We introduce an exact Monte Carlo approach to the statistics of discrete
quantum systems which does not rely on the standard fragmentation of the
imaginary time, or any small parameter. The method deals with discrete objects,
kinks, representing virtual transitions at different moments of time. The
global statistics of kinks is reproduced by explicit local procedures, the key
one being based on the exact solution for the biased two-level system.Comment: 4 pages, latex, no figures, English translation of the paper

A. F. Elesin

A. Lagendijk

B. V. Svistunov

E. L. Pollock

G. G. Batrouni

I. S. Tupitsyn

J. E. Hirsch

N. Metropolis

N. V. Prokof’ev

R. Blankenbecler

R. M. Fye

R. P. Feynman

W. Krauth

English

arXiv

We propose an exact Monte Carlo approach for the statistics of discrete quantum systems that does not employ the standard partition of the imaginary time into a mesh and does not contain small parameters. The method operates with discrete objects — kinks, describing virtual transitions at different moments in time. The global statistics of the kinks is reproduced by exact local procedures, the main one being based on the known solution for an asymmetric two-level system

Prokof\u27ev, Nikolai

Svistunov, Boris

Tupitsyn, I

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1996Exact quantum Monte Carlo process for thestatistics of discrete systemsNikolai Prokof 'evUniversity of Massachusetts - Amherst, prokofev@physics.umass.eduBoris SvistunovUniversity of Massachusetts - Amherst, svistunov@physics.umass.eduI TupitsynFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationProkof 'ev, Nikolai; Svistunov, Boris; and Tupitsyn, I, "Exact quantum Monte Carlo process for the statistics of discrete systems"(1996). JETP Letters. 1186.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/1186arXiv:cond-mat/9612091v1  9 Dec 1996Exact Quantum Monte Carlo Process for the Statistics of Discrete SystemsNikolai V. Prokof’ev, Boris V. Svistunov, and Igor S. TupitsynRussian Research Center “Kurchatov Institute”, 123182 Moscow, Russia.We introduce an exact Monte Carlo approach to the statistics of discrete quantum systems whichdoes not rely on the standard fragmentation of the imaginary time, or any small parameter. Themethod deals with discrete objects, kinks, representing virtual transitions at different moments oftime. The global statistics of kinks is reproduced by explicit local procedures, the key one beingbased on the exact solution for the biased two-level system.PACS numbers: 05.30.Ch, 02.50.Ng1. Quantum Monte Carlo (MC) simulation is the most powerful, if not the only available method of getting exactresults for complex systems [1–7], where analytic solutions are not possible and exact diagonalization methods do notwork because of the enormous Hilbert space. One conventionally starts from the general expression for the thermalaverage 〈A 〉 = TrAe−βH/Z(β), where Z(β) = Tre−βH is the partition function for the Hamiltonian H , and convertsthe trace into the Feynman path integral [8]. To sum over all trajectories with the weighting function e−S definedby the action S the imaginary time interval [0, β] is divided into Nβ = β/∆τ smaller intervals of width ∆τ , and theactual trajectory is described by specifying the system state |αk〉 at each time τk = k∆τ where k = 0, 1, ..., Nβ. Toensure summation over trajectories with the near-optimal action, the complete sum is replaced by the stochastic sum(the known Metropolis algorithm [9]); in the most commonly used schemes the probabilities W1,2 to have the twotrajectories {αk}1,2 in the sum are related asW1/W2 = eS2−S1 . (1)After some trajectory is accepted to the statistics, the MC process generates small variation to this trajectory whichis either accepted or rejected according to Eq.(1).An approximate treatment of noncommuting operators originating from the artificial discretization of the timevariable at points where the trajectory changes, i.e., when |αk〉 6= |αk+1〉 ( in what follows we call such points“kinks”), results in an intrinsic error ∼ ∆τ2 (see, e.g., [10]). Taking smaller and smaller values of ∆τ eventuallyeliminates this intrinsic error, but then the algorithm starts accumulating statistics too slowly. Consider, as a typicalexample, the Hamiltonian of a particle on a latticeH = −t∑<ij>a†iaj +∑iUini , (2)where a†i creates a particle on site i, t is the hopping matrix element, ni = a†iai, and < ij > conventionally denotesnearest neighbor sites. In this case the set of states {α} may be chosen to describe particle coordinate. The typicalseparation in time between two adjacent kinks is of order 1/t and independent of ∆τ , so that for small ∆τ thereare some 1/(t∆τ) ≫ 1 time intervals between the kinks. Now, if the MC procedure will suggest to create a newkink-antikink pair then the corresponding variation of the trajectory will be most probably rejected since the ratioWnew/Wold ∼ (t∆τ)2 is proportional to the square of a small parameter. On another hand, when the MC proceduresuggests to shift the already existing kink to the nearest point in time, the corresponding variation of the trajectory isaccepted with probability ∼ O(1). Thus, in average, it takes some 1/(t∆τ)2 attempts to create a new kink - anti-kinkpair, and 1/(t∆τ) attempts to move the kink to the nearest position in time. To conclude, the algorithm is becomingprogressively ineffective when ∆τ is reduced.In this paper we prove that for any Hamiltonian with discrete Hilbert space it is possible to arrange an exact MCprocess, i.e., when both the description of the trajectories in time and their actions contain no approximations at all,and which is at least 1/(t∆τ)2 times faster than the conventional scheme in accumulating statistics.First, we observe that for discrete systems like (2) any trajectory may be easily described using continuous timevariable: it is sufficient to define the state at τ = 0 and finite number of points in time {τ1, τ2, . . . , τn} where the statehas changed from some |αk−1〉 to |αk〉. Within this description the action of the trajectory is known exactly, sinceone may formally think about taking the limit ∆τ → 0 while calculating S within the standard approach. When weturn to the problem of generating new kink-antikink pairs, we formally face an incontestable obstacle: the probabilityto accept the trajectory with one extra kink-antikink pair is formally zero, Wnew/Wold ∼ (t∆τ)2 → 0, so that thealgorithm does not allow to change the number of kinks at the level of comparing two particular trajectories. This1obstacle, however, can be circumvented, if one knows an exact integrated statistics of trajectories with a given numberof kinks. One may decide then about the number of kinks on the trajectory by relating these integrated probabilities.In what follows we demonstrate explicitly how to find this integrated statistics for arbitrary discrete system in anytime window of width τo by relating it to the statistics of the biased two-level system.2. Let H0 and V be the diagonal and non-diagonal parts of the Hamiltonian, H , in a chosen representationcorresponding to the full set, {α}, of eigenstates of H0, and H0 | α 〉 = Eα | α 〉. Then statistical operator can beordinarily related to the Matsubara evolution operator, σ, in the interaction representation, i.e., we write e−βH =e−βH0σ withσ = 1−∫ β0dτ V (τ) + . . .+ (−1)m∫ β0dτm · · ·∫ τ20dτ1 V (τm) · · ·V (τ1) + . . . , (3)where V (τ) = eτH0V e−τH0 . Without loss of generality and in accordance with typical forms of Hamiltonians ofinterest, V can be written as a sum of elementary terms, Qs, whose action on any function from the set {α} resultsin another function from this set:V =∑sQs ; Qs | α 〉 = −qγα(s) | γ 〉 (γ = γ(s, α)) . (4)Since V is Hermitian, for any s in the sum (4) there exists s′ such that Qs′ = Q†s. Then we may rewrite Eq. (3) incomponents (below Eαγ ≡ Eα − Eγ):σαγ = δαγ +∑s∫ β0dτ qαγ(s)eτEαγ + . . .+∑s1,...,sm∫ β0dτm · · ·∫ τ20dτ1 qαν(sm)eτmEαν · · · qλγ(s1)eτ1Eλγ + . . . . (5)Note that there is no additional summation over the indexes of the intermediate complete sets (labeled by Greekletters) since these are defined in a unique way by configurations of (s1, s2, . . . , sm).We confine ourselves to the case of finite-range interaction, which is defined by the requirement that for each terms1 of elementary operators {Qs} there exists only a finite number of terms s2 for which the condition[Qs1(τ1), Qs2(τ2) ] = 0 (6)is not met. In the case of finite-range interaction the structure of the series (5) is drastically simplified, the simplifi-cation being of crucial importance for practical realization of our algorithm. From (6) it follows that up to irrelevantchange in indexing energies and matrix elements, one may pay no attention to the chronological order of Qs1(τ1) andQs2(τ2) in the evolution operator. This suggests representing of a general term of the series (5) in the following form.First, we introduce the notion of a “kink of kind s”, which is characterized by a time τ , a matrix element qαγ(s), anda diagonal-energy difference Eαγ . The former two we will refer to as parameters of the kink. It is essential that (i) toobtain parameters of a kink one need not know explicitly the whole state | α 〉, or | γ 〉 - local information is enough;(ii) to set a particular structure of a term in Eq. (5), including the chronological order of all noncommuting operators,it is enough to specify for each kink its associated neighbors, i.e. nearest in time noncommuting kinks.Now we are ready to introduce a stochastic process directly evaluating Eq. (5). For simplicity, we assume that allqαβ(s) are positive real numbers. [In many particular alternative cases a straightforward generalization is possible,but usually at the expense of convergence.] Summations and integrations in Eq. (5) then can be regarded, up toa normalizing factor, as an averaging over a statistics of different configurations of kinks, each configuration beingdefined by a certain number of kinks of certain kinds, their associations and particular positions in imaginary time.Our aim is to organize a stochastic process of examining this statistics by generating different kink configurations inaccordance with their weights. The process will consist of a number of independent sub-processes, responsible formodifications of particular types. The simplest sub-process is the random shift of the position of a kink in imaginarytime without interchange with associated neighbors. The weight function for this sub-process is just exp(τEαγ), τis the position of the kink, Eαγ is the corresponding energy difference. It is clear that this process is far from beingcomplete since it does not change neither the number of kinks nor their associations.The sub-process which is capable of doing so, and which will play the key part in the whole procedure, is the processof creation/annihilation of a kink - anti-kink pair. First we define the notion of the appropriate window (with respect2to kink - anti-kink pairs of the kind s - s′), as the interval of imaginary time of some chosen length τ0 where (i) thereis either one and only one kink - anti-kink pair of the kind s-s′ (case A), or no such pairs at all (case B), and (ii)there are no kinks associated with any kink of the pair. Now we notice that the total contribution of configurations ofthe class A (integrated over the positions of the pair inside the window and integrated and summed over all possibleconfigurations of the rest of the kink pattern) differs from the total contribution of the configurations of the class Bby the factorI(2)αγ (s) =∫ τa+τ0τadτ2∫ τ2τadτ1 qαγ(s′)eτ2Eαγqγα(s)eτ1Eγα , (7)where τa - is a randomly chosen left boundary of the time window. Elementary integration yieldsI(2)αγ (s) =| qγα(s) |2E2αγ[eτ0Eαγ − τ0Eαγ − 1]. (8)This allows us to weigh these two classes. Having found an appropriate window we renew its contents by the followingrule. First, we delete a pair s - s′, if there is one. Then with the probability wA we create another kink - anti-kinkpair s - s′, and with the probability wB = 1− wA leave it as it is.wA/wB = I(2)αγ (s) . (9)In the former case we also randomly choose the positions of the kink and anti-kink, in accordance with the distributionfollowing from their exponential factors e(τ2−τ1)Eαγ .It is worthnoting that the kink - anti-kink procedure actually relies on an important general fact, easily seen fromEq. (5), that the local statistics of kink - anti-kink chains, corresponding to virtual transitions between two states,| α 〉 and | γ 〉, is exactly the same as for the biased two-level system with the bias energy equal to Eαγ and tunnelingamplitude equal to qαγ(s). The unnormalized probabilities of having exactly n kink-anti-kink pairs in the appropriatewindow are given then byI(2n)αγ (s) =(| qγα(s) |2 τ20 )nn!√pi2e−xIn−1/2(x)− In+1/2(x)(2x)n−1/2, (10)where x = τ0Eαγ/2, and Iν(x) is the modified Bessel function. Therefore, in principle a more general weightingprocedure is possible, involving transformations of the kink - anti-kink chains of arbitrarily large length, which firstdeletes the whole chain of the kink - anti-kink pairs s - s′ from the selected appropriate window and then createsanother chain according to Eq.(10). This, however, can hardly have practical importance because of small probabilityto find an appropriate window of large length.Generally speaking, the reason why the kink - anti-kink process in certain cases turns out to be incomplete is thatthere may exist more than one way to get from a state | α 〉 to a state | γ 〉. Consider, for example, a typical situationcorresponding to the following two kink chainsqγλ(s2) eτ2Eγλ qλα(s1) eτ1Eλα (case C) , (11)qγν(s4) eτ4Eγν qνα(s3) eτ3Eνα (case D) . (12)To obtain relative weights of classes C and D one could proceed in direct analogy with the kink - anti-kink processes,comparing the integral contributions. It is more reasonable, however, to take advantage of the fact that the number ofkinks is the same in both cases. This allows us to take the differential limit. Namely, we compare only configurationswhen τ1 and τ3 (τ2 and τ4) vary in an infinitesimal interval dτ around τ1 = τ3 = τa (τ2 = τ4 = τb). As a result, wecome to a Metropolis-type discrete sub-process of replacing the pair (s1, s2) by the pair (s3, s4) and vice versa withprobabilities wD and wC (respectively), satisfyingwDwC=qγν(s4)qνα(s3)qγλ(s2)qλα(s1)exp [ (τb − τa)Eλν ] . (13)All the above-considered sub-processes deal with the “bulk” of the kink configuration. Finally, we introduce an“edge” sub-process, which, on one hand, changes one of the indexes of σαγ (for definiteness, γ into λ), and, on anotherhand, changes the number of kinks by one. The appropriate window for the kink s is now defined as an interval [0, τ0],3where there is only one kink (case E) or zero (case F ) kinks of the kind s, and no kinks associated with s . In closeanalogy with the kink - anti-kink sub-process, in the new configuration with the probability wE there will be a kink,and with the probability wF = 1− wE it will be absentwEwF= qγλ(s)∫ τ00dτ eτEγλ . (14)The position of the kink in the former case is chosen in accordance with the factor eτEγλ .To illustrate the above-described approach, let us formulate it for bosonic Hubbard models [and closely relatedto them (by Holstein-Primakoff transformation) spin lattices]. In the site representation, the diagonal part of theHamiltonian, H0, involves interparticle interactions and an external potential, if any. The nondiagonal part, V ,corresponds to the intersite hopping, and terms Qs describe hopping from one particular site to another. The mosttransparent case is a hard-core one- dimensional system, especially if the quantities of interest are just potential andkinetic energies. As is easily seen, in this case the kink - anti-kink sub-process is enough to reproduce the statistics. Toenhance the convergence, however, it is useful to utilize also the sub-process of the random shift of the kink positionand the discrete sub-process of the kink - anti-kink pair reversal, i.e., Eq. (13) with s2 = s′1, s4 = s1, s3 = s′1. Ind ≥ 2 the discrete two-kink sub-process Eq. (13) is essential to reproduce all possible trajectories of a particle. If oneis interested in non-local (both in space and imaginary time) off-diagonal correlators or, say, the system behavior ina ring geometry with a global gauge phase, he should take advantage of the edge sub-process Eq. (14).It should be noted that the requirement that the interaction be of finite-range character is not of fundamentalnature. In principle our process is applicable to interactions of arbitrarily long range, and, in particular, may workin momentum representation - if this is crucial for this or that reason. The only limitation which arises here is thesmall size of the appropriate window τ0.We note also, that our approach might prove to be the most effective and very accurate method for MC simulationsof continuous systems if one first approximate them by lattice models. For one thing, in the case of harmonic oscillatorthe corrections for discreteness are known to be extremely small.3. All the above mentioned ideas were realized in a code allowing calculations of the one-particle properties for theHamiltonian (2). The results were compared with those obtained by the standard Monte Carlo world-line algorithm,based on finite ∆τ [11]. We found that, e.g., for the periodic potential Ui = (−1)iU and discrete harmonic oscillatorUi = Ui2 the discrete-time code with ∆τ = 0.1/t was more then two orders slower in accumulating statistics (orrequired > 102 longer computation time in order to achieve the same accuracy) than ours. This result is hardlysurprising, because, as discussed above, the “bottleneck” in the standard procedure is connected with the unlikelyprocess of the kink-anti-kink pair creation (with probability (t∆τ)2 ∼ 10−2). No matter how simple are our testmodels their description, nevertheless, contains all the essential features of simulation of much more complicatedobjects.We are grateful to V. Kashurnikov for his permanent interest and numerous discussions. This work was supportedby Grant No. INTAS-93-2834-ext [of the European Community] and partially by the Russian Foundation for BasicResearch [Grant No. 95-02-06191a].[1] J.E. Hirsch, D.J. Scalapino, R.L.Sugar, and R. Blankenbecler, Phys. Rev. Lett., 47, 1628 (1981); J.E. Hirsch, R.L.Sugar,D.J. Scalapino, and R. Blankenbecler, Phys. Rev. B, 26, 5033 (1982).[2] R. Blankenbecler, D.J. Scalapino, and R.L.Sugar, Phys. Rev. B, 24, 2278 (1981); ibid. 24, 4295 (1981); J.E. Hirsch, Phys.Rev. B, 31, 4403 (1985).[3] A. Lagendijk and B. De Raedt, Phys. Rev. Lett., 49, 602 (1982); H. De Raedt and A. Lagendijk, Phys. Rep. 127, 233(1985), and references therein.[4] E.L. Pollock and D.M. Ceperley, Phys. Rev. B, 36, 8343 (1987).[5] W. Krauth, N. Trivedi, and D. Ceperly, Phys. Rev. Lett., 67, 2307 (1991).[6] G.G. Batrouni, R.T. Scalettar, and G.T. Zimanyi, Phys. Rev. Lett., 65, 1765 (1990); ibid. 66, 3144 (1991); G.G. Batrouni,R.T. Scalettar, Phys. Rev. B, 46, 9051 (1992).[7] A.F. Elesin and V.A. Kashurnikov, Zh. Exp. Teor. Fiz., 106, 1773 (1994); V.A. Kashurnikov, Phys. Rev. B, 53, 5932(1996).[8] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).[9] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth et al., J. Chem. Phys. 21, 1087 (1953).[10] R.M. Fye, Phys. Rev. B, 33, 6271 (1986).[11] We thank V.A. Kashurnikov and A.V. Krasavin for providing us with the world-line Monte Carlo code, based on thechecker-board algorithm [1].4

Exact quantum Monte Carlo process for the statistics of discrete systems

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Exact Quantum Monte Carlo Process for the Statistics of Discrete Systems

Exact Quantum Monte Carlo Process for the Statistics of Discrete Systems

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