Despite several attempts, no efficient cluster algorithm has been constructed
for CP(N-1) models in the standard Wilson formulation of lattice field theory.
In fact, there is a no-go theorem that prevents the construction of an
efficient Wolff-type embedding algorithm. In this paper, we construct an
efficient cluster algorithm for ferromagnetic SU(N)-symmetric quantum spin
systems. Such systems provide a regularization for CP(N-1) models in the
framework of D-theory. We present detailed studies of the autocorrelations and
find a dynamical critical exponent that is consistent with z = 0.Comment: 14 pages, 3 figure

Affleck

Azcoiti

B.B. Beard

Beard

Bietenholz

Brower

Burkhalter

Caracciolo

Chandrasekharan

D'Adda

Eichenherr

Evertz

Golo

Haldane

Harada

Hasenbusch

Imachi

Jansen

Kawashima

M. Pepe

Novikov

S. Riederer

Seiberg

Swendsen

Troyer

U.-J. Wiese

Wess

Wiese

Witten

Wolff

English

arXiv

Beard, Bernard

Pepe, Michele

Riederer, Stéphane

Wiese, Uwe-Jens

Bern Open Repository and Information System (BORIS)

PoS(LAT2005)113An Efficient Cluster Algorithm for CP(N-1) ModelsMichele Pepe, Stéphane Riederer∗†, Uwe-Jens WieseBern University, SwitzerlandE-mail: pepe@itp.unibe.ch, riederer@itp.unibe.ch, wiese@itp.unibe.chBernard B. BeardChristian Brothers University, Memphis, USAE-mail: bbbeard@cbu.eduWe construct an efficient cluster algorithm for ferromagnetic SU(N)-symmetric quantum spinsystems. Such systems provide a new regularization for CP(N− 1) models in the framework ofD-theory, which is an alternative non-perturbative approach to quantum field theory formulatedin terms of discrete quantum variables instead of classical fields. Despite several attempts, noefficient cluster algorithm has been constructed for CP(N−1) models in the standard formulationof lattice field theory. In fact, there is even a no-go theorem that prevents the construction of anefficient Wolff-type embedding algorithm. We present various simulations for different correla-tion lengths, couplings and lattice sizes. We have simulated correlation lengths up to 250 latticespacings on lattices as large as 640×640 and we detect no evidence for critical slowing down.XXIIIrd International Symposium on Lattice Field Theory25-30 July 2005Trinity College, Dublin, Ireland∗Speaker.†This work was supported in part by the Swiss Nationalfonds for Scientific Research.c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/PoS(LAT2005)113Efficient Cluster Algorithms for CP(N-1) Models Stéphane Riederer1. Standard Formulation of CP(N−1) ModelsThe manifold CP(N−1) = SU(N)/U(N−1) is a (2N−2)-dimensional coset space relevantin the context of the spontaneous breakdown of an SU(N) symmetry to a U(N− 1) subgroup. Inparticular, in more than two space-time dimensions (d > 2) the corresponding Goldstone bosonsare described by N×N matrix-valued fields P(x) ∈CP(N−1) which obeyP(x)2 = P(x), P(x)† = P(x), TrP(x) = 1. (1.1)For d = 2 the Hohenberg-Mermin-Wagner-Coleman theorem implies that the SU(N) symmetrycannot break spontaneously. Correspondingly, similar to 4-dimensional non-Abelian gauge the-ories, the fields P(x) develop a mass-gap nonperturbatively. Motivated by these observations,D’Adda, Di Vecchia, and Lüscher [1] introduced CP(N − 1) models as interesting toy modelsfor QCD. The corresponding Euclidean action is given byS[P] =∫d2x 1g2Tr[∂µP∂µP], (1.2)where g2 is the dimensionless coupling constant. Note that this action is invariant under globalΩ ∈ SU(N) transformationsP(x)′ = ΩP(x)Ω†, (1.3)and under charge conjugation C which acts as CP(x) = P(x)∗.2. D-Theory Formulation of CP(N−1) ModelsIn this section we describe an alternative formulation of field theory in which the 2-dimensionalCP(N − 1) model emerges from the dimensional reduction of discrete variables — in this caseSU(N) quantum spins in (2+ 1) space-time dimensions. The dimensional reduction of discretevariables is the key ingredient of D-theory, which provides an alternative nonperturbative regular-ization of field theory. In D-theory we start from a ferromagnetic system of SU(N) quantum spinslocated at the sites x of a 2-dimensional periodic square lattice. The SU(N) spins are representedby Hermitean operators T ax = 12λ ax (Gell-Mann matrices for the triplet representation of SU(3)) thatgenerate the group SU(N) and thus obey[T ax ,Tby ] = iδxy fabcT cx , Tr(T ax T by ) =12δxyδab. (2.1)In principle, these generators can be taken in any irreducible representation of SU(N). However,as we will see later, not all representations lead to spontaneous symmetry breaking from SU(N) toU(N−1) and thus to CP(N−1) models. The Hamilton operator for an SU(N) ferromagnet takesthe formH =−J ∑x,iT ax Tax+ˆi, (2.2)where J > 0 is the exchange coupling. By construction, the Hamilton operator is invariant underthe global SU(N) symmetry, i.e. it commutes with the total spin given byT a = ∑xT ax . (2.3)113 / 2PoS(LAT2005)113Efficient Cluster Algorithms for CP(N-1) Models Stéphane RiedererThe Hamiltonian H describes the evolution of the quantum spin system in an extra dimensionof finite extent β . In D-theory this extra dimension is not the Euclidean time of the target theory,which is part of the 2-dimensional lattice. Instead, it is an additional compactified dimension whichultimately disappears via dimensional reduction. The quantum partition functionZ = Trexp(−βH) (2.4)(with the trace extending over the Hilbert space) gives rise to periodic boundary conditions in theextra dimension.The ground state of the quantum spin system has a broken global SU(N) symmetry. Thechoice of the SU(N) representation determines the symmetry breaking pattern. We choose a totallysymmetric SU(N) representation corresponding to a Young tableau with a single row containing nboxes. It is easy to construct the ground states of the SU(N) ferromagnet, and one finds spontaneoussymmetry breaking from SU(N) to U(N−1). Consequently, there are (N2−1)−(N−1)2 = 2N−2massless Goldstone bosons described by fields P(x) in the coset space SU(N)/U(N−1) =CP(N−1). In the leading order of chiral perturbation theory the Euclidean action for the Goldstone bosonfields is given byS[P] =∫ β0dt∫d2x Tr[ρs∂µP∂µP−2na2∫ 10dτ P∂tP∂τP]. (2.5)Here ρs is the spin stiffness, which is analogous to the pion decay constant in QCD. The secondterm in eq.(2.5) is a Wess-Zumino-Witten term which involves an integral over an interpolationparameter τ .For β = ∞ the system then has a spontaneously broken global symmetry and thus masslessGoldstone bosons. However, as soon as β becomes finite, due to the Hohenberg-Mermin-Wagner-Coleman theorem, the symmetry can no longer be broken, and, consequently, the Goldstone bosonspick up a small mass m nonperturbatively. As a result, the corresponding correlation length ξ =1/m becomes finite and the SU(N) symmetry is restored over that length scale. The question arisesif ξ is bigger or smaller than the extent β of the extra dimension. When ξ  β the Goldstoneboson field is essentially constant along the extra dimension and the system undergoes dimensionalreduction. Since the Wess-Zumino-Witten term vanishes for field constant in t, after dimensionalreduction the action reduces toS[P] = βρs∫d2x Tr[∂µP∂µP], (2.6)which is just the action of the 2-d target CP(N−1) model. The coupling constant of the 2-d modelis determined by the extent of the extra dimension and is given by1g2= βρs. (2.7)Due to asymptotic freedom of the 2-d CP(N − 1) model, for small g2 the correlation length isexponentially large, i.e.ξ ∝ exp(4piβρs/N). (2.8)Here N/4pi is the 1-loop coefficient of the perturbative β -function. Indeed, one sees that ξ  βas long as β itself is sufficiently large. In particular, somewhat counter-intuitively, dimensional113 / 3PoS(LAT2005)113Efficient Cluster Algorithms for CP(N-1) Models Stéphane Riedererreduction happens in the large β limit because ξ then grows exponentially. In D-theory one ap-proaches the continuum limit not by varying a bare coupling constant but by increasing the extentβ of the extra dimension. This mechanism of dimensional reduction of discrete variables is genericand occurs in all asymptotically free D-theory models [2, 3]. It should be noted that (just like inthe standard approach) no fine-tuning is needed to approach the continuum limit.3. Path Integral Representation of SU(N) Quantum Spin SystemsLet us construct a path integral representation for the partition function Z of the SU(N) quan-tum spin ferromagnet introduced above. In an intermediate step we introduce a lattice in the Eu-clidean time direction, using a Trotter decomposition of the Hamiltonian. However, since we aredealing with discrete variables, the path integral is completely well-defined even in continuousEuclidean time. Also the cluster algorithm to be described in the following section can operatedirectly in the Euclidean time continuum [4]. Hence, the final results are completely independentof the Trotter decomposition. In 2 spatial dimensions (with an even extent) we decompose theHamilton operator into 4 termsH = H1 +H2 +H3 +H4, (3.1)withH1,2 = ∑x=(x1,x2)xievenhx,i, H3,4 = ∑x=(x1,x2)xioddhx,i. (3.2)The individual contributionshx,i =−J T ax T ax+ˆi, (3.3)to a given Hi commute with each other, but two different Hi do not commute. Using the Trotterformula, the partition function then takes the formZ= limM→∞Tr{exp(−εH1)exp(−εH2)exp(−εH3)exp(−εH4)}M . (3.4)We have introduced M Euclidean time-slices with ε = β/M being the lattice spacing in the Eu-clidean time direction. Inserting complete sets of spin states q ∈ {u,d,s, ...} the partition functiontakes the formZ = ∑[q]exp(−S[q]). (3.5)The sum extends over configurations [q] of spins q(x, t) on a (2+1)-dimensional space-time latticeof points (x, t). The Boltzmann factor is a product of space-time plaquette contributions withexp(−s[u,u,u,u]) = exp(−s[d,d,d,d]) = 1,exp(−s[u,d,u,d]) = exp(−s[d,u,d,u]) = 12[1+ exp(−εJ)],exp(−s[u,d,d,u]) = exp(−s[d,u,u,d]) = 12[1− exp(−εJ)]. (3.6)In these expressions the flavors u and d can be permuted to other values. All the other Boltzmannfactors are zero, which implies several constraints on allowed configurations.113 / 4PoS(LAT2005)113Efficient Cluster Algorithms for CP(N-1) Models Stéphane Riederer4. Cluster Algorithm for SU(N) Quantum FerromagnetsLet us now discuss the cluster algorithm for the SU(N) quantum ferromagnet. Just like theoriginal SU(2) loop-cluster algorithm [5, 6], the SU(N) cluster algorithm builds a closed loopconnecting neighboring lattice points with the spin in the same quantum state, and then changesthe state of all those spins to a different randomly chosen common value. To begin cluster growth,an initial lattice point (x, t) is picked at random. The spin located at that point participates intwo plaquette interactions, one before and one after t. One picks one interaction arbitrarily andconsiders the states of the other spins on that plaquette. One of the corners of this interactionplaquette will be the next point on the loop. For configurations C1 = [u,d,u,d] or [d,u,d,u] the nextpoint is the time-like neighbor of (x, t) on the plaquette, while for configurations C2 = [u,d,d,u]or [d,u,u,d] the next point is the diagonal neighbor. If the states are all the same, i.e. for C3 =[u,u,u,u] or [d,d,d,d], with probabilityp =12[1+ exp(−εJ)] (4.1)the next point on the loop is again the time-like neighbor, and with probability (1− p) it is thediagonal neighbor. The next point on the loop belongs to another interaction plaquette on whichthe same process is repeated. In this way the loop grows until it finally closes.5. Critical slowing down in the continuum limitIn order to determine the efficiency of this algorithm one has to study its critical slowing downwhen one approaches the continuum limit. We have used a multi-cluster algorithm for an SU(3)quantum ferromagnet which corresponds to a CP(2) model. As an observable, we have chosen theuniform magnetization which gives the cleanest signal,M = ∑x,t(δq(x,t),u−δq(x,t),d). (5.1)The autocorrelation time τ of the magnetization is determined from the exponential fall-off of theautocorrelation function. The simulations have been performed at fixed ratio ξ/L≈ 2.5, for latticesizes L/a = 20, 40, 80, 160, 320, 640 and the corresponding correlation lengths ξ/a = 8.87(1),16.76(1), 32.26(3), 64.6(1), 123.4(2), 253(1). Remarkably, the autocorrelation time doesn’t changewhen one varies the size of the system and the correlation length and stays close to τ ≈ 1 sweep.This is a strong indication for an almost perfect algorithm where the critical slowing down is com-pletely eliminated.6. ConclusionsDue to a no-go theorem [7], so far no efficient cluster algorithm has been developed forCP(N − 1) models in the usual Wilson formulation. In the D-theory formulation, one has beenable to perform simulations using a multi-cluster algorithm for large correlation lengths and with acorresponding autocorrelation time of about one sweep. Remarkably, there is almost no variation113 / 5PoS(LAT2005)113Efficient Cluster Algorithms for CP(N-1) Models Stéphane Riedererof the autocorrelation time when one spans a factor of about 30 in the correlation length. The crit-ical slowing down of the algorithm in the continuum limit is hence completely eliminated. Theseresults can be compared to the ones obtained with the efficient multigrid algorithm [8]. Our methodhas the advantage to obtain autocorrelation times more than 20 times smaller for similar correlationlengths, the multi-cluster algorithm is in addition straightforward to implement.With the D-theory regularization, it has recently also been possible to simulate CP(N−1) modelsat non-trivial θ -vacuum angle [9] which is normally impossible due to a severe sign problem.References[1] A. D’Adda, P. Di Vecchia, and M. Lüscher, An 1/N expandable series of nonlinear sigma model withinstantons, Nucl. Phys. B146 (1978) 63.[2] R. Brower, S. Chandrasekharan, and U.-J. Wiese, QCD as a quantum link model, Phys. Rev. D60(1999) 094502 [hep-th/9704106].[3] R. Brower, S. Chandrasekharan, S. Riederer, and U.-J. Wiese, D-Theory: Field quantization bydimensional reduction of discrete variables, Nucl. Phys. B693 (2004) 149 [hep-lat/0309182].[4] B. B. Beard, U.-J. Wiese, Simulations of discrete quantum systems in continuous Euclidean time,Phys. Rev. Lett. 77 (1996) 5130 [cond-mat/9602164].[5] H. G. Evertz, G. Lana, and M. Marcu, Cluster algorithm for vertex models, Phys. Rev. Lett. 70 (1993)875-879 [cond-mat/9211006].[6] U.-J. Wiese and H.-P. Ying, A determination of the low-energy parameters of the 2-d Heisenbergantiferromagnet, Z. Phys. B93 (1994) 147 [cond-mat/9212006].[7] S. Caracciolo, R. G. Edwards, A. Pelissetto, A. D. Sokal, Wolff type embedding algorithm for generalnonlinear sigma models, Nucl. Phys. B403 (1993) 475-541 [hep-lat/9205005].[8] M. Hasenbusch and S. Meyer, Testing accelerated algorithms in the lattice CP(3) model, Phys. Rev.D45 (1992) 4376-4380.[9] B.B. Beard, M. Pepe, S. Riederer, U.-J. Wiese, Study of CP(N−1) theta-vacua by cluster-simulationof SU(N) quantum spin ladders, Phys. Rev. Lett. 94 (2005) 010603 [hep-lat/0406040].113 / 6

Efficient Cluster Algorithm for CP(N-1) Models

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