)−Y(!)h(!)Y(!)) (1.40)=1(1− lim!→0Y(!)h(!))(1.41)where h(!) is the Fourier transform of the charge density correlation function (1.37).This perturbative expansion in powers of k0 should be valid for weak k0. Renormalizationgroup arguments [7, 14] show that in fact k0 flows to 0 provided that the dimensionless friction given in (1.39) is less than one. We then have the dimensionless mobilityR0 = 1 for Q 1: (1.42)The region where the periodic potential does not flow to zero is not accessible to theperturbation theory. The approach to this side of the “phase diagram” has been to insteadwork in the strong potential limit. In this sort of tight-binding limit, the particle lives mostlyin harmonic oscillator levels localized in the wells of the potential. The possibility of motion isprovided by tunneling through the barrier between minima; these are instantons [6].A single instanton tunnels from, for example x = 0 to x = v0 through a barrier of heightk0(1− cos(2.xRv0). In an inverted potential formulation, we obtain the action of the classicalpath associated with this (this is the WKB phase):s =∫ a00yx√2bk0(1− cos(2.xRv0)) = 4v0.√bk0: (1.43)The paths associated with these jumps have the formf(t) =2v0.tan−1 z!0t (1.44)with !0 =2a0√2k0Rb the effective harmonic oscillator frequency in the base of the wells. A11good approximation to a multi-instanton path is constructed as a sum of jumps of the type(1.44). For n jumps at times t1P : : : P tn in direction zj = ±1 we haveq(t′) =n∑j=1zjf(t− tj)P (1.45)which we may Fourier transform (using an integration by parts) to getq(!) =i!h(!)∑jzjzi!tj (1.46)where h(!) is the Fourier transform of yytf(t). Note that since the path (1.44) is a smoothed-outstep function, its derivative h(t) is a smoothed-out delta function. When integrated alongsidefunctions that vary on time scales longer than 1R!0, h(t) ≈ v0(t).The effective action is infinite unless the paths start and end at the same position, whichallows us to impose charge neutrality∑j zj = 0. For the n instantons described by (1.45) weget effective actionhzff = ns+12∫y!2.Y−1(!)q(!)q(−!) + i∫yt′F (t′)q(t′)= ns+ 12∑j;kzjzk∆(tj − tk) + i∫y!2.F˜ (−!)h(!))!∑jzjzi!tj (1.47)where ∆(t′) is the inverse Fourier transform of∆˜(!) =|h(!)|2|!| : (1.48)As mentioned above, on time scales shorter than 1R!0, the function h(t) acts as a deltafunction v0(t). In this regime we then have∆˜(!) ≈ v20|!| =2.~|!| (1.49)12and we see that the double sum term in the effective action (1.47) has the same form as thatin (1.38) with the replacement → 1R.Calculating the generating functional using the effective action (1.47) by integrating overtimes ti we obtain an expression very similar to (1.34), but with no quadratic term in the sourceF . The lack of quadratic term yields a mobility (calculated from the two point correlationfunction as in (1.36) that lacks the leading classical behaviour of (1.41):iB = lim!→0∆˜(!)Σ(!): (1.50)Just as ∆˜(!) and Y˜(!) are very similar on long time scales, so are the factor Σ(!) and h(!).This leads to an approximate duality between the mobility in the weak potential limit givenby (1.41) and the mobility iB in the tight-binding limit described above:()0= 1− iB(1R)0P (1.51)with defined in (1.39). The implication of this, given the renormalization argument thatR0 = 0 for Q 1, is thatR0 = 0 for S 1: (1.52)In (1.42) and (1.52) we see that there is a sudden transition from classical dissipativeparticle behaviour ( = 1R) to localized tight-binding behaviour ( = 0) as the friction passes through 1.A finite-temperature approach to this problem was undertaken by Fisher and Zwerger [7].Their approach involved the full evolution of the density matrix according to the Caldeira-Leggett prescription. An outline of their approach and a generalization of it to more complicatedlattices is investigated in chapters 4 and 5.13EB4 hhy gtring hhyory ConnywtionThe connection between Caldeira-Leggett dissipative quantum mechanics and string theorywas first pointed out by Callan and Thorlacius [4]. An essential technique in string theoryfor dealing with world-sheets of complicated topology is to cut up the world-sheet into various“fixtures” of simple topology (for example a disk or a cylinder). Calculations may be done oneach fixture, and the results sewn together in a prescribed fashion.When dealing with closed strings, which are closed loops, cutting the world-sheet intofixtures results in the severing of some of the strings. The severed closed strings become openstrings with endpoints that live on the boundary of the fixture. The behaviour of these openstrings is influenced by interactions in the bulk of the fixture, but these degrees of freedomcan be integrated away to leave a vounxury stuty. The boundary state is a functional of theboundary field degrees of freedom. Two adjacent fixtures are re-linked by taking the productof their boundary states; this is equivalent to a functional integral over their shared (boundary)degrees of freedom.Without developing too much formalism (some of which will be covered in chapters 2 and3), the boundary state for free bosons without interactions is|B〉frzz = exp(−∞∑m=11m−m˜−m)|0〉 (1.53)where m and ˜m are mode operators for the string fields in the bulk of the world-sheet and|0〉 is the famous ha(2P X) invariant vacuum, annihilated by the positive (m S 0) modes. Inthe presence of the gauge field the boundary state is modified to [3]|B〉 = exp( ∞∑m=11m−m˜−m)∫Dm(s) exp (−hrzg − h0 − hV − hls) |0〉 (1.54)where the path integral is over the field configurations on the boundary, with s ∈ [0P 2.]14parameterizing the boundary (which is assumed to be periodically identified). The variousaction terms h areh0[m] =18.′∫ 20ys2.∫ 20ys′2.(m(s)−m(s′))sin2 12(s− s′)(1.55a)hV[m] =i2∫ 20ys2.V(m(s))ym(s)ys(1.55b)hls[m] =∫ 20ys2.(s) ·mP where (s) =∞∑m=1i(˜−mz−ims + −mzims) (1.55c)hrzg[m] =∫ 20ys2.12b(m˙(s))2: (1.55d)The non-local term h0 is very similar to the dissipative term in (1.25), and in fact can beobtained from this other by enforcing 2.-periodicity of the paths q(t′) to rewrite the secondintegral. Alternatively, we may take the limit of the h0 term as the boundary length goes toinfinity, and recover the (s− s′)2 denominator seen in (1.25).The term hV is the topological term arising from the gauge field in the bulk. We may freelytake this to be zero now that we have used it to extract the other terms. (The inclusion of thisterm in the problem described in chapters 2 and 3 has been discussed in [10]).The linear source term hls we choose to ignore, since its function is to share the boundaryfields back into the bulk modes, and may be treated as a c-number and shifted away. The massterm hrzg is necessary to regulate the theory, which is the role of the identical mass term inthe DQM model once friction has been introduced.The conclusion we draw from this analogy is that calculations done with the DQM action(1.25) may be relevant to calculations in string theory for models of the type (1.54). Conversely,to find the boundary state in string theory (perhaps by some other means than the formula(1.54)) is also to perform the path integral in (1.54). A calculation based on the string theoryside can thus provide insight into condensed matter problems.To construct the Schmid model, we can consider the free bosonic field theory of the closed15strings in the bulk with a periodic interaction on the boundary. By finding the boundarystate for the theory, we are evaluating the path integral expression (1.54) with the periodicinteraction included. We end up working with the theory defined by actionh =14.∫ ∞−∞y∫ 0y((Um)2 + (Um)2)− ∫ ∞−∞y(g2zim(;0) +g¯2z−im(;0)): (1.56)This is the boundary sine-Gordon model, and is discussed in chapter 3.Since string theory and thus the boundary state formalism is developed in the context of con-formal field theory, any condensed matter problem being studied by these methods will inheritsymmetries from the string model. In condensed matter systems, extreme reparametrizationinvariance tends to arise at the critical points of phase transitions (between localized and delo-calized phases for example), so it is these critical theories that might be approached using thestring theory analogy.16FB Fryy Bosoniw Fiyld hhyoryConformal field theories arise in a variety of contexts, most famously in string theory and in thestatistical mechanics of condensed matter systems. The basic feature of conformal theories is atotal insensitivity to the parametrization of the underlying space (and thus is often associatedwith scale invariance). This feature leads to tight constraints on correlation functions andlimits the spectrum of possible theories (by dictating the number of space-time dimensions instring theory, for example).The theory of a free boson in 2 dimensions is one of the simplest conformal field theories,and is the basis for many others. In this chapter we review the important features of freebosonic field theory that will be applied in order to find the partition function of the boundarysine-Gordon model. At the end of the chapter we also touch on the subject of free fermion fieldtheory. The material here is mainly drawn from [12] and [8].FBE Awtion unx Yquutions oz aotionLet us consider fields m(P ) which live on the world-sheet parametrized by the spatial co-ordinate and the euclidean time . In string theory, the fields m represent coordinates inspace-time, so the field represents a time-dependent embedding of a one-dimensional entity inspace-time, or string.Our world sheet will live on a strip in the P plane: ∈ (−∞P∞)P ∈ [0P .]: (2.1)17Greek indices will be used to index the fields, and these will always contract with the Euclideanmetric, g, = , . Roman indices, as in Ua, refer to the coordinates (P ) which also contractwith a Euclidean metric. The action that interests us (for now) is the one that minimizes thearea of the string:h =14.′∫ ∞−∞y∫ 0yUamUam: (2.2)We may obtain the classical equations of motion by varying the fields and integrating byparts:h =14.′∫y2 (−(2UaUam)m + Ua(Uamm)) (2.3)This gives an equation of motionUaUam = 0P (2.4)and we must impose boundary conditions to ensure that the total derivative term is zero aswell. This is usually done by either forcing the derivative term normal to the boundaries to bezero (the Neumann boundary condition), or by preventing the variation of m at the boundariesby fixing it to some specific coordinate (the Dirichlet boundary condition).Since we have boundaries at = 0 and = . the Neumann condition reduces toUm(0P ) = 0Um(.P ) = 0: (2.5)For the Dirichlet condition, fixing the endpoints of the string is equivalent to asking that the18derivative of the field ulong the boundary be zero. The Dirichlet condition is thenUm(0P ) = 0Um(.P ) = 0: (2.6)Note that these are the most common boundary conditions but not the only possibilities. Whenwe later add boundary interactions to the action, our boundary conditions will become morecomplicated.Expectation values are defined by a functional integral of the fields, weighted by the expo-nential of the action. The expectation value of the operator F is〈F〉 =∫Dmzh[muF [m] (2.7)where the functional integral is over all field configurations that satisfy the specified boundaryconditions.It will be advantageous to work in an alternative coordinate basis defined byw = + i w = − i (2.8)(it is quite standard to use the complex variable z here, but we wish to reserve that symbolfor the radial quantization coordinate transformation). This is simply a linear coordinatetransformation, and w and w should be treated as independent variables. Since and arereal, however, it will always be true that w = w∗. We will commonly abbreviate ww = |w|2.Our fields m(wPw) are functions of the two new independent variables. All of our previous19expressions can be rewritten by applyingUw = U − iU (2.9)Uw = U + iUP (2.10)which gives us an actionh =12.∫yw yw UwmUwm (2.11)and equations of motionUwUwm = 0: (2.12)The equation of motion tells us that the fields m are harmonic functionsm(wPw) = ma(w) +mg(w) (2.13)with the “left-moving” field ma holomorphic (Uwma(w) = 0), and the “right-moving” mg(w)antiholomorphic (Uwmg = 0). Since a derivative with respect to w (or w) makes m holomorphic(or, respectively, antiholomorphic), we can omit the dependence on w (or w), and write simplyUwm(w) (or Uwm(w)).In quantum field theory, the equation of motion holds only as an opyrutor yquution, meaningthat expectation values involving UwUwm are zero as long as there are no other field operatorsnear w or w. In a path integral formulation, this is easily obtained using the properties offunctional derivatives:20UwUw 〈m(wPw) : : :〉 =∫Dm z−h[muUwUwm(wPw) : : :=∫Dm . z−h[mum(wPw): : := 0 (2.14)The final line in this equation only follows if : : : contains no field operators at wPw; in thiscase the functional derivative is a total derivative and its integral is 0.The complications that arise when field operators coincide are dealt with in the next section.FBF bormul crxyringWe now consider the two-point function, or propagator, 〈m(wPw)m,(0P 0)〉. Now allowingwPw to approach 0, we find the “naive” equation of motion is adjusted as follows:UwUw 〈m(wPw)m,(0P 0)〉 =∫Dm z−h[muUwUwm(wPw)m,(0P 0) : : :=∫Dm . z−h[mum(wPw)m,(0P 0)= .∫Dm((z−h[mum,(0P 0))m(wPw)− z−h[mu m,(0P 0)m(wPw))= −.,(2)(wPw) (2.15)The propagator is thus the green function for the operator UwUw (which is just the twodimensional Laplacian if we return to P coordinates):〈m(wPw)m,(0P 0)〉 = − 12, ln |w|2 (2.16)A normal ordered operator is one that, in each of its constituent fields, satisfies the equation21of motion. An example is the normal ordered two point function::m(wPw)m,(0P 0) : = m(wPw)m,(0P 0) + 12, ln |w|2 (2.17)Then the expectation value of :m(wPw)m,(0P 0) : satisfiesUwUw 〈:m(wPw)mnu(0P 0) :〉 = 0 (2.18)for all w, since the double derivative of the ln |w|2 term will exactly cancel the delta functionfrom (2.15). The normal ordered operator is well-defined even if its constituent fields arecoincident; :m(wPw)m,(wPw) : is a meaningful operator.An unordered operator differs from its normal ordered form by the sum of all possiblecontractions of its constituent fields. For the two-point function, this means:m(wPw)m,(0P 0) : = m(wPw)m,(0P 0) − 〈m(wPw)m,(0P 0)〉(2.19)Informally, the normal ordered operator is the sum of all possible ways of pairing fields inthe unordered operator and replacing pair with the corresponding propagator. We may expressthis more formally: let F({wiP wi}) be an unordered product of local field operators at wiP wi,for i = 1P : : : P n. Then let:F : = exp(−∫y2w∫y2w′〈m(wPw)m,(w′P w′)〉 m(wPw)m,(w′P w′))F= exp(∫y2w∫y2w′ ln |w − w′|2 m(wPw)m(w′P w′))F :(2.20)22(Here the integration measures are abbreviated, e.g. y2w = yw yw.) This will guarantee thatUwiUwi 〈:F :〉 = 0 for each of F ’s coordinates wi.FBG cpyrutor droxuwt YxpunsionWe have already seen in (2.15) that the product of two local operators may become ill-definedas the coordinates approach each other on the world-sheet. This can be dealt with by usingthe operator product expansion (O.P.E.) which approximates the product of nearby operatorsas a sum of local operators. For a set of operators {F}, for each pair Fi and Fj there is aneighbourhood of w about w0 such thatFi(wPw)Fj(w0P w0) =∑kxkij(w − w0P w − w0)Fk(w0P w0) (2.21)where the coefficient functions xij(wPw) are holomorphic in w and antiholomorphic in w in thisneighbourhood of 0 except possibly at 0.O.P.E.s are used to move the singularity of the operator product into the singularity of theotherwise holomorphic/antiholomorphic coefficient functions. For that reason it is usually onlythe singular terms of the O.P.E. that are of interest. Normal ordering can be used to identifythe singular terms (since the normal ordered form of the operator satisfies the naive equationsof motion, only the contractions of the operator will be present as non-singular terms). Forexample, consider the product of two field operators at neighbouring points (wPw) and (0P 0).m(wPw)m,(0P 0) = :m(wPw)m,(0P 0) : − 12, ln |w|2 (2.22)The normal-ordered product satisfies (2.16), and thus is well behaved at 0 and can be expanded23in a Taylor series.m(w)m,(0) = −12, ln |w − w′|2+ :m(0)m,(0) :+∞∑k=11k!(:Ukwm(0)m,(0) : wk+ :Ukwm(0)m,(0) : wk)∼ −12, ln |w − w′|2 (2.23)The first line above is the complete O.P.E., with all fields evaluated at 0. The second lineintroduces the equivalence ∼, which indicates that the expressions on either side are equal upto non-singular terms (so that we may drop all “well behaved” terms). Note that in the powerseries expansion the mixed derivative terms are zero by the equation of motion (which appliesto the normal ordered product).FB4 Conzormul hrunszormutionsOur action should be invariant under world-sheet reparametrization. Going from coordinatesw to w′ = f(w) for some analytic f and requiring that the new fields satisfym ′(w′P w′) = m(wPw) (2.24)means that the action naively written in the new coordinatesh′ =∫yw′ yw′ Uw′m ′(w′)Uw′m′(w′) =∫yw yw Uw′m(w)Uwm(w) = h (2.25)is equal to the original action. Thus any change of world-sheet coordinates does not changethe basic theory; this is conformal invariance. In boundary conformal field theory, the world-sheet boundaries will move with the transformation but the boundary operators appearing inthe action and thus the derived boundary conditions must retain their form under conformal24transformations. The set of boundary operators that is conformally invariant is not the sameas the set of operators that are conformally invariant in the bulk of the space.The world-sheet energy-momentum tensor is a set of Noether currents that arise fromvariation of world-sheet coordinates. Operating locally, it generates infinitesimal coordinatechanges. In the wPw coordinates it can be shown that the energy-momentum tensor has onlytwo independent components. The first,i (w) = − :UwmUwm :P (2.26)generates transformations in w, whilei (w) = − :UwmUwm :P (2.27)generates transformations in w. In the next section we discuss how these tensors may be usedto determine the conformal transformation properties of field operators.FB5 drimury Fiylxs unx Conzormul DimynsionA primary field A(wPw) is a field that, under a conform transformation w ′ = f(w) transformsaccording toA′(w′P w′) =(Uw′Uw)−h(Uw′Uw)−ehA(wPw)P (2.28)where (hP h˜) are called the conformal weights of the primary field A.From (2.24), for example, we know thatm is a primary field with weights (0,0). The energy-momentum tensor is useful for identifying and determining the weights of primary fields. It canbe shown that the O.P.E. of the energy-momentum tensor acting on a primary field satisfying25(2.28) isi (w)A(0P 0) ∼ hw2A(0P 0) + 1wUwA(0P 0) (2.29)Using this we may confirm, for example, that Uwm(w) is a primary field of weights (1,0).We accomplish this by finding the O.P.E. of i (w)Um(w0), by expanding the normal orderingof i (w) and then re-normal ordering the resulting operator products (this is equivalent tocross-contracting the fields in i (w) with the Um(w0) but it is instructive to work it out veryexplicitly):i (w)Um(wo) = − :Um,(w)Um,(w) : Um(wo)= − limw′→wUwUw′(m,(w′P w′)m,(wPw) + 12,, ln |w′ −w|2)Uwom(wo)= − limw′→w(12,,1(w − w′)2 Um(wo)+ UwUw′Uwo[:m,(w′)m,(w)m(w0) : − 12,, ln |w′ − w|2m(wo)−12, ln |w − wo|2m,(w′)− 12, ln |wo − w′|2m,(w)])∼ 1(w − wo)2 Um(w) (2.30)We want local operators at wo, so we Taylor expand m(w) = m(wo)+(w−wo)Um(wo)+ : : :to geti (w)Um(wo) ∼ 1(w − wo)2m(wo) +1w − wo Um(wo); (2.31)from which we conclude that Um is a primary field of weights (1P 0) (since Um has no wdependence, it trivially has weight h˜ = 0).26It is also important for our purposes to know the weights of an exponential of the fields:i (w) :zkm(wo) : = − :Um,(w)Um,(w) ::zikm(wo) := − :Um,(w)Um,(w)zikm(wo;w0) :− (−12 ikUw ln |w − wo|2)(−12 ikUw ln |w − wo|) :zikm(wo) :− 2Um(w)(−12 ikUw ln |w −wo|) :zikm(wo) :∼( |k|2R4(w − wo)2 +1w − woUwo):zikm(wo) : : (2.32)The O.P.E. has terms from contracting both or just one of the fields in i (w) with the ex-ponential. Acting with i (w) gives a similar result, and we find that : zikm: has weights(kkR4P kkR4).FB6 fuxiul euuntizutionIt is often advantageous to perform the coordinate transformationw→ z = zw w → z¯ = zw: (2.33)Referring to our original coordinates, circles about the origin in z coordinates are lines ofconstant ; = −∞ maps to z = 0, =∞ maps to z = ∞.In radial quantization it is very advantageous to make use of a connection between operatorsand states. Consider an operator very close to z = z¯ = 0. This position corresponds to = −∞,and thus in the original P coordinates it establishes an “ingoing” state for our subsequentcalculations. A path integral on the z world-sheet with the operator at the origin correspondsto a path integral on the strip (w world-sheet) with some initial state defined at = −∞. Thisrelationship will be explored further in the next chapter.27FBK aoxy YxpunsionsSince Um(z) satisfies U¯(Um(z)) = 0 everywhere except perhaps at z = 0, it has a Laurentexpansion about 0. This is customarily expanded asUm(z) = −i√12∞∑m=−∞kz−k−1 (2.34a)where the k are operators. (Note that in the original P coordinates, this is just a Fourierdecomposition; so it should not surprise us if the k turn out to act as harmonic raising andlowering operators.) Similarly,Um(z¯) = −i√12∞∑k=−∞˜kz¯−k−1: (2.34b)These can be integrated to give the general mode expansion for m(zP z¯) = ma(z) +mg(z¯):ma(z) =1√2(xa − ipa ln z + i∑n=0nnz−n)(2.35a)mg(z¯) =1√2(xg − ipg ln z¯ + i∑n=0˜nnz¯−n)(2.35b)The 0 and ˜0 mode operators have been reinterpreted as momenta. It can be shown, forexample by careful commutation of contour integrals that extract the various modes from thefield m, that the operators xaP paP a satisfy commutation relations[xaP pa] = i and [nP −n] = nP (2.36)with all other commutators zero. The analogous relations hold for the right-moving modes:[xgP pg] = i and [˜nP ˜−n] = nP (2.37)28and the left- and right-moving modes commute with each other.The vacuum states consist of eigenstates |kaP kg〉 of the momentum operatorspa |kaP kg〉 = ka |kaP kg〉 pg |kaP kg〉 = kg |kaP kg〉 (2.38)are annihilated by positive mode oscillators;n |kaP kg〉 = ˜n |kaP kg〉 = 0 for n S 0 (2.39)FBL Fryy Fyrmion hhyoryThe discussion of fermions will not be as deep as the discussion of bosons. Free fermionicfield theory has many similarities with the bosonic development, and several more complicatedaspects.A free fermion (wPw) = a(w) + g(w) has actionh =i2.∫ ∞−∞y∫ 20y( †a(w)Uw a(w) + †g(w)Uw g(w)): (2.40)where w = + i is on the Euclidean cylinder.Since only fermion bilinears must be periodic under → + 2., there are two sectors offermions, depending on whether they are periodic (Ramond) or antiperiodic (Neveu-Schwarz)in their spatial coordinate. Explicitly, → + 2. =⇒ ( aP g) → (− aP− g) “Neveu-Schwarz” (NS)( aP g) → ( aP g) “Ramond” (R) (2.41)with all fields evaluated at P The resulting mode expansions, in radial quantization z = zw,29z¯ = zw, are a(z) =∑n nz−n †a(z) =∑n †nz−n (2.42) g(z¯) =∑n ˜nz¯−n †g(z¯) =∑n ˜†nz¯−n (2.43)where the index n ranges over integers or half-odd integers depending on the sector:NS sector n− 12 ∈ ZR sector n ∈ Z:(2.44)We will refer to the modes nP †n as left-moving, and the ˜nP ˜†n modes as right-moving. Wewill frequently use the index r to sum over half-odd integers in the NS sector.The modes have anticommutation relations{ nP †−n}= 1{ ˜nP ˜†−n}= 1P (2.45)and all other inter-mode anticommutators are 0.The vacuum state is different in the Ramond and Neveu-Schwarz sectors. In both cases, allpositive mode (n S 0) oscillators annihilate the vacuum and negative mode operators (n Q 0)create excited states. In the Ramond sector, the n = 0 modes complicate the notion of vacuumsomewhat; we define our Ramond sector vacuum |0〉g such that ˜†0 |0〉g = 0 |0〉g = 0 (2.46)In the Neveu-Schwarz sector the Hamiltonian isHch =∞∑r=12(r −r r + r ˜−r ˜r)− 112: (2.47)30The Hamiltonian for the Ramond sector isHg =∞∑n=1(n −n n + n ˜−n ˜n)− 16: (2.48)We will ultimately be fermionizing a bosonic system, and most of our difficult work willinvolve the fermion operators.31GB Boundury ginyAGordon aodylIn this chapter we will calculate the partition function of the boundary sine-Gordon modelat the self-dual radius. The theory is free in the bulk, while the periodic interaction at theboundaries modifies the boundary conditions. The results of this section, along with severalextensions of the model, have been published in [10]. The approach was previously applied tothe rolling tachyon problem [11]We will first develop basic ideas about calculating partition functions using boundary states.In a path integral expression for the partition function, the field configurations are restrictedto be periodic in the euclidean time; at the same time we have to impose our boundary con-ditions at spatial extremes of the world-sheet. To re-enter familiar string theory territory, weinterchange the spatial and time coordinates so that the spatial component is periodic whilethe boundary conditions are recast as incoming and outgoing states.The success of this approach comes from fermionizing the system. In free field theory,fermions are obtained as exponentials of bosonic fields, with the exponent containing a factorof iR√2 to ensure the correct conformal weights for the fermions. Since the bosonic boundaryinteractions are constrained to have the right conformal scaling dimensions, the exponentialsappearing in the action appear without the factor 1R√2. The disparity is remedied by intro-ducing an additional boson field n with trivial boundary conditions, and forming new bosonfields (m ± n )R√2. The new fields can be fermionized, and the boundary interaction is stillmarginal.Upon fermionizing, the momenta of the bosons are related to fermion number. Since fermion32number is discrete, so must the boson momenta be. This leads to a requirement that the targetspace of the boson be compactified at a certain radius. Once this has been ensured, the partitionfunctions may be calculated in the fermion system.GBE hhy aoxylThe boundary sine-Gordon model is the model of a free bosonic field m on a 2d space, subjectto a periodic interaction at the boundary of the space. As in section (2.1), we let world-sheetEuclidean time parameter run over all real values, while the spatial coordinate is restrictedto the interval [0P .]. The sinusoidal interaction is present at the = 0 boundary, and weimpose either Neumann or Dirichlet conditions at the = . boundary. The action is thenh =14.∫ ∞−∞y∫ 0yUamUam −∫ ∞−∞y(g2zim(;0) +g¯2z−im(;0)): (3.1)One would typically take g = g¯ to make the Hamiltonian hermitian. We need not insist onthis for now, however. The equations of motion are as in (2.4). However, the derivative termin (2.3) that leads to the Neumann boundary condition equation is modified toUm(P 0) + ig2zim(;0) − i g2z−im(;0) = 0: (3.2)This is the new boundary condition at = 0.GBF Bounxury gtutysThe thermal partition function can be obtained in a path integral formalism fromZ =∫Dmz−H (3.3)33where the path integral is over configurations of m(P ) that are periodic in the direction(remember is already the Euclidean time) with period = 1Ri . In this problem we mustalso enforce the boundary conditions at = 0 and = ..In string theory we are much more accustomed to being the periodically identified coor-dinate, while is subject to notions of “in-coming” and “out-going” states. For this reasonwe redefine our coordinates, using conformal invariance. In particular we want to make the-periodic into a 2.-periodic ′:′ =2. ′ = − 2.Now ′ is 2.-periodic and our boundaries are at ′ = 0 and −2.2R. So ( ′P ′) parametrize afinite length of Euclidean cylinder.In the Boltzmann factor (3.3), the conformal transformation to ′P ′ will preserve the formof the double integral over Hamiltonian density H, but the integration limits are adjusted:H =∫ 0y∫ 0y H =∫ 20y′∫ 22R0y ′ H = 2.2∫ 20y′H = 2.2H ′: (3.4)So our partition function (3.3) becomesZ =∫D′m exp(−H) (3.5)where = 2.2R and the prime on the integration measure indicating that the paths arerestricted to satisfy the boundary conditions at ′ = 0 and . We may let m roam freely overall paths, provided that we enforce the boundary conditions some other way. Let Ψ1[m] be afunctional of the path that enforces the boundary condition at ′ = 0, and Ψ2[m] do the same34at ′ = . ThenZ =∫DmΨ2[m] exp(−H)Ψ1[m]:= 〈B1| exp(−H) |B2〉 (3.6)where |Bi〉 are the vounxury stutys corresponding to functionals Ψi[m] that enforce the bound-ary conditions [12]. To do this they must satisfyBi |Bi〉 = 0 (3.7)where Bi are the operators obtained from the left hand side of the boundary condition equation(3.2).We also need the form of the new Hamiltonian H ′. It is obtained in the canonical fash-ion from the free action in the new coordinates. The action preserves its form (other thanintegration limits) under the conformal transformation, so the new Lagrangian isa′ =14.∫ 20y′UamUam(3.8)and the new Hamiltonian is thenH ′ =ULU(U ′m)− a=14.∫ 20y′((U ′m)2 − (U′m)2)=12.∫ 20y′((zUzm(z))2 + (z¯Uzm(z¯))2)(3.9)where now the derivatives are with respect to z = z′+i′ , z¯ = z′−i′ . The fields Uzm(z) and35Uzm(z¯) have mode expansions given in (2.34a) and (2.34b), and we may may integrate themvery easily by realizing that ′ ∈ [0P 2.] is a contour integral in the complex plane. Then withy′ = yzRiz = yz¯Riz¯ we findH ′ =∮yz2.iz(Uzm(z))2 +∮yz¯2.iz¯(Uzm(z¯))2= 12p2a +12p2g +∞∑n=1(−nn + ˜−n˜n)− 112 : (3.10)(Here the 1R12 comes from the commutation of n past −n which results in a∑∞n=1 n. Therequirement of modular invariance demonstrates that this sum is best interpreted as (−1) =−1R12, with the Riemann zeta function.)GBG Auxiliury BosonIn order to facilitate fermionization, we must introduce an additional bosonic field n to thesystem [9, 13]. Our free action, on the cylinder in ′P ′ space is thenho =14.∫ ∞−∞y ′∫ 20y′ (UamUam + Uan Uan ) : (3.11)where we may extend the ′ domain to ±∞ since our boundary conditions will be enforced bythe boundary states. To make n easy to deal with, we give it Dirichlet boundary conditions atboth boundaries.Our real purpose is to change the scale of interaction term, however. We perform a unitaryrotation on the fields mPn to get1 =m + n√22 =m − n√2: (3.12)36The free part of the action (3.11) becomesho =14.∫ ∞−∞y ′∫ 20y′ (Ua1Ua1 + Ua2Ua2) : (3.13)Our mode expansion for the new fields i(zP z¯) = ia(z) + ig(z¯) has the same form as (2.35),explicitlyia(zP z¯) =1√2<ia − i.ia ln z +∑n6=01nnz−n (3.14)and similarly for ig.We will also write the consequences that simple boundary conditions on m and n have fori. For the case of the boundary state |cPY〉 where m has Neumann boundary conditions andn has Dirichlet boundary conditions,(ma −mg)|cPY〉 = 0(na + ng)|cPY〉 = 0 =⇒ (1a − 2g)|cPY〉 = 0(2a − 1g)|cPY〉 = 0 (3.15)with all fields evaluated at = 0, arbitrary .Similarly the boundary state |YPY〉, for the case ofm and n both having Dirichlet boundaryconditions at = 0, satisfies(ma −mg)|YPY〉 = 0(na − ng)|YPY〉 = 0 =⇒ (1a − 1g)|YPY〉 = 0(2a − 2g)|YPY〉 = 0: (3.16)GB4 FyrmionizutionFermionization is a process by which the bosonic fields of a system are combined into operatorsthat have fermionic commutation relations. If the inherited properties of the derived fermionscan be shown to follow from some fermionic action, then expressions obtained in the fermionic37theory (in particular the form of the partition function) are valid in the bosonic context aswell. To perform our computation in the fermion system, we will need the Hamiltonian, theboundary conditions, and thus the boundary state must be obtained in terms of the fermionicoperators.For the boson fields i, the fermionic field operators are [8, 11] 1a(z) = 1a :z−√2i1a : †1a(z) = :z√2i1a : †1a 2a(z) = 2a :z√2i2a : †2a(z) = :z−√2i2a : †2a 1g(z¯) = 1g :z√2i1R : †1g(z¯) = :z−√2i1R : †1g 2g(z¯) = 2g :z−√2i2R : †2g(z¯) = :z√2i2R : †2g:(3.17)The operators iH (where H is the handedness, a or g) are cocyclys, and are necessary in thetwo boson case to ensure that the 1 and 2 fields anticommute (without it, the two kindsof fermions would commute instead). It is enough to use bosonic momentum operators in thecocycles [11]:1a = 1g = exp−i2 (.1a + .1g + 2.2a + 2.2g)2a = 2g = exp−i2 (.2a + .2g) (3.18)In resolving the anticommutator of, for example, 1a and 1g, factors of z±iR2 are obtainedwhen commuting the cocycle from one operator through the exponential factor of the other.The exponential in the boundary condition (3.2), acting on the boundary state |BPY〉 can38then rewritten in the following way:g2zim |BPY〉 = g2zi(m+n )|BPY〉=g2zi√21 |BPY〉=g2zi√21azi√21R |BPY〉=g′2:zi√21a ::zi√21R : |BPY〉=g′2 †1a 1g|BPY〉 (3.19)with all fields evaluated at the boundary.In the first line of this equation, the introduction of n is allowed since m and n areindependent and zin |BPY〉 = |BPY〉 from n ’s Dirichlet boundary condition. In the third line,the left- and right-moving parts of 1 are independent and the exponential can be factored.The normal ordering in the fourth line, however, introduces an infinite constant that mustbe absorbed into the coupling constant. This is a standard coupling renormalization. We willcontinue using the symbols g and g¯ in the fermion theory, but we should remember that theseare renormalized versions of the original couplings. In a similar fashion we obtaing¯2z−im |BPY〉 = g¯′2 †2a 2g|BPY〉 (3.20)The operator product in bosonic operators can be used to show that, for a bosonic holo-morphic field m(z), [8]:zim(z)z−im(z) : = iUzm(z): (3.21)So our bosonic field derivatives may be expressed as fermion bilinears. In terms of the original39m and n fields we then have: †1a 1a(w) : =√2iUw1a(w) (3.22a): †2a 2a(w) : = −√2iUw2a(w) (3.22b): †1g 1g(w) : = −√2iUw1g(w) (3.22c): †2g 2g(w) : =√2iUw2g(w) (3.22d)The derivative operator in the boundary condition (3.2) is expressed as a fermion bilinearusing (3.22):Um = (Uw + Uw)1√2(1 + 2)= 1√2Uw (1a + 2a) +1√2Uw (1g + 2g)= 12i(: †1a 1a : − : †2a 2a : − : †1g 1g : + : †2g 2g :)(3.23)Introducing the vector notation a = 1a 2a †a = ( †1a †2a)(3.24)and similarly for g and †g, along with the Pauli matrices i we can write the completeboundary equation in fermion variables as(: †a3 a : − : †g3 g : + .g †a(1 + 3) g − .g¯ †a(1− 3) g)|BPY〉 = 0: (3.25)40In a similar fashion, the analogue of the Dirichlet boundary condition (2.6) for n is(: †a a : − : †g g :)|BPY〉 = 0 (3.26)GB5 Boson momyntu unx zyrmion numvyrsThe boson momenta .ia and .ig can be obtained from the spatial integral of the derivativeof the boson fields i; this follows from the mode expansion (2.34a). In radial coordinates wehave.ia =∮yz2.i(i√2Uzia)= −∮yz2.i(−1)i : †ia ia :=∑r=1( †i;−r i;r − i;−r †i;r)Neveu Schwarz∑n=1( †i;−n i;n − i;−n †i;n)+ †i;0 i;0 − 12 Ramond(3.27a)and similarly.ig =−∑r=1 ( †i;−r ˜i;r − i;−r ˜†i;r) Neveu Schwarz−∑n=1 ( †i;−n ˜i;n − i;−n ˜†i;n)− ˜†i;0 ˜i;0 + 12 Ramond. (3.27b)Since the fermion numbers are integers (in the Neveu Schwarz sector) or half-odd-integers(in the Ramond sector), this limits the spectrum of the momenta in the bosonic theory. The wayto handle this is to compactify the boson theory, which identifies target field values separatedby 2.g where g is the compucticution ruxius.From the bosonic mode expansions (2.35a) and (2.35b), the parts of m(P ) linear in and are−i√2((pm;a + pm;g) + (pm;a − pm;g)i) (3.28)41which leads to the definition of total momentum p and wrapping number w:p = pa + pg w = pa − pg (3.29)Since taking → + 2. must map m to an equivalent m (in the sense that m ∼ m + 2.), thewrapping number for m must satisfy1√2w = mg =⇒ w =√2gm (3.30)with m ∈ Z. The total momentum, is quantized according to2.g 1√2p = 2.n =⇒ p =√2gn (3.31)with n ∈ Z. At g = 1, √2pm;a and√2pm;g are integers and either both are even or both areodd. The same must apply to n , and the implication for the bosons is that 21;aP 21;gP 22;aPand 22;g all must be integers with the same parity. The significance of this is that our partitionfunction expression (which originated with a trace over all possible states) must respect theserelationships between momenta, and thus between fermion number as described in (3.27). Thismeans that our partition function must be the sum of fermionic expressions for the Ramondand Neveu-Schwarz sectors:Z = Zch + Zg: (3.32)By compactifying at g = 1, we are guaranteed that the half-integral spectrum of thefermion number operators spans the same values as the boson momenta; and by separatingthe partition function into the sum over the two fermion sectors we are enforcing the relationshipbetween those boson momenta.42GB6 Gluing rylutionsHaving established that the boson momenta are integers, we are set to establish another im-portant set of relationships for the fermion fields acting on boundary states. Using the “gluingrelations” (3.15) for the boson fields at the boundary, we obtain gluing relations for the fermionoperators. For example, acting on the |cPY〉 state with the vector †a(0P ) gives 1a 2a |cPY〉 =1a :z−√2i1a :2a :z√2i2a : |cPY〉 = 1a :z√2i2R :2a :z−√2i1R : |cPY〉 (3.33)with all fields in this expression evaluated at ( = 0P ). Since the boson momenta have beenrestricted to integers, we have †1a2g|BPY〉 = |BPY〉 and thus 1a 2a |cPY〉 = 1a :z√2i2R : †1a2g2a :z−√2i1R : †2a1g |cPY〉 = −i2g :z√2i2R :−i1g :z−√2i1R : |cPY〉= −i 2g 1g |cPY〉 (3.34)A similar computation can be performed for †a and †g. The result is the two gluingrelations for the vectors (3.24):( †g(0P ) + †a(0P ) i1)|cPY〉 = 0 ( g(0P ) + i1 a(0P ))|cPY〉 = 0P (3.35)where 1 = ( 0 11 0 ) is the Pauli matrix acting left on the row vector †a or right on the column a.If instead we impose Dirichlet boundary conditions on m, this leads to the much simpler43relations( †g + †a)|YPY〉 = 0 ( g − a)|YPY〉 = 0: (3.36)GBK Fyrmion vounxury stutyWe have found the expression for the partition function as a matrix element (3.6) of boundarystates, and we have developed an equivalent fermionic theory with Hamiltonian (2.47) or (2.48)and boundary conditions (3.25). We will now establish the form of the boundary states infermion variables. Then the computation of the partition function will be possible.The action for the pair of fermions 1P 2 can be written exactly as (2.40), though now thefield operators, such as a, represent the vectors defined in (3.24):h =i2.∫ ∞−∞y∫ 20y( †a(w)Uw a(w) + †g(w)Uw g(w)): (3.37)This action is invariant under the unitary transformation a → j a and †a → †aj−1 (3.38)and similarly (and independently) for g. These symmetries have corresponding currents [8]Jaa =12 : †aa a : (3.39)where a are Pauli matrices acting on the vectors a and †a. In fact[ aP Jaa] =12a a [JaaP †a] =12 †aa: (3.40)For a vector a of angles, the currents Jaa generate finite SU(2) transformations according44toz−iaJaa az+iaJaa = j a (3.41a)z−iaJaa †az+iaJaa = †aj−1 (3.41b)where j = ziaaR2.Note that the gluing relations (3.35) for the |cPY〉 and |YPY〉 states appear related by thissort of rotation, in particular it can be shown that|cPY〉 = ziJ1a |YPY〉 (3.42)We will attempt to construct the boundary state |BPY〉 that satisfies the boundary conditions(3.25) as an intermediate rotation of the |cPY〉 state in the 1 direction. Our ansatz is thusthat|BPY〉 = exp(−iaJaa)|cPY〉 (3.43)for some vector of (possibly complex) angles (a).Using (3.41b) in (3.35) we obtain gluing relations for the boundary state:0 = z−iaJaa( †g(0P ) + †a(0P )i1)ziaJaa |BPY〉=( †g(0P ) + †a(0P )j−1i1)ziaJaa |BPY〉 (3.44)and similarly0 =( g(0P ) + i1j a(0P ))|BPY〉: (3.45)Applying these to the boundary state equation (3.25) gives an equation for j . Beginning45with the boundary state equation, we apply (3.41a) to change g to a:0 =(: †a3 a : − : †g3 g :+ .g †a(1 + 3) g − .g¯ †a(1− 3) g)|BPY〉=(: †a3 a : − : †g3(−i1j a) :+ .g †a(1 + 3)(−i1j a) − .g¯ †a(1− 3)(−i1j a))|BPY〉 (3.46)We will suppress normal ordering for a while. We can then separate the a vector on the right,and anticommute the fermion operators left (note the introduction of indices v to handle thevector product). The anticommutators of the aP †a fields vanish because the matrix betweenthem is traceless. The first two terms on the right hand side of (3.46), then, satisfy( †a3 a − †g3(−i1j a))|BPY〉 = − (2j a)a( †a3j−12 − †g)a|BPY〉= − (2j a)a( †a3j−12 + i †aj−11)a|BPY〉= †a(3 − j−13j) a|BPY〉 (3.47)and so the complete boundary state equation is0 = †a[3 − j−13j + .g(2 − i1)j + .g¯(2 + i1)j] a|BPY〉: (3.48)The matrix j is then obtained by solving0 = 3j−1 − j−13 + .g(2 − i1) + .g¯(2 + i1)P (3.49)46to obtainj =√1− .2|g|2 −i.g−i.g¯√1− .2|g|2 (3.50)Here, the diagonal elements of j are fixed by demanding that j be unitary. This also forcesg¯ = g∗.The boundary state for the full boundary condition is a rotation of the Neumann and thusof the Dirichlet boundary states. In the two possible fermion sectors, the Dirichlet-Dirichletboundary state |YPY〉, which satisfies (3.16), is|YPY〉ch = 2−12∞∏r=12exp( †−r ˜−r + ˜†−r −r)|0〉ch (3.51)|YPY〉g = 2−12∞∏n=1exp( †−n ˜−n + ˜†−n −n)exp( †0 ˜0)|0〉g : (3.52)The operators r, etc., are vectors containing the modes of the a and g fermion operatorvectors (3.24), for example i (z) =∑n i;nz−n n ≡ 1;n 2;n : (3.53)We are interested in the action of the Hamiltonian on the boundary state. Since the currentsJaa commute with the Hamiltonian, we can move them left,z−H |BPY〉 = z−Hz−iaJaaziJ1a |YPY〉 = z−iaJaaziJ1az−H |YPY〉:and work with the |YPY〉 state instead. The action of the Hamiltonian on the Dirichlet-Dirichletstate can be examined mode by mode. We will work it out carefully in the Neveu-Schwarz47sector. Combining (2.47) and (3.52) we havez−HcS |YPY〉ch = zR61√2∞∏r=12exp(−Har − Hgr ) exp(Yr) |0〉ch (3.54)withHar = r( †−r r + −r †r)Hgr = r( ˜†−r ˜r + ˜−r ˜†r) (3.55)Yr = ( †−r ˜−r + ˜†−r −r):For different values of r S 0, the operators in (3.55) commute with each other. For equalr, we have[Har P Yr] = rYr [Hgr P Yr] = rYr (3.56)and thusz−(Har +HRr )Yr = Yrz−(Har +r)z−(HRr +r) = Yrz−2rz−(Har +HRr ) (3.57)and finallyz−(Har +HRr )zYr = zz−2rYrz−(Har +HRr ): (3.58)Since Har |YPY〉 = Hgr |YPY〉 = 0, (3.54) becomesz−HcS |YPY〉ch =zR6√2∞∏r=12exp(z−2r( †−r ˜−r + ˜†−r −r))|0〉ch (3.59)48Re-rotating the left-moving fields givesz−HcS |BPY〉ch =zR6√2∞∏r=12exp(z−2r(i †−rj−11 ˜−r − i ˜†−r1j †−r))|0〉ch (3.60)Not surprisingly, the result in the Ramond sector is simplyz−HR |BPY〉g =zR3√2∞∏n=1exp(z−2n(i †−nj−11 ˜−n − i ˜†−n1j †−n))exp(i †0j−11 ˜0) |0〉g (3.61)GBL Yvuluution oz thy durtition FunwtionUsing the boundary states found in the last section, we can construct the fermion theory matrixelements associated with the boson partition function. Continuing from above, the NS sectorpartition function isZch = 〈YPY| z−HcS |BPY〉=1√2〈0|ch∞∏r′=12exp( †r′ ˜r′ + ˜†r′ r′)× zR6√2∞∏r=12exp(( †−rb−1 ˜−r + ˜†−rb †−r))|0〉ch (3.62)where we have abbreviatedb = i1j b−1 = − ij−11: (3.63)and = z−2r.49The exponentials can be broken up and grouped:Zch =zR62〈0|ch∞∏r=12(z †re rz †−rb−1 e −r)(ze †r rz e †−rb †−r)|0〉ch : (3.64)Then we can obtainze †r rz e †−rb †−r |0〉 =(1 + ˜†r r + ˜†1;r 1;r ˜†2;r 2;r)(1 + ˜†−rb †−r++ 2b11b22 ˜†1;−r †1;−r ˜†2;−r †2;−r+ 2b12b21 ˜†1;−r †2;−r ˜†2;−r †1;−r)|0〉= (1 + trb + 2 detb) |0〉 (3.65)where in the first line, the first two terms in each set of brackets are still in our vector notation,while the remaining terms have been broken down into their components. Since detb = 1, itseigenvalues are and −1 with =.(g + g¯)2+ i√1− .(g + g¯)2: (3.66)Then our factor from (3.65) is(1 + trb + 2 detb) = 1 + z−2r( + −1) + z−4r= (1 + z−2r)(1 + −1z−2r): (3.67)The other pair of exponentials in (3.64) works out the same way and the partition functionisZch =zR62∞∏r=12(1 + z−2r)2 (1 + −1z−2r)2(3.68)50Using the Jacobi triple product identity,∞∑k=−∞zkqn2=∞∏n=0(1− q2n+2)(1 + zq2n+1)(1 + z−1q2n+1)P (3.69)we haveZch =12zR6(∑nnz−n2∞∏k=111− z−2k): (3.70)By a similar procedure in the Ramond sector we obtainZg =12zR6(∑n(n+12)z−(n+12 )2∞∏k=11(1− z−2k))2: (3.71)As explained leading up to (3.32), the partition function for the bosonic system is obtainedas the sum of the Ramond and Neveu-Schwarz partition functions. In this sum we face thecombination[∑nnz−n2]2+[∑nn+12 z−(n+12 )2]2=∑n;mn−m(z−(n2+m2) + z−((n+12 )2+(n+12 )2))=∑n;mn−mz−12(m−n)2(z−12(m+n)2+ z−12(m+n+1)2)=∑n;n′nz−12n2z−12n′2: (3.72)51Then our partition function readsZ = 12zR6( ∞∏k=111− z−2k)2∑m;nz−12(m−n)2m−n(z−12(m+n)2+ z−12(m+n+1)2)=(1√2zR12∑nz−12n2∞∏k=11(1− z−2k))×(1√2zR12∑nnz−12n2∞∏k=11(1− z−2k)): (3.73)The first bracketed term is independent, and is exactly the bosonic partition function ofthe free Dirichlet boson n . We may remove it to get the desired partition function for m only:Zm =1√2zR12∑nnz−12n2∞∏k=11(1− z−2k) : (3.74)Recalling that = 2.2R is our physical parameter, we use Poisson resummation to get∑nnz−12n2=∑nz−12n2+n ln =√2.∑kz−22a(k−i ln R2) =√.∑kz−(k+) (3.75)where from (3.66) we know =−i ln 2.=12.cos−1(12.(g + g¯)): (3.76)We can also writezR12∞∏k=11(1− z−2k) =[(i.)]−1=[√.(− .i)]−1=√2.zR24∞∏k=11(1− z−k)(3.77)where () is the Dedekind eta function (for which the identity (−1R) = (−i)1R2() holdsgenerally).52Combining these results, we have the partition function for the compact boson subject toa periodic potential on one boundary and a Dirichlet boundary condition on the other:ZB;Y = zR24∑nz−(n+)∞∏k=11(1− z−k) : (3.78)GBM ConwlusionsThe expression for the partition function (3.78) is not new, nor is the idea of attacking theproblem using fermionization [13]. However, our approach differs in the way that the renormal-ization of the coupling constants is performed. In the reference [13], the coupling is redefinedinside an exponential of the boundary condition, leading to a relationship between their cou-plings g′P g¯′ and the ones obtained here:sin2 .√g′g¯′ = .2gg¯ andg′g¯′=gg¯: (3.79)Our calculation, which involves a more direct renormalization of the couplings, in some sensejustifies their renormalization arguments and provides an alternative approach for constructingboundary states using periodic operators.Ultimately, the realization of a system described by the boundary sine-Gordon model at theself-dual radius is physically very difficult. At the self-dual radius, other interactions besides theperiodic boundary potential become marginal; in particular the vulk operator zi:m: is rylyvuntand destroys any hope of maintaining the critical boundary theory.An interesting extension, discussed in [10], involves compactification at rational radii g =nRm. The boundary state and partition function are then obtained by fermionizing and pro-jecting out combinations of fermion number that coincide with the boson momenta at therational radius. Perhaps these rational CFTs might be more easily realized in real systems.Future work using this approach should be possible, in particular it should be possible to53apply it to the dissipative Hofstadter model [5], where there are two bosonic degrees of freedom,a periodic potential for each, and a magnetic field term that couples the two directions.544B aovility ut nity tympyruturyIn this chapter we return to the condensed matter context and consider two other importantresults relating to the Schmid model. The first is a finite temperature analysis of the one-dimensional problem, and the second is a renormalization group analysis of the zero temperatureproblem on a two-dimensional lattice.4BE bonAzyro tympyrutury upprouwhFisher and Zwerger [7] attacked the Schmid model at arbitrary temperature, obtaining inter-esting results for the temperature dependence of the mobility. Their approach was also veryexplicit in its development of the duality between the weak potential and tight-binding limitsof the problem. In anticipation of generalizing the results to two dimensions, we present somedetails of their calculations.Working at finite temperature requires the full density matrix approach to Caldeira-Leggettdissipative quantum mechanics, presented in section 1.2. We will use much of the notation fromthat section. We work towards an expression for the mobility as defined in (1.5). We are nowworking in real time t.Our applied force F is now a constant that we include in the potential:k (q) = −k0 cos(2.qRv0)− Fq (4.1)Putting this into the expression for the reduced density matrix (1.16) involves the difference in55particle actionsh0[q]− h0[q′] = h0[x+ 12y]− h0[x− 12y]= b∫ t0yt′x˙(t′)y˙(t′) +∫ t0yt′∫ t0yt′Fy(t′)+∫ t0yt′[cos 2a0 (x+12y)− cos 2a0 (x− 12y)](4.2)where as before, x = 12 (q+q′) and y = q−q′. Each cosine is then expanded in a Coulomb gas asin section 1.3. The first cosine is represented as a sum over n charges i = ±1 and the secondis indexed by n′ charges zj = ±1. For each nP n′ and times tiP tj we have charge distributions/(t) =2.~v0n∑i=1i(t− ti) /′(t) = 2.~v0n∑j=1zj(t− tj): (4.3)We are interested in the classical probability distribution e (m; t) for the position of theparticle as a function of time, which is obtained as the diagonal component of the reduceddensity matrix /ˆ(qP q′; t) described in (1.16)e (m; t) ≡ /ˆ(mPm; t) (4.4)=∞∑n=0∞∑n′=0(ik2~)n (−ik2~)n′ ∑i;zj=±1∫yt1 : : :∫ytn∫yt′1 : : :∫yt′n′∫yx0∫yy0 /ˆ(x0 +12y0P x0 − 12y0; 0) ×G(/P /′;x0P y0) (4.5)where the propagator corresponding to charge distributions / and /′ isG(/P /′;x0P y0) =∫ mx0Dx∫ 0y0Dy exp[i~∫ t0yt′(bx˙y˙ + xy˙ + Fy + x(/− /′)− 12y(/+ /′))− h2[y]]: (4.6)56The path integrals may be performed to yieldG(/P /′;x0P y0) =b2.~y(t)exp(i~bmy˙xl∣∣∣t0+ i~∫ t0yt′[F − 12(/− /′)]yxl(t′)− h2[yxl])(4.7)wherey(t) = 1 (1− zt)P (4.8) = RbP (4.9)and the classical path yxl(t′) solvesy¨xl − y˙xl = (/′ − /)Rb (4.10)subject to yxl(0) = y0 and yxl(t) = 0.Note that if the charge distribution is not neutral, then the integral in h2 generates termslinear in t which effectively kill that configuration’s contribution to the probability distribution.Thus we may restrict our attention to the case of “neutral” total charge distributions, whichmeans n+ n′ is even and∑i −∑ zj = 0.To go further, we take the spatial Fourier transform of e (m),e˜ (P t) ≡∫ymzime (mP t)P (4.11)and use it as a generating function for the expectation value of the position at time t:m¯(t) ≡ 〈m(t)〉 =∫ym me (m) = −i yye˜ (P t)∣∣∣∣=0(4.12)In the generating function (4.11), the integral over m produces a delta function in y0, and57the resulting x0 integral involving the initial particle density matrix has no dependence on (which comes from the initial state having zero total momentum; the details are given theappendix of [7]). The result is that the expectation value of m(t) is〈m(t)〉 = Ft− 1〈12∫ t0yt′(/+ /′)〉0(4.13)where the average on the right hand side is a weighted average over the set of configurationsof the charges:〈V〉0 ≡∞∑n;n′=0n+n′ even(ik2~)n (−ik2~)n′ ∑i;zj=±1neutral∫yt1 : : : yt′n′V exp(Ω[yp]): (4.14)The influence phase isΩ[yp] =i~∫ t0yt′[F − 12(/+ /′)]yp(t′)− h2[yp]: (4.15)with yp(t′) the particular solution to (4.10) for the given charge distributions,yp(t′) =v0 n∑i=1zih(t′ − ti)−n′∑j=1jh(t′ − tj) : (4.16)The dimensionless friction is defined in (1.39), andh(t′) = (t′) + (−t′)zt′ (4.17)is the Green function for the operator 1y2yt2− yyt .Using (4.13) in the expression for the mobility (1.5) we get0= 1− limt→∞1Ft〈12∫ t0yt′(/+ /′)〉0: (4.18)58As in Schmid’s original calculation, we can show a duality between this expression and anexpression originating in a tight-binding approximation. The approach of Fisher and Zwergeris to reorganize and relabel the terms in (4.18) so that the smoothed-out paths become series oftight-binding hops, and the smoothness is absorbed into the form of the bath spectrum J(!).We begin by defining paths qs and q′s on a tight-binding lattice with lattice constant v˜0 =v0R according toqs(t) = v˜0n∑i=1zi(t− ti) = 1∫ t0yt′/(t′) (4.19a)q′s(t) = v˜0n∑i=1i(t− t′i) = 1∫ t0yt′/′(t′): (4.19b)Now by creating sum and difference paths xs(t) =12 (qs + q′s) and ys(t) = qs − q′s we canrewrite (4.13) and (4.15) as functionals of these sharp tight-binding trajectories instead of yp.The result is that (4.15) becomesΩ = iF~∫ t0yt′ys(t′) + iΦ [xsP ys] (4.20)where iΦ is defined in (1.17), and the superscript indicates that instead of an ohmic spectrumgiven by (1.22), the weighted density of states is to be taken asJ(!) =!1 + (!R)2: (4.21)The expectation value term in the expression for the mobility (4.18) can then be writtenlimt→∞1Ft〈12∫ t0yt′(/+ /′)〉0= limt→∞1Ft〈xs〉 (4.22)=tw0(4.23)59where the average 〈·〉 indicates the use of weight (4.20) with spectral function (4.21). Themobility in the weak potential limit with lattice spacing v0 is thus related to the mobilityiB of the same particle on a tight-binding lattice with spacing v˜0 = v0R. This disparityin lattice spacing v0 ↔ v0R is equivalent, given the definition of , to ↔ 1R. The otherparameter of the mobility is the force F ; this is regrouped into energy change due to a singlehop = Fq0. Then we may write the duality equation for the mobility(P )0= 1− iB(1RP R)0: (4.24)Working in the tight-binding framework, Fisher and Zwerger calculate iB as a function oftemperature to order k 2, by looking at all possible “one-blip” paths. This involves summingover all paths with n = n′ = 1, 1 = z1. The four contributing paths are writtenyw = v˜0((t′ − t1)− (t′ − t′1)) (4.25)xw = v˜0((t′ − t1) + (t′ − t′1)) (4.26)where and each take on the values ±1. For these simple paths the integrals can be simplifiedto leave0= 1− 2.k20~∫ ∞0yt sin(tR~) sin[(2R)f˜1(t)] exp[−(2R)f˜2(t)] (4.27)wheref˜1(t) =∫ ∞0y!sin!t!(1 + (!R)2)(4.28)f˜2(t) =∫ ∞0y!1− cos!t!(1 + (!R)2)coth(12~!): (4.29)This is enough to reproduce the zero-temperature result of Schmid. In taking the → ∞60limit it is necessary to replace the cut-off function (1 + (!R)2)−1 with an exponential cut-offz−t; then the integrals can be performed to give()0= 1− .2Γ( 2 )(k0~)2( ~)2(1R−1)exp(−R~)P (4.30)where = Fv0 is the potential energy difference between adjacent minima. The mobility thenclearly has a critical dependence on , and it can be seen that as the applied force F → 0 (thelinear mobility limit), the coefficient of k 20 goes to zero as long as Q 1. The perturbationexpansion breaks down for S 1, but the duality argument implies an abrupt transition in thedimensionless mobility to 0 as passes through 1.The finite temperature expression is also used by Fisher and Zwerger to divine the behaviourof the mobility with temperature. In particular they note in the weak-potential limit where = mu0 at zero temperature, there is a drop in the mobility as the temperature rises abovezero before it starts to approach its classical value of 1R.4BF miAKuny Gynyrulizution to FxYi and Kane [15] looked at the renormalization behaviour of a generalized two-dimensionalSchmid model. From the lattice vectors {X} they construct an arbitrary periodic potential thatcontains Fourier components corresponding to each reciprocal lattice vector G. They eliminatethe friction as a parameter in favour of the lattice spacing, and show using Schmid’s renormaliza-tion approach that the perturbative stability of the mobility depends (in this parametrization)only on the lengths of the shortest lattice vector and the shortest reciprocal lattice vector.Having stated the problem in this way, they apply their simple rules to more general cases,such as the triangular and hexagonal lattice.Beginning from (1.26), we rescale the particle coordinate and define imaginary time pathr() = q()√R2. = (qRv0)√, with the dimensionless friction defined in (1.39). Now the61free parameter of the system is the lattice spacing, corresponding to√. We may also go totwo dimensions, in which case the particle coordinate is the vector x() and the action, adaptedfrom (1.26), ish[x] = 12∫y!|!|z|!|c |x(!)|2 −∫yx∑GvGz2iG·r(): (4.31)In lieu of the mass term, we now have an exponential which enforces a short time cut-off x. Thecosine potential has been generalized to include Fourier components vG = v∗−G at all reciprocallattice vectors G (here the reciprocal lattice is defined as all vectors g such that for any latticevector X, g ·X is an integer).The duality between weak potential and tight binding limits is established very explicitlyby considering the dual action of the tight-binding model; for a momentum space path q()the tunnelling amplitudes tR between sites separated by X leads to the dual actionh[q] = 12∫y!|!|z|!|c |q(!)|2 −∫yx∑RtRz2iR·k(): (4.32)From Schmid’s analysis in 1.3 we know that in the absence of any potential (all vG = 0) wehave dimensionless mobility R0 = 1. In the tight-binding case, in the absence of tunnelling(tR = 0) we have localization R0 = 0. The stability of these couplings is obtained by astandard renormalization. For the weak potential limit, renormalization of the couplings vGgives flow equationsyvG = (1− |G|2)vG (4.33)and thus if all reciprocal lattice vectors are sufficiently small then the couplings vG flow to zero;62i.e.|Gmin| S 1 =⇒ 0= 1 (4.34)Analogously, the flow equations in the tight-binding limit send all tunnelling couplings tR tozero provided that the lattice spacing is sufficiently large:|Xmin| S 1 =⇒ 0= 0 (4.35)For a square lattice we will once again have the transition from free particle to tight bindingbehaviour as passes through 1. This is only the case for certain geometries however, where|Xmin| = |Gmin|−1. For a general Bravais lattice, there is some structure factor Σ whichdescribes the relationship between the shortest scales of the lattice and its reciprocal:|Xmin| =√Σ|Gmin|−1: (4.36)For lattice spacing |Xmin| =√, the reciprocal lattice spacing is |Gmin| =√ΣR. Theperturbative stability of the R0 = 0 regime is still guaranteed for S 1, but the R0 = 1limit is now stable for Q Σ.For a general Bravais lattice in two dimensions, we may take a basis {X1PX2} such that X1is the shortest non-trivial lattice vector and X2 is the shortest lattice vector that is not parallelto X1. The relative length of the two basis vectors is ≡ |X2|R|X1| S 1, and the anglebetween them. (It can be shown that in order to satisfy the other restrictions, cos ≤ 12 ,with equality in the case of a rhombic lattice). The reciprocal lattice has the same structure,but oriented differently and with structure factor Σ = 1R2 sin2 . For the square lattice, thisrecovers h = 1, while for an equilateral triangular lattice we have Σtri = 4R3.For the equilateral triangular lattice, in the region where 1 Q Q 4R3, both the tight-63binding and zero-potential limits are stable. (This implies that there is then some intermediateunstable fixed point as well.)Very interesting consequences result from Yi and Kane’s extension of their argument to thecase of non-Bravais lattices. They note in particular that changing the sign of the equilateraltriangle potential yields its dual lattice, which is hexagonal. Since only the sign of the potentialhas changed, the structure of the reciprocal lattice is the same. However, the lattice constantof the triangular lattice is a factor of√3 larger than the nearest neighbour separation on thehexagonal lattice, and thus the structure factor is Σhzx = 4R9. Now, for 4R9 Q Q 1, nyithyrof the two perturbative limits is stable. The result is that for these values of there must existstable intermediate fixed points 0 Q R0 Q 1.The behaviour predicted by Yi and Kane for non-square lattices is an interesting and poten-tially observable phenomenon. In the next chapter we investigate the two-dimensional problemfrom the perspective of Fisher and Zwerger, and attempt to generalize their approach to ageneral two dimensional Bravais lattice.64IB durtiwly aovility on u BruvuisLuttiwyHere we apply the methods of Fisher and Zwerger to the problem of the mobility of a particlemoving on a two dimensional Bravais lattice subject to dissipation. The lattice is modelledusing a potential that is a sum of plain waves oriented along a small number of reciprocallattice vectors. The generalization of Fisher and Zwerger’s expressions from the 1d case isstraightforward, though cumbersome.We then proceed to expose the duality between the weak potential and tight-binding mo-bilities, commenting on subtleties that arise in two dimensions. For the case of a triangularlattice, there is the possibility of terms arising at third order in the perturbation expansion inthe potential strength. A useful parameterization and expressions for these terms is developed.5BE fyzormulution in two ximynsionsWe will consider a general two-dimensional Bravais lattice, with an associated potential thatis a sum of cosine plane waves along certain reciprocal lattice vectors. We will denote thesereciprocal lattice vectors as gw, and generally use the index w to sum over the correspondingpotential components. Note that in the simplest rectangular lattice potential, the g w wouldconsist of two elements, each parallel to a rectangular axis of the lattice. For an equilateraltriangular lattice, however, it is necessary to take three cosine plane waves (corresponding tothe presence of six equivalent nearest neighbours in the triangle’s reciprocal lattice).65We write our potential, as a function of the particle position q, ask (q) =∑wkw cos (2.q · gw) : (5.1)We anticipate that this will need to be expanded in a Coulomb gas,exp(i~∫ t0k [q])=∏wexp(−ikw2~∫ t′0yt′(z2igb·q + z−2igb·q))=∏w ∞∑nb=0(−ikw2~)nb ∑bi =±1∫ t0yt′ : : : yt′nb exp(− ih∫ t0ys q(s) · w(s))P (5.2)and as usual we will need a corresponding construction for the forward-going potential k [q ′].We now have component charge densities that are vectors parallel to their associated g w,w(s) = 2.~gw∑iwi (s− twi)′w(s) = 2.~gw∑jzwj(s− twi)P (5.3)and the total charge density is just the sum of these:(s) =∑ww ′(s) =∑w′w: (5.4)The early steps of the problem factor exactly, and the initial expressions from section 4.1may be immediately adapted by replacing the paths qP q ′P xP y with their vector equivalents, withdot products forming the necessary linear combinations. This is true of the particle positionprobability distribution (4.5), although the propagator (4.7) picks up another prefactor ofbR2.~y(t). Ultimately the behaviours in the two dimensions are not independent becausethe two components of the charge distributions are correlated (assuming a non-rectangularlattice).66We once again encounter the issue of charge neutrality; unless the distributions and ′have the same net charge the h2[y] will kill the contribution to the probability distributione (X). As a result, the neutrality condition∫ t0yt′(′(t′)− (t′)) = 0 (5.5)is imposed on the charge distributions.Without any new complications, we come to the expression for the linear mobility analogousto (4.18):ij0= ij − limt→∞Fj1t〈12∫ t0yt′(/+ /′)i〉0∣∣∣∣F=0: (5.6)The average 〈·〉0 involves all the gw contributing to the potential〈Vi〉0=∏w∑nb;n′b(ikw2~)nb (−ikw2~)−n′b ∑b;zb∫ t0yt1 : : : yt′nbVi exp(Ω[yp]) (5.7)with the weight Ω the obvious generalization of (4.15), and the particular solution yp(s) to thevector version of (4.10) given concisely asyp(s) =1b∫ys′ h(s− s′)(′(s′)− (s′)): (5.8)The function h(s) is defined in equation (4.17).5BF hightAvinxing xuulityWe can rewrite this as a tight-binding expression and expose the duality as in section 4.1. Theonly complications are in dealing with the structure factor Σ (defined in section 4.2) for non-square lattices, and the relative orientation of the tight-binding lattice to the original lattice.67We give our original (real-space, not tight-binding) lattice a basis {g1Pg2} whose memberssatisfy the relations imposed on X1 and X2 in section 4.2, namely that they open at acuteangle , |g2|R|g1| = ≥ 1, and cos ≤ 12 . The structure factor is Σ = 1R2 sin2.We may then take the reciprocal lattice basis {g1P g2} to satisfygi · gj = ij (5.9)where ij =[0 1−1 0], which implies|g1| =1|g2| sin |g2| =1|g1| sin: (5.10)Just as in the one-dimensional case, the particular solution yp(s) is the basis for our tight-binding expansion. We then take basis {g˜1P g˜2} for our tight-binding lattice asg˜i =|g1 × g2|gi =2.~gi: (5.11)where in two dimensions the dimensionless friction coefficient involves the area of the unit cell: ≡ |g1 × g2|2.~=|g1||g2| sin2.~: (5.12)Now we may produce tight-binding pathsqs(t′) =∑wg˜w∑iwi (t− twi) q′s(t′) =∑wg˜w∑izwi(t− twi) (5.13)and note that this definition leads to1∫ t′0ys (s) = q(t′) (5.14)68as desired. We can then rewrite (5.6) asij0= ij − ijiB0(5.15)withijiB =1 limt→∞Fj1t〈xs(t)〉∣∣∣∣F=0(5.16)the mobility on the tight binding lattice defined in (5.11) and the average (5.7) is taken withrespect to the revised spectrum (4.21).Since the reciprocal lattice, and thus the tight-binding lattice, is rotated relative to theoriginal lattice, directional information should be extracted with care.The relation between the dimensionless friction and the corresponding quantity ˜ of thetight binding lattice is˜ =|g˜1 × g˜2|2.~=sin: (5.17)5BG gywonx orxyr wontrivutionsIn the original one-dimensional analysis, the zero-temperature stability of the weak-potentiallimit was shown to second order by showing that the coefficient of k 20 in (4.27) is zero as longas ′ Q 1. For ′ S 1 the perturbation expansion breaks down. (We use ′ to refer to thedimensionless dissipation parameter in the 1d problem.)The dependence on ′ in the 1-d calculation comes from the integrals in the influence phase(1.16), which involves products of the paths x and y. These paths, on the tight-binding lattice,each contribute a factor q˜0 = q0R′ which combine with the other factors to leave a residual1R′. In two dimensions, the paths in the order k 2w term each contribute |g˜w| = |gw|R, and69combine with the surrounding factor R2.~ to produce, in the case of k1,2.~( |g1|)2=1( |g1||g2| sin)=√Σ(5.18)From this we conclude that at zero-temperature, k1 = 0 for Q√Σ. For the case of k2, theexponential factors combine to give1 sin2 √Σ(5.19)and thus at i = 0, k2 = 0 for Q sin2 √Σ.When |g1| 6= |g2|, the limits on are different, and this suggests that the behaviourrenormalizes differently along the two directions, or that it is simply determined by the smallerbound. When |g1| = |g2|, such as for the equilateral triangular lattice,√Σ = 1R sin and theselimits on are the same. (In the triangular case there would also be a k3 term which wouldhave the same features).Upon accounting for the different definition of used in this section and 4.2, we are notsurprised to find that the limit Q 1R sin agrees with Yi and Kane’s renormalization groupargument for the stability of the R0 = 1 fixed point in the equilateral triangular lattice.5B4 hhirx orxyr wontrivutionsA potentially interesting aspect of the triangular lattice is the possibility for paths of odd order.For a reasonably constructed triangular lattice, the potential will consist of 3 terms where anytwo of the associated reciprocal lattice vectors gw form a basis for the reciprocal lattice, andthe third can be formed from some combination g3 = ±g1± g2. Then there are many paths oflength 2 and length 1 that end at the same point. Here we will work out some of the details ofthe third order contributions.Our potential in this case consists of components in three directions, g i for i = 1P 2P 3, with70g1 = g2 + g3. One contribution to the average position at time t results from the tight-bindingpathsqs(t′) = g1(t′ − t1) (5.20)q′s(t′) = g2(t′ − t2) + g3(t′ − t3): (5.21)This contribution is weighted by the potential strengths(ik12)(−ik22)(−ik32)=−ik1k2k38: (5.22)The integrals in the influence phase contribute terms depending on the relative positions of thetimes ti. Assuming that t1 Q t2 Q t3 we have corresponding centre and difference vectors xs(t′)and ys(t′) satisfying (here we suppress the subscript s)x(t′) =0 P t′ Q t112g1 P t1 Q t′ Q t212 (g1 + g2) P t2 Q t′ Q t3g1 P t3 Q t′(5.23a)y(t′) =g1 P t1 Q t′ Q t2g1 − g2 P t2 Q t′ Q t30 P otherwise(5.23b)Defining 1 = t2 − t1, 2 = t3 − t2 our weighting factor for this path is z with (here wecombine equations (4.20), (1.17) and (1.19) and extend them to two dimensions)Ω = iΩ1 + iΩ2 + Ω3 (5.24)71withiΩ1 =i~∫ t0yt′F · ys(t′) (5.25)iΩ2 = −2i~∫ t0yt′∫ tt′ys y(s) · x(t′)I (s− t′) (5.26)Ω3 = −1~∫ t0yt′∫ t′0ys y(s) · y(t′)g(s− t′): (5.27)Working these out one by one for the particular paths 5.23, for all t1 Q t2 Q t3 Q t we haveiΩ1 =i~(∫ 10yt′F · g1 +∫ 20yt′F · (g1 + g2))=i~F · (g11 + (g1 + g2)2); (5.28)iΩ2 = − i~[∫ t2t1yt′∫ t2t′ys+∫ t2t1yt′∫ t3t2ys+∫ t3t2yt′∫ t3t′ys]y(s) · x(t′)I (s− t′)= − i~((g1 · g1)∫ 10yt′∫ 1t′ys I(s− t′) + g1 · (g1 − g2)∫ 10yt′∫ 20ys I(s− t′ − 2)+(g1 + g2) · (g1 − g2)∫ 20yt′∫ 2t′ys I(s− t′))= − i~((g1 · g1)V0(1) + g1 · (g1 − g2)V1(1; 2) + (g1 + g2) · (g1 − g2)V0(2))(5.29)where we have defined the required double integrals of I asV0() =∫ 0yt′∫ t′ys I (s− t′) (5.30)V1(1; 2) =∫ 10yt′∫ 20ys I (s− t′ − 1): (5.31)72Similarly, we haveΩ3 = −1~∫ t0yt′∫ t′0ys y(s) · y(t′)g(s− t′)= −1~((g1)2B0(1) + g1 · (g1 − g2)B1(1P 2)+(g1 − g2)2B0(2)): (5.32)whereB0() =∫ 0yt′∫ t′ys g(s− t′) (5.33)B1(1; 2) =∫ 10yt′∫ 20ys g(s− t′ − 1): (5.34)These expressions for Ωi represent only one of 72 similar paths contributing at order k3.The others are obtained by the following 4 independent operations:1. fyAordyring thy timys; t1 Q t2 Q t3 is only one of 6 possibilities. Reordering of thetimes makes a significant difference to the phases Ω2 and Ω3, particularly in the crossterms associated with V1 and B1.2. huking un ultyrnutivy singly vywtor into ; we chose g1 but there are 3 distinctoptions. This has no effect on factors Ω2 and Ω3 if the lattice is equilateral, since x andy are rotated in the same way and their dot products feel nothing. The force term Ω1does change, as does the final coordinate x(t) = gi that z is weighting.3. Chunging thy ovyrull purity; we get a new configuration by taking the negative of allthree vectors. There are 2 ways to do this; doing it flips the sign of Ω1 as well as the signof x(t) that z is weighting.4. Intyrwhunging thy forwurd und vuwkwurd puths and ′; this removes the asymme-try in putting only one vector in and two in ′. There are 2 choices for this assignment,73which changes the sign of Ω1 and Ω2. It also changes the sign of the contribution in theaverage, since (ik1)(−ik2)(−ik3) = −i∏w kw → (ik2)(ik3)(−ik1) = i∏w kw.Using index j = 1P : : : P 6 for the timing arrangements and k = 1P 2P 3 for the single vectorchoice (t′) = gk(t′− t1), we then have weights Ωjki (1P 2) associated with these six possibili-ties. Letting = ±1 represent the parity and = ±1 the forward or backward path assignment,we can write the order k 3 contribution to the expectation value of xs(t):〈xs(t)〉(3) = −ik1k2k38~3∑j;k∑=±1=±1 gk∫ t0yt1∫ t0y1∫ t1y2 exp(iΩjk1 + iΩjk2 + Ωjk3 ) (5.35)=−k1k2k32~3∑j;kgk∫ t0yt1∫ t0y1∫ t1y2 sin(Ωjk1 ) cos(Ωjk2 ) exp(Ωjk3 ) (5.36)(5.37)Having parametrized the terms and reduced the problem to a sum over 18 similar pieces,the integrals remain difficult to simplify because of the cross-coupling of the integration times1 and 2 in the V1 and B1 terms. It may be interesting to pursue this calculation further, toverify that the critical dependence on is not altered by these third order terms.5B5 Closing rymurksIn chapters 4 and 5 we have presented the core results of Fisher and Zwerger’s analysis ofa particle in a periodic potential subject to dissipation, and extended the analysis into twodimensions. This has included adapting the second order perturbative expressions for themobility at finite temperature to the case of an arbitrary Bravais lattice, as well as makinginvestigations into the form of the third order contributions (which appear in the case of atriangular lattice).74The generalization is consistent with the renormalization group arguments of Yi and Kanefor the zero temperature problem, but is formulated at finite-temperature and permits pertur-bative calculation of the non-linear mobility.75Bivliogruphy[1] A. O. Caldeira and A. J. Leggett. Influence of dissipation on quantum tunneling in macro-scopic systems. dhysB fyvB LyttB, 46(4):211–214, 1981.[2] A. O. Caldeira and A. J. Leggett. Path integral approach to quantum Brownian motion.dhysicu A gtutisticul aychunics unx its Applicutions, 121:587–616, 1983.[3] C. G. Callan, C. Lovelace, C. R. Nappi, and S. A. Yost. Loop corrections to superstringequations of motion. buclyur dhysics B, 308:221–284, 1988.[4] C. G. Callan and L. Thorlacius. Open string theory as dissipative quantum mechanics.buclyur dhysics B, 329:117–138, 1990.[5] Curtis G. Callan and Denise Freed. Phase diagram of the dissipative Hofstadter model.buclyur dhysics B, 374:543, 1992. arXiv:hep-th/9110046.[6] Sidney Coleman. Aspycts of gymmytryN gylyctyx Ericy lycturys. Cambridge UniversityPress, 1988.[7] Matthew P. A. Fisher and Wilhelm Zwerger. Quantum brownian motion in a periodicpotential. dhysB fyvB B, 32(10):6190–6206, 1985.[8] Paul Ginsparg. Applied conformal field theory: Lectures given at Les Houches summersession, June 28 - Aug. 5, 1988., 1988. arXiv:hep-th/9108028.[9] F. Guinea, V. Hakim, and A. Muramatsu. Diffusion and localization of a particle in aperiodic potential coupled to a dissipative environment. dhysB fyvB LyttB, 54(4):263–266,1985.[10] M. Hasselfield, G. W. Semenoff, Taejin Lee, and Philip Stamp. Boundary sine-Gordontheory revisited, 2005. arXiv:hep-th/0512219. Submitted to Annuls of dhysics.[11] Taejin Lee and Gordon W. Semenoff. Fermion representation of the rolling tachyon bound-ary conformal field theory. Journul of High Enyrgy dhysics, 2005(05):072, 2005. arXiv:hep-th/0502236.[12] Joseph Polchinski. gtring hhyory. Cambridge University Press, 1999.[13] Joseph Polchinski and La´rus Thorlacius. Free fermion representation of a boundary con-formal field theory. dhysB fyvB D, 50(2):R622–R626, 1994.76[14] Albert Schmid. Diffusion and localization in a dissipative quantum system. dhysB fyvBLyttB, 51(17):1506–1509, 1983.[15] Hangmo Yi and C. L. Kane. Quantum Brownian motion in a periodic potential and themultichannel Kondo problem. dhysB fyvB B, 57(10):R5579–R5582, 1998.77
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Localization of a particle due to dissipation in 1 and 2 dimensional lattices Hasselfield, Matthew 2006
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Title | Localization of a particle due to dissipation in 1 and 2 dimensional lattices |
Creator |
Hasselfield, Matthew |
Publisher | University of British Columbia |
Date Issued | 2006 |
Description | We study two aspects of the problem of a particle moving on a lattice while subject to dissipation, often called the "Schmid model." First, a correspondence between the Schmid model and boundary sine-Gordon field theory is explored, and a new method is applied to the calculation of the partition function for the theory. Second, a traditional condensed matter formulation of the problem in one spatial dimension is extended to the case of an arbitrary two-dimensional Bravais lattice. A well-known mathematical analogy between one-dimensional dissipative quantum mechanics and string theory provides an equivalence between the Schmid model at the critical point and boundary sine-Gordon theory, which describes a free bosonic field subject to periodic interaction on the boundaries. Using the tools of conformal field theory, the partition function is calculated as a function of the temperature and the renormalized coupling constants of the boundary interaction. The method pursues an established technique of introducing an auxiliary free boson, fermionizing the system, and constructing the boundary state in fermion variables. However, a different way of obtaining the fermionic boundary conditions from the bosonic theory leads to an alternative renormalization for the coupling constants that occurs at a more natural level than in the established approach. Recent renormalization group analyses of the extension of the Schmid model to a two-dimensional periodic potential have yielded interesting new structure in the phase diagram for the mobility. We extend a classic one-dimensional, finite temperature calculation to the case of an arbitrary two-dimensional Bravais lattice. The duality between weak-potential and tightbinding lattice limits is reproduced in the two-dimensional case, and a perturbation expansion in the potential strength used to verify the change in the critical dependence of the mobility on the strength of the dissipation. With a triangular lattice the possibility of third order contributions arises, and we obtain some preliminary expressions for their contributions to the mobility. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-18 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085809 |
URI | http://hdl.handle.net/2429/18616 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2006-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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