Bold Diagrammatic Monte Carlo technique for frustrated spin systems

S. A. Kulagin1,2, N. Prokof’ev1,3, O. A. Starykh4, B. Svistunov1,3, and C. N. Varney1

1Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA2Institute for Nuclear Research of Russian Academy of Sciences, 117312 Moscow, Russia

3Russian Research Center “Kurchatov Institute”, 123182 Moscow, Russia4Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

(Dated: November 16, 2012)

Using fermionic representation of spin degrees of freedom within the Popov-Fedotov approachwe develop an algorithm for Monte Carlo sampling of skeleton Feynman diagrams for Heisenbergtype models. Our scheme works without modifications for any dimension of space, lattice geome-try, and interaction range, i.e. it is suitable for dealing with frustrated magnetic systems at finitetemperature. As a practical application we compute uniform magnetic susceptibility of the anti-ferromagnetic Heisenberg model on the triangular lattice and compare our results with the bestavailable high-temperature expansions. We also report results for the momentum-dependence of thestatic magnetic susceptibility throughout the Brillouin zone.

PACS numbers: 02.70.Ss, 05.10.Ln

I. INTRODUCTION

Properties of geometrically frustrated spin systems invarious dimensions, geometries, and temperature regimesare at the heart of modern condensed matter physics.Here, frustration is a technical term which refers to thepresence of competing forces that cannot be simultane-ously satisfied. In numerous quantum antiferromagnetsfrustration often has a simple geometric origin. Localizedspins on two- and three-dimensional lattices with triangu-lar motifs, such as planar triangular antiferromagnet andthree-dimensional pyrochlore antiferromagnets, cannotassume energetically favorable antiparallel orientation.Of three spins forming a minimal triangle, and interact-ing via simple antiferromagnetic pair-wise exchange in-teraction, only two can be made antiparallel, leaving thethird one frustrated. In case of classical Ising spins, whichcan point up or down with respect to some axis, this leadsto an extensive ground-state degeneracy: for example, ina system of N Ising spins on a triangular lattice thereare ΩN = e0.323N configurations having the same (min-imal) energy. On a three-dimensional pyrochlore latticeof site-sharing tetrahedra, the (effectively) Ising spins ofDy2Ti2O7, Ho2Ti2O7 and Ho2Sn2O7 realize1 fascinat-ing spin ice physics2 where strong local ice rules (for anygiven tetrahedron, two of its spins must point in, and theother two - out) enforce long-ranged power-law correla-tions between spins3, in effect realizing artificial magneticfield and fractionally charged magnetic monopoles4!

Quantum spins can exploit this extensive degeneracyvia quantum-mechanical coupling between different con-figurations – their wave function can be thought of as alinear superposition of all degenerate microstates repre-sented by classical patterns of up- and down-spins. In thecase of strong coupling we may arrive at a quantum spinliquid (QSL)5,6 state in which spins never settle in oneparticular configuration and continue their explorationforever. It is clear that such a state encodes highly non-trivial correlations between different spins when flipping

of one spin induces that of its neighbors so that as awhole the spin system remains in the lowest-energy man-ifold. Extensive experimental7–20 and theoretical21–30

search for materials and models which may realize thisintriguing QSL state constitutes one of the main topicsof the quantum spin physics. Currently there are severalintensely researched materials that hold promise of real-izing the elusive spin liquid state. Among them, we men-tion two-dimensional spin-1/2 organic triangular antifer-romagnets EtMe3Sb[Pd(dmit)2]2

10,11,19 and κ-(BEDT-TTF)2Cu2(CN)3

8,19, spin-1 material NiGa2S416, a se-

ries of inorganic quasi-two-dimensional kagome latticeantiferromagnets: herbersmithite ZnCu3(OH)6Cl2

12,15,volborthite Cu3V2O7(OH)2·2H2O7,13, and vesignieiteBaCu3V2O8(OH)2

14,18, and a three-dimensional hyperk-agome antiferromagnet Na4Ir3O8

9.

Taking the system of frustrated spins to a finite tem-perature, where all experiments are done, adds thermalrandomness to the picture. At a finite temperature Teven classical spins can explore different microstates fromthe lowest-energy manifold. This too leads to a stronglycorrelated (although not necessarily phase-coherent, asin the case of quantum spins at T = 0) motion of spinswhich is often described by a term “cooperative param-agnet”. Even if the ground state of spin system is not atrue spin liquid, but instead is one of the many possibleordered states, the spins will (thermally) disorder at suf-ficiently high temperature, T ≥ T0, where T0 stands forthe ordering temperature. In the usual, non-frustratedmagnets the ordering temperature is determined by theexchange interaction energy J and coordination numberof the lattice z, T0 ∼ zS(S + 1)J . Quite generally, itis of the order of the Curie-Weiss temperature θcw whichis easily determined experimentally via high-temperaturebehavior of the spin susceptibility χ ∝ (T − θcw)−1. Inantiferromagnets θcw is negative, and, in the absence offrustration, its absolute value sets the scale at whichcorrelations between spins become pronounced. Thus,T0 ∼ |θcw|. Frustrated magnets are very different as

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there T0 |θcw| : despite experiencing strong interac-tions with each other, the spins can not “agree” on oneparticular pattern which would satisfy them all. In thecase of a true spin liquid T0 = 0 and the order neverarrives. In the majority of studied situations the orderdoes take place, T0 > 0, but only at a temperature muchlower than the naıve estimate provided by |θcw|. Thisresults in a wide temperature interval T0 < T < |θcw|where the spins are strongly interacting but remain ina disordered cooperative paramagnet state. In fact, theexistence of such temperature (and energy) window rep-resents a defining feature of the frustrated magnet, asargued by Ramirez31 who introduced the frustration pa-rameter f = |θcw|/T0 (it is not uncommon to find a sit-uation with f ∼ 100 or greater).

Strong suppression of the ordering temperature can bealso the consequence of close proximity to the quantumcritical point, separating the spin-disordered state (suchas a putative QSL) from the more usual ordered one.It is well established now32 that the finite-temperatureregion above the quantum critical point, known as thequantum-critical region, is as informative of the quantumstate of the many-body system as the unreachable T = 0ground state. The quantum-critical scaling of, say, thedynamic spin susceptibility contains information on thespin correlation length, dynamic exponent z and othercritical exponents characterizing spin systems of differentsymmetries and dimensions.

Importantly, there is a large number of high-precisionprobes—neutron and X-ray scattering, nuclear magneticresonance, muon spin rotation, susceptibility, magneti-zation and specific heat measurements—which allow usto address various aspects of strange and conflicting be-havior of frustrated quantum magnets experimentally ina wide range of energies and temperatures. Given thatin many important cases the correlated spin-liquid re-gion T0 < T < |θcw| occupies most of the experimentallyaccessible temperature interval, unbiased understandingof correlations and dynamics in this regime becomes themajor theoretical task.

Theoretical understanding of frustrated magnetism atfinite temperature is severely limited by the lack of nat-ural small parameter(s). As a result, possible analyti-cal approaches require one to study suitably ‘deformed’models, such as, for example, very popular and well de-veloped large-component (large-N) version of the Heisen-berg spin model on frustrated lattices. The small pa-rameter is then provided by 1/N , expansion in powersof which (about the N = ∞ limit) controls the calcu-lation. However the physical limit of S = 1/2 SU(2)lattice spins corresponds to rather small N : 3 in thecase of O(N) generalization and 1 in the case of Sp(N).Whether or not continuation of the results from N =∞to the physical value is reliable remains an open (andcase sensitive) question. Other popular ‘deformations’include SU(2)-to-U(1) symmetry reduction(s) and quan-tum dimer model approaches. While very insightful andinteresting in their own, applicability of these ‘modifica-

tions’ to the original problem is always an issue. Whileanalytical approaches are extremely useful in providingus with qualitative physical insights and understanding,in frustrated magnets they often fall short of quantitativedescription desired by experimentalists. Numerical ap-proaches are limited as well. Exact diagonalizations arerestricted to small systems (about 40 sites at most) dueto the exponentially large Hilbert space while standardquantum Monte Carlo techniques usually suffer from theinfamous ‘sign problem’. Powerful series expansion meth-ods often start to diverge in the most interesting regimeT0 < T < |θcw|. Variational tensor-network type meth-ods are also suffering from finite-size limitations and aremostly limited to ground state properties.

In this article we combine the most versatile theoreticaltool, Feynman diagrammatics, with the power of MonteCarlo sampling of complex configuration spaces. Thesimplest way of arriving at the diagrammatic techniquefor spins is to represent them by auxiliary fermions withimaginary chemical potential. This trick was introducedby Popov and Fedotov for spin-1/2 systems in Ref. 33.The ultimate strength of the diagrammatic approach ascompared, e.g. with high-temperature or strong couplingexpansions, comes from its self-consistent (skeleton) for-mulation with automatic summation of certain classesof graphs up to infinite order. This lead to better con-vergence properties and the possibility of obtaining re-liable results in the strong coupling regime. It turnsout that skeleton formulations can be easily implementedwithin the sampling protocols leading to the so-calledBold Diagrammatic Monte Carlo (BDMC)34 which ob-tains physical answers by computing contributions frommillions of graphs and extrapolates them to the infinitediagram order. Recently, this method was successfullyapplied to the normal state of the Fermi-Hubbard modelat moderate interaction strength35 and the strongly cor-related system of unitary fermions (the so-called BCS-BEC crossover problem)36. It was, however, never im-plemented for models of quantum magnetism, which, atleast at the formal level, can be also viewed as a fermionicsystem with strong correlations. Here we present the firstattempt to achieve an accurate theoretical description ofthe correlated paramagnetic regime within the BDMCframework.

In what follows in Sec. II we consider a spin-1/2 Heisen-berg type model at finite temperature and its auxiliaryfermion version which admits the standard diagrammaticexpansion. In Sec. III we formulate principles of theBDMC technique and the specific self-consistent schemefor dealing with skeleton diagrams based on fully dressedlines. We proceed with detailed description of the wormalgorithm for efficient sampling of the resulting config-uration space (updates, counters, and data processing)in Sec. IV. Our results for the triangular lattice anti-ferromagnet are presented, discussed, and compared tothe best available finite temperature simulations basedon numerical linked-cluster (NLC) expansions37 and ex-trapolations of high-temperature series 38 in Sec. V. We

3

conclude with broader implications of this work and fur-ther developments in Sec. VI.

II. MODEL AND ITS ‘BOLD-LINE’DIAGRAMMATIC EXPANSION

Consider the standard Heisenberg model

H =∑i,j

Jij ~Si · ~Sj , (1)

where ~S are quantum spin-1/2 operators. The dimensionof space, lattice geometry, and interaction range are as-sumed to be arbitrary. Now, the idea is to replace spindegrees of freedom with fermionic ones:

~Si →1

2

∑α,β

f†iα~σαβfiβ . (2)

Here fiβ is the second quantized operator annihilatinga fermion with spin projection β = ±1 on site i, and~σ are the Pauli matrices. As a result, we convert cou-pling between spins into the standard two-body inter-

action term Jij,αβγδ f†iαfiβf

†jγfjδ with matrix element

Jij,αβγδ = (1/4)Jij~σαβ · ~σγδ. By doing so we also in-crease the Hilbert space on every site from 2 to 4 byadding non-physical states with zero and two fermions.

The benefit of the auxiliary fermion representationdoes not require explanation: once the spin model ismapped onto a familiar problem of interacting fermionsone can employ numerous diagrammatic tricks to solve it.However, this raises an issue of eliminating contributionsfrom unphysical states to the answer in a manner con-sistent with the diagrammatic technique. Remarkably,there is a very simple way to achieve the goal by addingthe chemical potential term to the fermionic Hamilto-nian33 (see also Ref. 39)

HF =∑

ij,αβγδ

Jij,αβγδ f†iαfiβf

†jγfjδ − µ

∑i

(ni − 1) , (3)

with complex µ = −iπT/2 and ni =∑α f†iαfiα. The

added term commutes with the original Hamiltonianand has no effect on properties of the physical subspaceni = 1 whatsoever. Moreover, the grand canonicalpartition functions and spin-spin correlation functions ofthe original spin model and its fermionic version are alsoidentical because (i) physical and non-physical sites de-couple in the trace and (ii) the trace over non-physicalstates yields identical zero on every site. As a result, wearrive at a rather standard Hamiltonian for fermions in-teracting through two-body terms. Complex value of thechemical potential is essentially a zero price to pay forthe luxury of having the diagrammatic technique.

Formally, the entire setup is similar to the fractionalquantum Hall effect system because the non-interacting

part of Eq. (3) describes particles with zero dispersion re-lation, i.e. we start building the solution from the degen-erate manifold. There is one important difference though:in conventional fermionic systems both the Green’s func-tion and the spin-spin (or density-density) correlationfunction contain direct physical information about thesystem. In spin systems the Green’s function is ratheran auxiliary object which always remains localized on asingle site. This does not imply any pathological behav-ior yet since the physical degrees of freedom are spins,i.e. bilinear combinations of fermionic operators. Corre-spondingly, the main object of interest is not the Green’sfunction of the system but the spin-spin correlation func-tion, or magnetic susceptibility:

χ(i, j, τ) = 〈TτSzi (0)Szj (τ)〉 =1

3〈Tτ ~Si(0) · ~Sj(τ)〉 , (4)

where Tτ stands for the imaginary time ordering opera-tor.

To simplify the presentation below we consider latticeswith one atom per unit cell and make use of the latticetranslation invariance; otherwise one would need to keepan index enumerating different sites in the unit cell. Forthe same reason we do not place the system into the ex-ternal magnetic field to preserve the symmetry betweenup- and down-spins; in the presence of the magnetic fieldH0 one has to add spin-dependent real part to the chem-ical potential µ→ µα = −iπT/2− αH0 and take propercare of the spin index.

The diagrammatic technique itself for Eq. (3) is ab-solutely standard40. The perturbative diagrams areexpressed in terms of the non-interacting, or ’bare’,

Green’s functions (particle propagators), G(0)αβ(i, τ) ≡

δi,0δα,βG(0)(τ), and two-body interaction lines Ji−j,αβγδ.

The non-interacting Green’s function on a lattice in theimaginary time representation reads (below the Boltz-mann constant kB = 1)

G(0)(τ > 0) = − eµτ

1 + exp(µ/T )=

eµτ

i− 1, (5)

with conventional anti-periodic boundary conditionsG(0)(τ < 0) = −G(0)(1/T + τ). In what follows we com-pletely suppress the site index for purely local quantities.For the Heisenberg model the dependence of the interac-tion line on spin indexes is rather simple

Ji−j,αβγδ =Ji−j

4Mαβγδ ,

Mαβγδ = αγ δα,βδγ,δ + 2 δδ,αδγ,−αδβ,−α , (6)

The first term describes diagonal coupling between thespin densities on sites i and j, while the second spin-flipterm exchanges spin values, see Fig. 1. The magnitudeof the spin-flip process is fixed by the SU(2) symmetryof the problem. Even when interactions are screened bymany-body effects, see below, the retarded interactionhas exactly the same dependence on spin indexes. Anarbitrary perturbative diagram of order n contributing

4

to, say, the free-energy is obtained by (i) placing graph-ical elements depicted in Fig. 1 with some space/timevariables i, j, and τ ∈ (0, 1/T ) and connecting incomingand outgoing propagators with the same spin and siteindex to each other in such a way that all points in theresulting graph are connected by some path.

1( , )i

2( , )j

2 1( )4 i jJ

2 1

1( )

2 i jJ

Diagonal Flip‐flop

2( , )j

1( , )i

Spin-flip

FIG. 1. (Color online) Graphic representation of allowedinteraction processes for the Heisenberg model.

III. BOLD DIAGRAMMATIC MONTE CARLOSCHEME

The unique feature of diagrammatic expansions forpropagators is that there are no numerical coefficientsin the diagram weight depending on the diagram orderor structure (this is not true for other well-known seriessuch as virial, high-temperature, linked-cluster, strong-coupling, etc. expansions). This leads to the diagram-matic technique when certain infinite sets of diagrams,e.g. in the form of geometric series, are easily dealt withby algebraic means or reduced to self-consistently definedintegral equations.

In the skeleton technique, the diagrams are classifiedaccording to some rule which eliminates the need forcomputing repeated blocks of diagrams. In the simplestscheme which is used in this paper, one identifies theproper self-energy blocks which consist of diagrams whereall vertexes (points where two particle propagators andthe interaction line meet) remain connected by some pathwhen one removes any two lines of the same kind: twopropagator lines with the same spin index or two inter-action lines. We will refer to this set of diagrams asirreducible. The omitted diagrams are fully accountedfor by replacing bare propagators and interaction linesin the proper self-energy blocks with exact propagators,G(τ), and screened interactions, W (r, τ)M . The result-ing formulation is self-consistent and highly non-linearsince G and W depend on proper self-energies throughthe Dyson type equations. If Σ is the proper self-energyfor the particle propagator, and Π is an analogous quan-tity for the interaction line (better known as polarizationoperator) then (in Fourier representation (r, τ)→ (q, ωm)

for space-time variables)

G(m) =G(0)(m)

1−G(0)(m)Σ(m),

W (q,m) =J(q)

4− J(q)Π(q,m)≡ J(q)

4+ W (q,m) . (7)

where J(q) =∑r e

iqrJ(r). Due to fermionic/bosonic na-ture of propagators G/W we have different definitionsof the Matsubara frequency here, ωm = 2πT (m + 1/2)for (G,Σ) and ωm = 2πTm for (W,Π). Note also thatwe split the W function into two parts by separatingout the original coupling. This is done for technical rea-sons explained below; here we simply point out that inthe imaginary time domain this is equivalent to pay-ing special attention to the δ-functional contribution,W (q, τ) = 1

4J(q) δ(τ) + W (q, τ). Correspondingly, forgraphical representation of the diagrams we use wavylines for W and dashed vertical lines for bare coupling.The corresponding (rather standard in many-body the-ory) G2W -skeleton scheme is illustrated in Fig. 2. Themagnetic susceptibility Eq. (4) is directly related to thepolarization operator appearing in Eq. (7) and Fig. 2

χ(q, n) =Π(q, n)

4− J(q)Π(q, n). (8)

ˆ

ˆ

G

/4J

...

...

G

G

W

FIG. 2. (Color online) Typical low-order diagrams contribut-ing to the particle self-energy and polarization operator withinthe G2W -skeleton scheme (red color denotes spin-exchangecoupling). Diagrams for Σ and Π, in their turn, are used tocalculate fully dressed G and W functions, see Eq. (7).

To make connection with general rules of diagrammaticMC41,42 we formalize the problem at hand as computingquantity Q(y, s) (where y stands collectively for space,imaginary time, and spin variables, while s = 1, 2 labelsthe proper self-energy and the polarization operator sec-tors) from the series of multidimensional sums/integrals

Q(y, s) =∑nξ

∫dx1...dxndY D(n, ξ, xi;Y, s)δ(y − Y ) .

(9)

5

Here n = 0, 1, . . .∞ is the diagram order, ξ labels dif-ferent terms of the same order, xi are internal integra-tion/summation variables, and D is the diagram contri-bution to the answer. Formally, one can think of theset of skeleton diagrams for the free-energy of the systemwith one line marked as ’dummy’. When the dummy lineis removed from the graph the rest is interpreted as a di-agram for Σ(y) (in this case s = 1 and the dummy lineis the particle propagator) or Π(y) (in this case s = 2

and the dummy line is the W one). The last rule fol-lows from the fact that Π(y) is a continuous functionof time. In the BDMC approach one interprets Eq. (9)as averaging of ei arg(D)δ(y − Y ) over the configurationspace ν = (n, ξ, x1, . . . , xn;Y, s) with probability densityproportional to |D|.

The skeleton formulation does not cause any funda-mental problem for Monte Carlo methods and is easy toimplement for diagrams of arbitrary order. Essentially,at any stage in the calculation both G and W are con-sidered to be known functions (the calculation may startwith G = G(0) and W = J/4) while Eq. (7) is used fromtime to time to improve one’s knowledge about G and Wusing accumulated statistics for Σ and Π and fast Fouriertransform algorithms. Moreover, we have shown34 thatBDMC methods are more stable and have better con-vergence properties than conventional iterations. Oneimmediately recognizes that in the skeleton-type formu-lation (i) the number of diagrams to be sampled in a givenorder is dramatically reduced, (ii) the convergence of theskeleton series is likely to be different than that of thebare series, (iii) non-analytic and non-perturbative be-havior might emerge even from a finite number of termsdue to highly non-linear self-consistent formulation (werefer here to the famous mean-field BCS solution).

The other crucial advantage of BDMC over more con-ventional MC methods simulating finite clusters of spinsis that it deals directly with the thermodynamic limitof the system. In practice, in dimension d > 1 theerror bars often become too large before a reliable ex-trapolation to the thermodynamic limit can be done. Inthis sense, BDMC is not subject to the infamous sign-problem which is understood as exponential scaling ofcomputational complexity with the space-time volume ofthe physical system. Feynman diagrams do alternate insign and contributions from high-order diagrams canceleach other to near zero. However, this behavior is bettercharacterized as a ’sign-blessing’, not a sign-problem, be-cause it is crucial for convergence properties. With thenumber of graphs growing factorially with their order theonly possibility for obtaining series with finite conver-gence radius is to have sign-alternating terms such thathigh-order diagrams cancel each other (the sign-blessingphenomenon). For series with finite convergence radiusthere are numerous unbiased re-summation techniqueswhich allow one to determine the answer well outside ofthe convergence radius provided enough terms in the se-ries are known. Relatively small configuration space forskeleton diagrams allows one to establish if sign-blessing

takes place in a given model and to obtain accurate re-sults for diagram orders as large as 7-10 (depending onthe model).

It should be noted that in recent years the Popov-Fedotov trick has become popular within the functionalrenormalization group (PFFRG) framework. It hasbeen applied to frustrated J1 − J2 model on squarelattice43, planar J1 − J2 − J3 antiferromagnet44, spa-tially anisotropic triangular antiferromagnet45, and hon-eycomb lattice antiferromagnet with competing interac-tions46. These studies also attempt at attacking theproblem using Feynman diagrammatic series but are rad-ically different in the technical implementation. WhilePFFRG is also based on sums of subsets of selected di-agrams to infinite order, it does not offer a convenientway to check for convergence of final results as moreand more diagrams are retained. This is the most severedrawback of PFFRG; after all the major problem withexisting theories is reliable estimate of the accuracy. ThePFFRG spectral functions are very broad in energy43

and appear to underestimate ordering fluctuations. Thisalso leads to significant rounding of susceptibility peaksand makes identification of different phases difficult. Inaddition, these studies are typically focused on the zero-temperature phase diagram, not the finite-temperaturecooperative paramagnet state.

IV. NORMALIZATION ANDWORM-ALGORITHM UPDATES

To simplify notations let us write the diagrammaticcontribution from the configuration space point ν asDν = eiϕνDν and call the non-negative function Dν theconfiguration ’weight’. Within the G2W -skeleton schemeDν is given by the modulus of a product which runs overall lines

Dν = |∏lines

fline(ν)| , (10)

where fline(ν) stands for a collection of functions describ-ing various lines in the diagram. At this point we noticethat equation

Q(y, s) =∑ν

eϕνDνδ(y − Y ) . (11)

can be always interpreted as averaging over the proba-bility density distribution Pνs = Dν/Cs, where Cs is thenormalization factor, and thus sampled by MC methods:

Q(y, s) = Cs∑ν

eiϕν δ(y−Y )Pνs −→ Cs

MC∑ν

eiϕν δ(y−Y ) .

(12)In the last transformation we replace the full sum overthe configuration space with the stochastic sum over con-figurations which are generated from the probability den-sity Pνs. This is, of course, nothing but the standard MC

6

approach to deal with complex multi-dimensional spaces.We stress here that all configuration parameters are sam-pled stochastically, including the diagram order and itsstructure, making Diagrammatic MC radically differentfrom calculations which first create a list of all diagramsup to some high-order and then evaluate them one-by-one (often with the use of MC methods for doing theintegrals).

A. Normalization

The normalization constant C can be determined in anumber of ways:(i) using known behavior of Q(y, s) in some limiting case,for example Q(y → y0, s)→ Q0(s),(ii) through the exact sum rule,

∫dyQ(y, s) = Rs, if

available, or, more generically,(iii) by measuring the ratio between the contributions ofall diagrams in Eq. (9) and diagrams which are knowneither analytically or numerically with high accuracy. In-deed, imagine that one or several diagrams, say the lowestorder ones, are known and their integrated contributionto the answer is QN (s) =

∫dyQN (y, s). Let Ns be their

configuration space. Then the ratio Q(y, s)/QN (s) canbe measured in the MC simulation as[∫

dyQ(y, s)

]/QN (s) = (

MC∑ν

eiϕν )/(

MC∑ν

δν∈Ns eiϕν ) .

(13)This leads to

Cs =QN (s)

Zs, Zs =

MC∑ν

δν∈Ns eiϕν . (14)

The diagrams used for normalization are not necessar-ily the physical ones, i.e. they can be artificially “de-signed” to have simple analytic structure and added as aspecial sector to the configuration space ν, see Ref. 34.In the latter case, the diagrams contributing to Q(y, s)and Zs are mutually exclusive and physical contributionshave to be filtered by (1− δν∈Ns).

In the present study we use the modulus of the Hartreediagram to normalize statistics for Σ, and the modulus ofthe lowest order GG-diagram, see the first term in Fig. 2,to normalize statistics for Π:

QN (1) = ΣN =∑r

∑α

|J(r)

4| |G(τ = −0)| ,

QN (2) = ΠN =∑α

∫ 1/T

0

dτ |G(τ)G(−τ)| . (15)

Even though in the self-consistent scheme one does notknow the G(τ)-function analytically (it is tabulated nu-merically) it takes no time to compute the normalizationfactor ΠN with high accuracy. We take the modulus ofthe Hartree diagrams because in the absence of external

magnetic field the spin-up and spin-down contributionexactly cancel each other. Correspondingly,

Z1 =

MC∑ν

δν,(Hartree,s=1) ,

Z2 =

MC∑ν

δν,(n=1,s=2) . (16)

B. Worm-algorithm trick

We now proceed with the description of updates whichensure that all points in the configuration space are sam-pled from the probability density Pνs. Among many pos-sibilities we seek a scheme which involves the smallestnumber of lines in a single update and does not requireglobal analysis of the diagram structure. Such updatesare called ”local”. Typically, they are much more flexi-ble in design, are easier to implement, and lead to moreefficient codes by having large acceptance ratios. Thereader not interested in algorithmic details may proceeddirectly to the next Section.

An easy way to ensure that the diagram is irreducibleis to check that no two lines in the graph have the samemomentum—by momentum conservation laws the iso-lated self-energy blocks cannot change the line momen-tum. By creating a hash table where momenta of all linesare registered according to their values one can readilyverify that momenta of updated lines are not repeated inthe graph without looking at the graph topology or ad-dressing all other lines. This simple tool solves the prob-lem of performing local updates within the irreducibleset. It can be applied even if diagrams are sampled inthe real-space representation, as is done in this article, byattaching an auxiliary momentum variable to each lineand satisfying the momentum conservation law at eachvertex. In this case, the sole purpose of introducing aux-iliary momenta is an efficient monitoring of the diagramtopology and the Dν value is independent of them.

Since the trick with auxiliary momenta is based on mo-mentum conservation laws one is necessarily limited to ei-ther (i) performing updates on closed loops, i.e. partiallyabandoning the idea of local updates, or (ii) extendingthe configuration space to include diagrams which vio-late these conservation laws. The second strategy is theessence of the worm algorithm. One more reason for us-ing the worm algorithm approach is the spin projectionconservation law in the interaction process, see Eq. (6)and Fig. 1. It also requires that updates are performedonly on closed loops of interaction lines and propagators,unless one admits diagrams which violate the correspond-ing conservation law. It turns out that a straightforwardextension of the worm algorithm introduced in Refs. 47and 35 allows one to go around both hurdles.

The additional (unphysical) diagrams have the follow-ing structure. There are two special vertexes, or ’worms’,S and T , where momentum and spin conservation laws

7

are violated, see Fig. 3. In a given graph, any two ver-texes can be special if they are not connected by the in-teraction line. Conservation laws become satisfied if oneimagines a special line connecting S to T . This line isnot associated with any propagator and its mission is totransfer momentum pw and spin projection 1 (doted linein Fig. 3); otherwise special vertexes represent a drainand a source of momentum pw and spin projection 1, seeFigs. 3 and 4. It has to be realized that the spin conser-vation law is formulated for a pair of vertexes connectedby the interaction line, i.e. an interaction line contain-ing a worm on one of its ends is unphysical and can bedescribed by any suitable function F (r, τ).

1 2

( , 1)w zp s

S

T

F

F WW

G

FIG. 3. (Color online) Non-physical diagram with twospecial vertexes (worms) S and T (marked by blue circles).Momentum and spin conservation laws would be satisfied atall vertexes and interaction lines if one considers S/T as adrain/source of momentum pw and spin projection 1; the samerule is recovered by imagining a line (green dotted) which car-ries (pw, sz = 1) from S to T . If special vertexes were notpresent in the diagram, then this graph would contribute toΣ(τ1−τ2) after removing the dummy propagator line marked

by the cross (if cross marks the dummy W -line then the dia-gram is contributing to Π).

The worm algorithm idea is based on the observationthat updates performed with the use of special vertexescan always be made local, including non-trivial changesin the diagram topology and order as well as transforma-tions replacing diagonal interaction lines with spin-flipones. Even though unphysical diagrams are frequentlyencountered in the simulation process they are excludedfrom the statistics of Σ and Π which is accumulated onlyon the physical set of diagrams.

C. Updates

Below we describe the simplest ergodic updatingscheme. With trivial modifications and additional filtersfor proposals which would be rejected because they areincompatible with the allowed configuration space it can

T

,k

q,wk q p

,p ,p q

S,k

q,wk q p

,p ,p q

, 1w zp s

S

T

, 1w zp s

,k

q,wk q p

,p ,p q

,k

q,wk q p

,p ,p q

, 1w zp s , 1w zp s

FIG. 4. (Color online) Detailed structure of S and T ver-texes. Note, that the interaction line containing a worm onone of its ends is also unphysical and the spin conservationlaw applies to a pair of vertexes.

be made more efficient. This, however, would overwhelmthe presentation with minor programming details and wechoose not to discuss them here. For example, below wewill use the following algorithmic rules for dummy linesused to identify diagrams as Σ- or Π-type: (i) the dummyinteraction line cannot be removed or created in any up-date, (ii) if the dummy propagator originating from ver-tex A is modified by adding/removing an intermediatevertex C then A always remains the originating vertexof the new dummy line. These rules can be easily mod-ified to avoid fast rejections of updates in certain casesat the expense of using additional random numbers todeal with available choices and taking care of them in ac-ceptance ratios. Alternatively, the notion of the dummyline can be avoided altogether by designing an improvedestimator based on free-energy diagrams.Create - Delete The pair of complementary updates

Create - Delete switches between physical and unphysicalsectors by inserting/removing a pair of special vertexesconnected by the particle propagator. It is not allowedto have S = T or to have S and T being connectedby the interaction line, see an illustration in Fig. 5. InCreate, the particle propagator for update and the typeof special vertex to be placed at its origin (A → S orA → T ) are selected at random. One has to verify thatflipping the propagator spin is consistent with the wormrules or reject the proposal. The missing momentum pwat the worm vertex is selected at random. In Delete, oneselects a worm at random, checks that the outgoing par-ticle propagator arrives at the other worm, and proposesto remove worms from the diagram. The correspondingacceptance ratios are given by

RCreate =Dν′

Dν

u2u1

2n , RDelete =Dν′

Dν

u1u2

1

2n, (17)

where n is the diagram order and ν and ν′ are configu-ration space points before and after the update and Dν

8

is the product in Eq. (10). In the present case only onepropagator and two interaction lines are affected:

Dν′

Dν=

∣∣∣∣ GAB(α)FSCFTDGAB(−α)WACWBD

∣∣∣∣ , (18)

in Create and similarly, Dν′/Dν = |GWW/GFF |, inDelete with appropriate arguments for all functions in-volved. [If the line is labeled as W it can be either of

J or W type.] In what follows we will stop mentioningwhich vertexes determine function parameters since thesecan be easily recovered from the figures. Finally, assum-ing that the protocol of deciding which update shouldbe implemented next is random and based on assigningeach update some probability, ui, we mention the ratioof probabilities in the acceptance ratio (u1 for Create andu2 for Delete).

A ,k B

W W

C D

T,wk p S

F F

C D

Delete Create

A ,k B

W W

C D

S,wk p T

F F

C D

Delete Create

wp

wp

FIG. 5. Two cases for Create and Delete updates which in-sert/remove a pair of worms at the ends of the particle prop-agator.

Create-H - Delete-H This pair of complementary up-dates also switches between physical and unphysical sec-tors with an additional ingredient—it increases/decreasesthe diagram order by attaching a Hartree-type bubble tothe existing graph. According to the rule ’no two linesmay have the same momentum’ the diagram remains irre-ducible because one of the worms is placed on the bubblevertex. The overall transformation is illustrated in Fig. 6;the text below addresses to this figure with regards tothe procedure of selecting specific graph parameters. InCreate-H a particle propagator (going from vertex A tovertex B) and whether to place S or T on vertex B is de-cided at random. If the proposal is inconsistent with theworm rules it has to be rejected. Next, a new time vari-able for the intermediate vertex C is generated from theprobability density, t(τ), and a random decision is made

whether the new interaction line is of the J- or W -type.For the J-line (with τ ′ = τ) the position of the secondworm vertex in space is obtained from the normalizedX(r′) ∝ J(r′) distribution. For the W -line this positionis obtained from some designed probability distribution

Y (r′) while the time location is drawn from the prob-ability density, t(τ ′). The spin variable in the bubble,the bubble momentum variable p and the worm momen-tum pw are decided at random. In Delete-H a randomchoice is made what type of special vertex must be onthe Hartree bubble provided the overall topology of linesis identical to that on the r.h.s of Fig. 6. The proposalis to remove worms and the bubble from the diagram. Itis rejected if either the propagator originating from C orthe interaction line attached to C is the dummy one. Theacceptance ratios for these updates are (probabilities ofcalling Create-H are Delete-H are u3 and u4, respectively)

RCreate−H =Dν′

Dν

u4u3

23n

t(τ)

1/X(r′); (J)

1/Y (r′)t(τ ′); (W )(19)

RDelete−H =Dν′

Dν

u3u4

t(τ)

23(n− 1)

X(r′); (J)

Y (r′)t(τ ′); (W )(20)

with the diagram weight ratios Dν′/Dν given by|GGFFG/GW | and |GW/GGFFG| in Create-H andDelete-H, respectively (note that here n is the initial di-agram order). The simplest choices for probability dis-tributions in Eqs.(19) and (20) would be uniform distri-butions t(τ) = T , X(r′) = 1/z, and Y (r′) = 1/V , whereV is the total number of lattice sites. One can use otherfunctions for better acceptance ratio. The three updates

A ,k B

D

A ,k T

D

Delete‐H Create‐H ,wk p

C

S

,p

FW

F

' 'r

wp

A ,k B

D

A ,k

T

D

Delete‐H Create‐H ,wk p

CS

,p

FW

F

r r

r r

' 'r

wp

wpwp

FIG. 6. Two cases for Create-H and Delete-H updates whichinsert/remove a pair of worms and increase the diagram orderby adding a Hartree-type bubble.

described next represent a random diffusion of specialvertexes along the graph lines (this places S and T onany allowed pair of vertexes) supplemented by an updatewhich changes the graph topology.Move-P In this self-complementary update one se-

lects at random S or T and proposes to shift the se-lected worm along the incoming or outgoing propagatorline, deciding again randomly between the two choices.

9

Since all four case are identical in their implementationwe describe below an update shifting S along the outgo-ing propagator line to vertex B, see upper panel in Fig. 7,if B 6= S, of course. According to the rules, if B = T ,or D = T , or the spin of the outgoing line is down, theupdate is rejected. Finally, one has to check whether theproposal may lead to the irreducible diagram. For theupdate shown in Fig.7 the acceptance ratio is given by

RMove−P =

∣∣∣∣GWF

GFW

∣∣∣∣ . (21)

S=A ,k B

F W

C D

S,wk p A

W F

C D

Move‐P

Move‐I

S=A

F

B

q

A

F

S

wq p

FIG. 7. Upper panel: An illustration of the Move-P updateshifting S along the propagator line. This move changes themomentum of the propagator line and flips its spin as wellas the status of interaction lines between physical and un-physical. Lower panel: An illustration of the Move-I updateshifting S along the interaction line. This move changes theauxiliary momentum of the interaction line only.

Move-I In this self-complementary update one selectsat random S or T and proposes to shift the selected wormalong the interaction line to vertex B, see the lower panelin Fig. 7. One has to check whether the proposal maylead to the irreducible diagram. This update is alwaysaccepted since it changes only the auxiliary momentumvariable.

Commute All possible topologies in a graph of order ncan be generated by randomly connecting outgoing prop-agator lines to incoming ones. Having this observationin mind the self-complementary Commute update pro-poses to swap destination vertexes for propagator linesoriginating at S or T , see Fig. 8. This proposal is validonly if all vertexes have the same space coordinate andboth propagators have the same spin index. The cru-cial advantage of the worm algorithm at this point be-comes clear—momentum conservation law is satisfied byabsorbing the difference k− p into the worm momentumpw → pw + k − p. This update is accepted with ratio

RCommute =

∣∣∣∣GGGG∣∣∣∣ . (22)

In the lower panel of Fig. 8 we show a typical diagramchange produced by Commute. It always changes thenumber of fermionic loops in the graph and, in partic-ular, will transform a diagram with a bubble attachedto the worm vertex into the vertex-correction type dia-gram. This explains, to some extent, our design of up-dates Create-H and Delete-H.

S ,k A

Commute

r

T,p B

r

S Ar

T

,p

Br

,k

wp

S T S T

wp k p

FIG. 8. Upper panel: In the Commute update the diagramtopology is changed by re-directing propagators originating atspecial vertexes S and T to have their destination vertexes atB and A, respectively. Lower panel: typical transformationproduced by the Commute update.

Dummy To place the dummy line mark on any of the

G- or W -lines we select one of the vertexes at random,say vertex A, and make a random decision whether thenew dummy line should be the interaction line attachedto A (it has to be of the W type) or the propagator lineoriginating from A. The proposal is always accepted ifthe notion of the dummy mark does not change the valueof the function behind the line (which is assumed to bethe case here).

The above set of updates is sufficient for doing thesimulation. It can always be supplemented by additionalupdates which do not necessarily involve special vertexesbut reduce the autocorrelation time and lead to more ef-ficient sampling of the diagram variables. Moreover, anon-trivial check of the detailed balance for debuggingpurposes is only possible if the set of updates is overcom-plete. Below we describe several such updates.Insert-Remove An idea here is to increase/decrease

the diagram order by inserting/removing a ladder-typestructure. More precisely, in Insert we make a randomchoice between S and T to start the construction fromthe special vertex V1. Next we identify vertex A as thedestination vertex of the propagator originating from V1,and vertex B as the originating vertex for the propagatorwith the destination vertex V2, which is the other wormend, see the upper panel in Fig. 9. If A = V2 the up-date is rejected. The proposal is to insert new vertexesC (intermediate between V1 and A) and D (intermediatebetween B and V2) and to link them with the interaction

10

line of randomly chosen type, either J or W , and momen-tum q. The new time variables are generated from theprobability density t(τ). The momenta of new lines andthe worm are modified as described in Fig. 9 to satisfyconservation laws. Finally, if one of the propagators is thedummy line, the new dummy line has to have the sameoriginating vertex. In Remove we select one of the spe-cial vertexes, V1, at random and verify that the topologyof lines connecting it to the other special vertex, V2, aswell as lines parameters are consistent with Fig. 9 (upperpanel). If either C-A or D-V2 propagator is a dummy linethe update is rejected. The proposal then is to removevertexes C and D from the graph and update momentaof the lines accordingly. The acceptance ratios are givenby

RInsert =Dν′

Dν

u10u9

2

t(τ)

1; (J)

1/t(τ ′); (W )(23)

RRemove =Dν′

Dν

u9u10

t(τ)

2

1; (J)

t(τ ′); (W )(24)

where Dν′/Dν = |GGGGW/GG| in Insert and its inversein Remove.

V1 ,k A

Remove Insert

B ,p V2

wp

V1 ,k A

B ,p V2wp q

,p q

,k q

q

W

C

D

'

B

,k

A Dress Undress

C

,p

B,k

A

C,p

,k q

,p q

qW

D

E

'

FIG. 9. Upper panel: Increasing/decreasing the diagramorder using a pair of complementary updates Insert and Re-move. When the propagators V1-A and B-V2 are linked withthe new interaction line C-D carrying momentum q the closedloop for momentum conservation goes as V1-C-D-V2-V1. Thesame loop is used in the Remove update. Lower panel: Dia-gram transformation when vertexes are dressed and undressedwith interaction lines.

Dress-Undress One of the easiest updates to increasethe diagram order within the G2W -skeleton formulationis to dress an existing vertex with interaction line andconsider the smallest closed loop for transferring momen-tum. The Dress update starts from random selection ofvertex A and identification of vertexes B and C linked toit by propagator lines; if B = A the update is rejected(we pay no attention in this update whether one of the

vertexes is of a special type). The proposal is to add newvertexes D (intermediate between B and A) and E (inter-

mediate between A and C) and to link them with the Wline with random momentum q. The new time variablesare generated from the probability density t(τ). The mo-menta of new lines are modified as described in the lowerpanel of Fig. 9 to satisfy conservation laws. In Undresswe select vertex A at random, identify vertexes D, B, E ,and C using links along the propagator lines, and verifythat the topology of lines and their parameters are con-sistent with the dressed vertex configuration. If the D-Eline is not of the diagonal W type or one of the propaga-tors D-A or E-C is a dummy line, the update is rejected.The proposal is to remove vertexes D and E from thegraph. The acceptance ratios are

RDress =Dν′

Dν

u12u11

n

(n+ 1)t(τ)t(τ ′), (25)

RUndress =Dν′

Dν

u12u11

n

(n− 1)t(τ)t(τ ′), (26)

where Dν′/Dν = |GGGGW/GG| in Dress and its inversein Undress.Recolor An easy and efficient way to change the spin

index of propagator lines is to select a random vertexand use it to construct a closed loop by following thepropagator lines attached to it. If all propagators in theloop have the same spin index α it can be changed to−α with acceptance ratio unity in the absence of exter-nal magnetic field; otherwise, one has to use the ratioof products of all propagator lines after and before theupdate.

B

( )W F

C

D

Move‐T

S

A

B

( )W F

C

D

A'

F

A

Move‐T F

'

S

A

FIG. 10. An illustration of the Move-T update changing theimaginary time location of a randomly chosen vertex. One isfree to change the type of worm from S to T or to place it onany of the vertexes in both panels.

Move-T This self-complementary update is designedto sample time variables of the diagram without changingits order and topology. The proposal is to select one ofthe vertexes at random, let it be vertex A and updateits imaginary time variable from τ to τ ′ using probabilitydensity distribution t(τ ′), see Fig. 10 for two alternatives.

11

The interaction line attached to A cannot be of the J-type; whether it is the physical W -line or unphysical F -line does not mater. The acceptance ratio is given by

RMove−T =t(τ)

t(τ ′)

|W (F )/W (F )| |GG/GG| (generic)

|F/F | (bubble).

(27)

D. Diagram sign

With the dummy line removed, the diagram phase re-quired to compute the self-energy and polarization oper-ator using Eqs. (12) and (14) is determined by standarddiagrammatic rules (see also Eq. (10)):

ϕν =∑lines

arg(fline(ν)) + π(n+ l) , (28)

where l is the number of fermionic loops (one only needsto know whether it is even or odd). This phase is readilyrecalculated in updates without addressing the whole di-agram since l always changes its parity when Create-H,Delete-H, and Commute are accepted.

E. Satisfying the sum rule

The value of the spin-spin correlation function χ(r =0, τ = 0) = 〈(Sz)2〉 = 1/4, see Eq. (4), provides an im-portant sum rule in the Fourier space

T∑n

∫BZ

dq

8π2/√

3χ(q, n) = 1/4 , (29)

which can be used for modifying convergence propertiesof the self-consistent scheme as follows (the integral istaken over the Brillouin zone (BZ)). When the maximumdiagram order is fixed at N the sum rule is violated bysome amount which vanishes as N →∞. Since the finalresult is claimed after taking the limit, it is perfectlyreasonable to impose a condition that the sum rule isalways satisfied by scaling Π by an appropriate factor.This is exactly what is done in this article: after solvingthe Dyson Equation we check the value of χ(r = 0, τ =0), adjust the scaling factor for Π, and go back to solvingthe Dyson Equation again until the sum rule is satisfiedwith three digit accuracy.

V. TRIANGULAR LATTICE HEISENBERGANTIFERROMAGNET

In the diagrammatic formulation there is no concep-tual difference in the implementation of the numericalscheme for any dimension of space, lattice type, and in-teraction range. Thus, sign-problem free systems, e.g.the square/cubic lattice Heisenberg antiferromagnet with

nearest neighbor coupling, can be used for testing pur-poses since their properties are known with high degreeof accuracy (with reliable extrapolation to the thermo-dynamic limit) using path-integral and stochastic seriesexpansion MC methods. After passing such tests, weturn our attention to the triangular-lattice Heisenbergantiferromagnet (TLHA) which is a canonical frustratedmagnetic system with massively degenerate ground statein the Ising limit.

The most important question to answer is whether thesign blessing phenomenon indeed takes place, i.e. thereis a hope for obtaining accurate predictions in the strongcoupling regime by calculating higher-and-higher orderdiagrams despite factorial growth in the number of con-tributing graphs. In Fig. 11 we show comparison betweenthe calculated answer for the static uniform susceptibility

χu = χ(q = 0,m = 0) =

∫ 1/T

0

dτ∑r

χ(r, τ) , (30)

and the high-temperature expansion results37,38 atT/J = 2. This temperature is low enough to ensurethat we are in the regime of strong correlations be-cause χu is nearly a factor of two smaller than thefree spin answer χ0

u = 1/4T . On the other hand, thistemperature is high enough to be sure that the high-temperature series can be described by Pade approxi-mants without significant systematic deviations from theexact answer37,38 (at slightly lower temperature the bareNLC series start to diverge). We clearly see in Fig. 11that the BDMC series converges to the correct resultwith accuracy of about three meaningful digits and thereis no statistically significant change when more than ahundred thousand of 7-th order diagrams are accountedfor. [We recall that the number of topologically distinctdiagrams within the G2W -skeleton scheme was calcu-lated in Ref. 48; for the eight lowest orders they are1, 1, 6, 49, 542, 7278, 113824, 2017881.] The error bar forthe 7-th order point is significantly increased due to ex-ponential growth in computational complexity. The 4-thorder result can be obtained after several hours of CPUtime on a single processor.

Interestingly enough, when temperature is lowereddown to T/J = 1, which is significantly below the pointwhere the bare NLC series start to diverge (see Fig. 13),the BDMC series continue to converge (see Fig. 12).This underlines the importance of performing simulationswithin the self-consistent skeleton formulation.

In Fig. 13 we show results of the BDMC simula-tion performed at temperatures significantly below themean-field transition temperature. For all points we ob-serve extremely good agreement (essentially within ourerror bars) with the Pade approximants used to ex-trapolate the high-temperature expansion data to lowertemperature38. Within the current protocol of dealingwith skeleton diagrams we were not able to go to lowertemperature due to the development of singularity in theresponse function (and thus effective interaction W at

12

0 2 4 6 8

0.0645

0.065

0.0655

0.066

0.0665

0.067

u

NLC

T/J=2

FIG. 11. (Color online) Uniform susceptibility calculatedwithin the G2W -skeleton expansion as a function of the max-imum diagram order retained in the BDMC simulation (blackdots) for T/J = 2. The result of the high-temperature expan-sion (with Pade approximant extrapolation)38 is shown by redsquare and horizontal line.

0 2 4 6 8

0.08

0.082

0.084

0.086

0.088

u T/J=1

NLC

FIG. 12. (Color online) Uniform susceptibility as a functionof the maximum diagram order (black dots) for T/J = 1. Theresult of the high-temperature expansion (with Pade approx-imant extrapolation)38 is shown by red square and horizontalline. Its error bar is based on the difference between variousexpansion/extrapolation schemes.

the wave-vector Q = (4π/(3a), 0). When the denomina-tor 4−J(q)Π(q,m) in (7) is close to zero it becomes verydifficult to control highly non-linear sets of coupled inte-gral equations given finite statistical noise on the mea-sured quantity Π(q,m).

This is clearly seen in Fig. 14 where we show data for

the staggered susceptibility χ(Q, 0), defined as

χs = χ(Q,m = 0) =

∫ 1/T

0

dτ∑r

eiQ·rχ(r, τ) , (31)

along with the Curie law and the uniform susceptibility,on the double logarithmic scale.

0.1 1

0.04

0.05

0.06

0.07

0.08

0.09

u 8T

7T

BDMC

Pade

12S

13S

FIG. 13. (Color online) Uniform susceptibility as a functionof temperature (red dots) for the triangular Heisenberg an-tiferromagnet calculated within the BDMC approach. NLCexpansion results 37 based on triangles (labeled as 7T and8T) and sites (labeled as 12S and 13S) are shown alongwith two different Pade approximant extrapolations of high-temperature expansions38.

One of the advantages of our approach is the abilityto perform calculations of susceptibility at arbitrary mo-mentum. In Fig. 15 we show data for χ(q, 0) along theΓ − K −M − Γ trajectory in the Brillouin zone (BZ).Here Γ is the center of the BZ, K = Q = (4π/(3a), 0),

and M = (π/a, π/(√

3a)) is the mid-point on the faceof the hexagonal BZ (see Fig. 15). Results presentedin Figs. 14 and 15 are new because they are obtainedfor the static (zero Matsubara frequency) susceptibilityin the thermodynamic limit. It is useful to note that thestatic response is far more difficult to get within the NLCmethod which is suited for calculations of the equal timecorrelation functions, such as, for example, equal timespin structure factor. The exception is represented bythe uniform, or zero momentum, response which is basedon the total magnetization commuting with the Hamil-tonian. In addition, we can afford very high resolution inmomentum space which is not the case for calculationsbased on clusters of finite size.

It is clearly seen in Fig. 15 that around T/J = 1 sys-tem’s response is enhanced along the whole Brillouin zoneboundary indicative of the frustrated behavior. Only attemperatures below T/J = 0.5 it becomes evident that

13

1

0.1

1

Q

u

1

FIG. 14. (Color online) Staggered susceptibility at the wavevector Q as a function of temperature (black dots) plotted forcomparison along with the Curie-Weiss law (blue curve) anduniform susceptibility (red dots and line).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

T/J=1

T/J=2

T/J=0.5

T/J=0.375

BZ

KM

FIG. 15. (Color online) Static spin-spin correlation functionalong the characteristic trajectory in the Brillouin zone.

the system wants to develop correlations commensuratewith the K point. We confirm previous observation49

that even at T/J = 0.375 the spin correlation length,which can be estimated from the half-width of the peakaround K point, is still of the order of lattice constanta. This can be checked even more explicitly by lookingon the static spin correlations in real space. Figure 16shows that while the sign structure of short-range spa-tial correlations is consistent with the three-sublattice120 state, the magnitude of the correlations becomes

1 2 3 4

0.001

0.01

0.1

1

0 1

2

3

4

5

6

7

8

9

10

BA

A

A

A

AB

B

BB

C

C

C

C

A

BC

FIG. 16. (Color online) Modulus of the spin susceptibilityin real space at T/J = 0.375 on the logarithmic scale. Lat-tice points are enumerated according to their distance fromthe origin. Spins on red sites (0, 2, 5, 6, 10) are correlatedferromagnetically while spins on blue and black sites are cor-related anti-ferromagnetically with the spin at the origin. Atslightly higher temperature T/J = 0.5 black points (3, 7)start to correlate ferromagnetically with the spin S0 at theorigin, contrary to the classical ground state pattern depictedto the right.

exponentially small on the scale of a few lattice periods.Closer look also reveals dramatic suppression of correla-tions between site 0 (say, sublattice A) and sites 3 and 7both of which would belong to sublattice C in the per-fectly ordered 120 classical state. Moreover, at slightlyhigher temperature T/J = 0.5 the sign of correlationson sites 3 and 7 changes sign and turns ferromagnetic,similar to A-sublattice spins, though with much smalleramplitude. This temperature induced reversal of cor-relations is a remarkable effect specific for a frustratedsystem.

It has been noted some time ago that short wavelengthspin excitations contribute significantly to the finite tem-perature properties of triangular lattice antiferromagnetat not too low temperature50. This has to do with sub-stantial phase space volume these excitations occupy aswell as with their relatively weak dispersion51. It is con-ceivable that particularly weak correlations between sub-lattices A and C noted above have to do with these ex-citations as well. All of these features can be extractedfrom the retarded spin susceptibility χ(q, ω) calculationof which requires analytic continuation of our Matsubara-frequency susceptibility χ(q, ωm) to the real frequency.We plan to address this important issue in the near fu-ture.

VI. CONCLUSIONS

This paper describes novel approach to frustrated spinsystems. Obtained numerical results for the spin-1/2 tri-angular lattice Heisenberg model, Section V, show thepower and competitiveness of our approach in compari-

14

son with other well established numeric techniques.

Future work has to address the issue of performingsimulations at lower temperature in the regime charac-terized by the large correlation length. Technically, thistranslates into being close to zero in the denominatorof Eq. (7) for some momentum values. Progress in thisdirection should allow us to better describe the cross-over/transition from the cooperative paramagnet to thelong-range ordered (albeit frustrated) state.

Yet perhaps the most promising line of attack has todo with applying our technique to the geometrically frus-trated models that do not support magnetically orderedstate at all. In two dimensions this singles out quantumkagome lattice antiferromagnet which have recently be-ing shown to realize a long-sought Z2 spin liquid state27.This task will require extension of our approach to sys-tems with several (three in this case) spins in a unit cell.The added matrix complexity does not represent any fun-damental difficulty.

Moving one dimension higher brings one to the mostfrustrated antiferromagnet in the world – spin-1/2 py-rochlore antiferromagnet52–54. Pyrochlore Ising-likemodel realizes beautiful quantum spin-ice physics55 whilethe fate of spin-1/2 Heisenberg model is an open ques-

tion. It is widely believed that ‘cooperative paramagnet’region is most extended in this three-dimensional system.Unlike many lower-dimensional frustrated system, spin-1/2 pyrochlore is essentially not accessible by quantumMonte-Carlo technique due to large unit cell (4 spins).We believe that our Diagrammatic MC approach is there-fore uniquely suited for studying the finite-temperaturedynamics of this outstanding frustrated magnet.

Recently we have generalized the Popov-Fedotov trickto a universal technique of fermionization which leads toa well-defined standard diagrammatic technique for arbi-trary lattice spin, boson, and fermion system with con-straints on the on-site Fock states39. This developmentcreates a broader context for the present work: success-ful implementation of the BDMC method for models ofquantum magnetism may lead to the universal numer-ical tool for arbitrary strongly correlated lattice modelswithin the fermionization framework when diagrammaticexpansion do not involve large parameters.

We thank M. Rigol for communicating us data ob-tained within the NLC method. This work was sup-ported by the National Science Foundation under grantsPHY-1005543 (S.K., N.P., B.S., and C.N.V.) and DMR-1206774 (O.A.S.), and by a grant from the Army Re-search Office with funding from the DARPA.

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