Worm-type Simulation of Quantum Transverse-Field Ising Model

Chun-Jiong Huang,1, 2, 3 Longxiang Liu,1, 2, 3 Yi Jiang,4, ∗ and Youjin Deng1, 2, 3, †

1Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Shanghai, 201315, China

2CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China

3CAS-Alibaba Quantum Computing Laboratory, Shanghai, 201315, China4Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

(Dated: May 21, 2020)

We apply a worm algorithm to simulate the quantum transverse-field Ising model in a path-integral repre-sentation of which the expansion basis is taken as the spin component along the external-field direction. Insuch a representation, a configuration can be regarded as a set of non-intersecting loops constructed by “kinks”for pairwise interactions and spin-down (or -up) imaginary-time segments. The wrapping probability for spin-down loops, a dimensionless quantity characterizing the loop topology on a torus, is observed to exhibit smallfinite-size corrections and yields a high-precision critical point in two dimensions (2D) as hc =3.044 330(6), sig-nificantly improving over the existing results and nearly excluding the best one hc = 3.044 38(2). At criticality,the fractal dimensions of the loops are estimated as d`↓(1D) = 1.37(1)≈ 11/8 and d`↓(2D) = 1.75(3), consistentwith those for the classical 2D and 3D O(1) loop model, respectively. An interesting feature is that in 1D, boththe spin-down and -up loops display the critical behavior in the whole disordered phase (0 ≤ h < hc), having afractal dimension d` = 1.750(7) that is consistent with the hull dimension dH = 7/4 for critical 2D percolationclusters. The current worm algorithm can be applied to simulate other quantum systems like hard-core bosonmodels with pairing interactions.

I. INTRODUCTION

The quantum transverse-field Ising model (QTFI) is atextbook model in quantum many-body physics and playsan important role in quantum phase transition1 and quan-tum information science2. The one-dimensional (1D) QTFIcan be solved exactly3, and it has been widely used totest theoretical or numerical methods4,5 and to study novelquantities like entanglement entropy6,7 and quantum fidelitysusceptibility8. In higher dimensions, analytical results arescarce, and one has to rely on numerical or approximate meth-ods. Many methods have been developed, including transfermatrix method9, series expansion8,10, continuous-time MonteCarlo approach5,11–15, tensor renormalization group method16,density matrix renormalization group17, projected entangled-pair states18, and machine learning method19,20 etc. Neverthe-less, to obtain a high-precision critical point still remains tobe a challenging task. To our knowledge, the best estimatesof the critical point for the 2D QTFI are 3.044 2(4) in Ref. 8and 3.044 38(2) in Ref. 5, respectively achieved by stochas-tic series expansion (SSE) and continuous-time Wolff clustermethods.

In this work, we apply a worm algorithm to simulate QTFIin 1D and 2D. It is shown that as pointed out in Ref. 21 and 22,the worm algorithm exhibits efficiency comparable to clus-ter schemes. A high-precision estimate of the square-latticecritical point is obtained as 3.044 330(6), significantly im-proving the existing results and nearly excluding the best one3.044 38(2)5. In the path-integral representation for the cur-rent worm algorithm, a configuration can be regarded as a setof non-intersecting loops constructed by “kinks” for pairwiseinteractions and spin-down (or -up) imaginary-time segments.Rich geometric properties are observed for these loops. In par-ticular, it is found that in 1D, the loops over a wide parameter

range exhibits scaling behaviors that are in the universalityclass of the classical 2D percolation. Deep theoretical under-standing is desired. Further, a variety of physical quantities,including the magnetic and the fidelity susceptibilities, are ex-amined.

The rest of the paper is organized as following. Section IIexplains the current path-integral representation for the QTFIand the formulation of the worm algorithm. The numericalresults are presented in Sec. III. A brief summary is given inSec. IV.

II. WORM ALGORITHM

Consider a d-dimensional lattice, the Hamiltonian of theQTFI is conventionally written as

H = −t∑〈i j〉

σziσ

zj − h

∑i

σxi , (1)

where σαi (α = x,z) are Pauli matrices, 〈i j〉 represents near-est neighboring sites, t > 0 is the ferromagnetic interactionstrength and h is the transverse field. Taking the σz spincomponent as the expansion basis for the path-integral rep-resentation, one can map a d-dimensional QTFI onto a (d+1)-dimensional classical system, for which each lattice site has acontinuous line of spin segments; see Fig. 1 (a) for an exam-ple. This continuous dimension is called the imaginary-time(τ) direction, along which the spin state σz

i can be flippedby the σx

i operator but must satisfy the periodic condition.The length of the τ dimension is the inverse temperatureβ = 1/kBT (the Boltzmann constant is set as kB = 1 fromnow).

To formulate a worm algorithm that is effective for configu-rations of closed loops, we choose the external-field direction

arX

iv:2

005.

1006

6v1

[co

nd-m

at.s

tat-

mec

h] 2

0 M

ay 2

020

2

(a)

τ

x

(b)

τ

x

M

I

x

τ

(c)

FIG. 1. (Color online) Illustration of path-integral configurations. (a)for Eq. (1). Red (gray) segments represent the spin-up (-down) state.(b) a G configuration for Eq. (8), having an open path with the twoends marked as I and M. Blue (green) lines are for pairing (hopping)kink. (c) a sketch of Z configuration on the 1D torus, with a non-local loop of winding numbers (Wτ = 4,Wx = 1). For simplicity,the lattice structure is not shown.

as the expansion basis and rewrite Hamiltonian (1) as

H ≡ K + U = −t∑〈i j〉

σxi σ

xj − h

∑i

σzi . (2)

As a result, the expansion is still along the σz direction, andU and K are respectively the diagonal and the non-diagonalterm. The pairwise interactions K = −t

∑〈i j〉 σ

xi σ

xj can be fur-

ther expressed in terms of the raising and lowering spin oper-ators, σ± = (σx ± iσy)/2, as

K ≡ K1 + K2 (3)

= −t∑〈i j〉

(σ+i σ−j + h.c) − t

∑〈i j〉

(σ+i σ

+j + h.c) .

The term K1 flips a pair of opposite spins and thus the totalmagnetization is conserved along the τ direction, while K2flips a pair of spins of the same sign. We note that with theHolstein-Primakoff transformation, bi(b

†

i ) = σ−i (σ+i ) and thus

ni ≡ b†i bi = (σzi + 1)/2, the QTFI can be mapped onto a hard-

core Bose-Hubbard (BH) model with Hamiltonian

H = −t∑〈i j〉

(b†i b j + h.c.) − t′∑〈i j〉

(b†i b†j + h.c.) − µ∑

i

ni , (4)

where t′ = t, the particle number ni = 0, 1, and the chemicalpotential µ= 2h. In the language of the hard-core BH model,K1 accounts for the hopping of a particle, and K2, which si-multaneously creates/deletes a pair of particles, represents thepairing of two neighboring bosons. For convenience, we shallrefer to K1 and K2 as the hopping and the pairing term, respec-tively.

With Eq. (3), the partition function of Hamiltonian (2) canbe formulated in the Feynman’s path-integral representation(also called the worldline representation) as

Z = Tr[e−βH

]=

∑α0

〈α0|e−βH |α0〉 (5)

= limdτ= β

nn→∞

∑{αi}

〈α0|e−Hdτ|αn−1〉 · · · 〈α1|e−Hdτ|α0〉

=∑α0

∞∑N=0

∫ β

0

∫ β

τ1

· · ·

∫ β

τN−1

N∏k=1

dτk F(t, t′, h)

with the integrand function

F(t, t′, h)= tNh t′Np exp(−∫ β

0U(τ)dτ) , (6)

whereNh andNp are respectively the number of hopping andpairing kinks (N = Nh + Np), |αi〉 = |σz

1, σz2, · · · , σ

zN〉 is

an eigenstate in the σz basis (N is the total number of lat-tice sites), and the term with U acts as a potential energy.Moreover, Eq. (5) can be graphically viewed as the summa-tion/integration over configurations in the (d + 1)-dimensionalspace-time {i, τ}, of which the statistical weight is

WZ(t, t′, h) =

N∏k=1

dτk F(t, t′, h) . (7)

In such a representation, each lattice site has a line of spin seg-ments, and at imaginary time τk (k = 1, 2, · · · ,N), a pair ofneighboring spins is simultaneously flipped either by a hop-ping term K1 or by a pairing term K2. We shall call them thehopping or the pairing kink, respectively. Starting from an ar-bitrary space-time point (i, τ), one would construct a closedloop by following spin-up (-down) segments and kinks. Thus,a configuration effectively consists of closed loops.

An important ingredient of the worm algorithm is then toextend the configuration space {Z} for Eq. (5) by includingtwo defects. For the QTFI, the extended configuration space{G} is for the spin-spin correlation function of the Pauli matrixσx:

G(i,m, τI, τM)=Tr

[Tτ

(σx

i (τI)σx

m(τM)e−βH

)], (8)

where Tτ is the τ-ordering operator. In addition to closedloops, a path-integral configuration in the {G} space containsan open path with two ending points; see an example inFig. 1(b). We shall refer to the ending points as “Ira” (I) and“Masha” (M), and denote their coordinates in the space time as(xI, τI) and (xM, τM). The statistical weight of a G configura-tion can be written as

WG =dτIdτMωG

N∏k=1

dτk F(t, t′, h) , (9)

where F is given by Eq.(6) and ωG is an arbitrary positiveconstant. When I coincides with M, the open path forms aclosed loop, and the G space is reduced to theZ space.

The full configuration space for the worm-type simulationcorresponds to the combination of the G and theZ space. Forergodicity, the simulation must be able to change the spatialand imaginary-time location of any kink as well as of de-fects (I, M), and to switch back and forward between the Zand G space. We adopt the following three update schemes:(a) Create/Annihilate defects (I, M), (b) Move imaginary-timeof defect M, and (c) Insert/Delete a kink. The first operationswitches between theZ and theG space by creating or annihi-lating a pair of defects (I, M). The second updates the τM value,and the third changes the xM value by inserting or deleting akink. Except “Create defects (I, M)”, the other update schemes

3

apply in the G space, and each of them is chosen with a prioriprobability given before simulation.

(a) Create/Annihilate defects (I, M). If the current config-uration is in the Z space, “Create defects (I, M)” is the onlypossible update scheme. One randomly picks up a point(xI, τI) from the whole space-time of volume βN = βLd,draws a uniformly distributed imaginary-time displacementδ ∈ [−τa/2, τa/2) and δ , 0 with range τa ∈ O(1)23, assignsxM = xI and τM = mod(τI + δ, β), and flips the spin state be-tween defects I and M. The β-periodicity is taken into accountby the modular function. As illustrated in Fig. 2(a), the types(hopping or pairing) of kinks between defects I and M are in-terchanged during this operation.

Action “Annihilate defects (I, M)”, the reverse operation of“Create defects (I, M)”, is chosen with a priori probabilityAain the G space. It changes a G configuration into a Z oneby annihilating defects (I, M) and flipping the spin state in-between. This is possible only if I and M are on the sameworldline xI = xM and their imaginary-time displacementmin {|τI − τM|, β − |τI − τM|} ≤ τa/2.

Accordingly, the detailed-balance condition reads

dτIβN

dτMτa·WZ · Pcrea = Aa ·WG · Panni , (10)

where Pcrea (Panni) is the acceptance ratio for the “Create de-fects” (“Annihilate defects”) operation, WZ (WG) is the statis-tical weight for the configuration before (after) the creation ofdefects, and dτI/(βN) and dτM/τa account for the probabilityof choosing the space-time location for I and M, respectively.

Making uses of Eqs. (7) and (9), the acceptance probabili-ties for the Metropolis filter can be calculated as

Pcrea = min[1,Aa τa

βNωG

Fnew

Fold

](11)

Panni = min[1,

1Aa

1τa

ωG

βNFnew

Fold

],

where Fnew and Fold, given by Eq. (6), is respectively for theconfiguration after and before the corresponding operation.Note that the statistical-weight change, Fnew/Fold, is mainlydetermined by the random displacement |δ| ≤ τa/2. As a re-sult, the range τa should be of O(1), and further, the accep-tance probabilities in Eq. (11) can be optimized by tuning τa.

A natural choice for the relative weight is ωG = βN, sincethe acceptance probabilities in Eq. (11) then hardly dependon L and β. Physically, this is because that the spin-spin cor-relation function G(i,m, τ

I, τM) has the space-time translation

invariance so that the statistical weight of a G configurationshould be normalized by a factor 1/(βN). Further, with thischoice, the number of Monte Carlo steps between two adja-cent creations of (I, M), called the worm-return time, measuresthe ratio of theG space over theZ space, and exactly gives thedynamic magnetic susceptibility of the QTFI which is statedexplicitly in Sec. III C.

(b) Move imaginary time of defect M. The operation, reverseto itself, is chosen with probability Ab in the G space. Onerandomly draw a random time-displacement δ ∈ [−τb/2, τb/2)and δ , 0, assigns τ′M = mod(τM + δ, β) for the new temporal

I M

(a) Create/Annihilate defects (I, M)

I M I M

(b) Move imaginary time of defect M

I M I

M

(c) Insert/Delete a kink

FIG. 2. (Color online) Update schemes.

location of defect M, and flips the spin state inbetween; seeFig. 2(b). The types of inbetween kinks are also interchanged.The acceptance probability is

Pmove = min {1, Fnew/Fold} . (12)

(c) Insert/Delete a kink. Each operation is chosen withprobability Ac in the G space. In “Insert a kink”, one ran-domly chooses one of the zd = 2d neighboring worldlines ofxM, say x′M, and updates the spatial location of M as (xM, τM)→(x′M, τM). Meanwhile, one inserts a kink k between worldlinesxM and x′M at imaginary time τk = mod(τM + δ, β), with a ran-dom displacement δ ∈ [−τc/2, τc/2) and δ , 0. Further, thespin states between τM and τk, on both xM and x′M, are flipped.Due to the Z2 symmetry of the Ising model, the types of in-between kinks, linking xM and x′M, remain unchanged. How-ever, the types of inbetween kinks are interchanged if they linksome other worldlines to xM or x′M. An example is illustratedin Fig. 2(c).

In the reverse operation, “Delete a kink”, one also picks upa random neighboring worldline x′M and moves M as (xM, τM)→(x′M, τM). Further, one counts the number of kinks nk thatconnect worldlines xM and x′M in the imaginary-time domain[τM − τc/2, τM + τc/2). If no kink exists nk = 0, the opera-tion is rejected. Otherwise, one randomly picks up one of thenk kinks and deletes it, and meanwhile flips the spin stateson both worldlines between τM and the imaginary time of thedeleted kink. Besides types of kinks linking xM or x′M to otherspins are changed as well.

Algorithm 1 Worm algorithmBEGIN: Given aZ configuration.loop

if it is aZ configuration thenchoose the “Create defects (I, M)” operation

elsechoose an operation with its a priori probability except “Cre-ate defects (I, M)”

end ifcalculate the acceptance probability P and carry out the opera-tion with probability P

end loop

4

The detailed balance condition of this pair of operationsreads

Ac ·1zd·

dτk

τc·W · Pinse = Ac ·

1zd·

1nk·W+ · Pdele , (13)

where Pinse (Pdele) is for the acceptance ratio for “Insert akink” (“Delete a kink”). The statistical weights W and W+,given by Eq. (9), are respectively for the configuration beforeand after inserting a kink. The infinitesimal dτk on the left-hand side is cancelled by W+, which has one more kink. Theacceptance probabilities are then

Pinse = min[1,

τc

nk+1Fnew

Fold

](14)

Pdele = min[1,

nk

τc

Fnew

Fold

],

where nk denotes the number of inbetween kinks for the cur-rent configuration. The denominator nk + 1 in Pinse is due tothat a kink is inserted.

The worm algorithm is then formulated as in Alg. 1, inwhich priori probabilitiesAa +Ab +2Ac = 1. It is mentionedagain that the acceptance probabilities in the updates can beoptimized by tuning the ranges of random τ-displacement, τa,τb and τc. As its analog for the classical Ising model whichcarries out a weighted random walk over the lattice, the defectM in this quantum Monte Carlo method effectively performs arandom travel in the whole spacetime and a symmetric updatefor the spin state passed by it.

For the conventional BH model which has neither the pair-ing term nor the Z2 symmetry, the interchange between thehopping and pairing kink cannot be allowed. For “Cre-ate/Annihilate defects” and “Move imaginary-time of M”, theabove illegal updates can be avoided when performing theseoperations only within a larger spin segment. In “Insert/Deletea kink”, the simplest remedy is that the proposed update is re-jected as long as it leads to an illegal configuration, giving aprice that the acceptance probabilities is decreased by a factorof O(1). As a more sophisticated remedy, one can reformulatethe operation in a way such that no illegal configuration wouldbe introduced.

Finally, for the computational efficiency, it is important toimplement hash tables such that each operation is done withinO(1) CPU time.

III. NUMERICAL RESULTS

In the absence of the external field (h = 0), the spin-up and-down spin states are fully balanced in the QTFI (2). As hturns on, the system evolves into a disordered phase with thespin-down state being suppressed and the spin-up order stillnot formed (h < hc). It enters into the spin-up ordered phase(h > hc) through a second-order quantum phase transition.The critical point is exactly known as hc/t = 1 in 1D andnumerically determined as hc/t ≈ 3.044 in 2D (square lattice).Without loss of generality, the pairwise interaction is set ast=1 from now unless stated explicitly.

0.0

0.2

0.4

0.6

0.8

0.0 0.4 0.8 1.2 1.6

(a)

1D

0.0 0.4 0.8 1.2 1.6 2.0

(b)

0.4

0.5

0.6

0.99 1.00 1.01

0.06

0.09

0.12

1.00 1.10 1.20

0.3

0.5

0.7

0.99 1.00 1.01

R↑

h

R↓

h

64128256

0.0

0.2

0.4

0.6

0.8

1.0

0.0 2.0 4.0

(c)

2D

0.0 2.0 4.0 6.0

(d)

R↑

h

R↓

h

163264

FIG. 3. (Color online) Wrapping probabilities R↑ and R↓ versus thetransverse field h. The error bars are much smaller than the size ofpoints. The vertical black lines indicate the critical point. (a) (b) arefor 1D, and (c) (d) are for 2D. The inset plot of (a) shows the curvenear hc = 1.

Using the worm algorithm, we simulate the 1D and 2D QT-FIs with linear lattice size L and inverse temperature β = L;the choice of β = L is due to the dynamic critical exponentz = 1 for the QTFI. Periodic boundary conditions are ap-plied along each spatial direction, so that the lattice is essen-tially a torus. The linear size is taken up to L = 512 in 1Dand L = 128 in 2D, and no severe critical slowing down isobserved. A variety of geometric and physical quantities issampled. To locate the phase transition hc, we make use ofthe topological properties of the non-intersecting loops on thetorus, instead of the scaling behavior of physical quantitieslike the magnetic susceptibility.

A. Critical point

Given aZ configuration, we record how many timesW`i ≥

0 each loop ` winds around the ith direction (i = 1, 2, · · · , d),and calculate the total winding numberWi =

∑`W

`i from all

the loops; see Fig. 1(c) for an illustration. The path-integralconfiguration is said to wrap along the ith direction as long asWi > 0; namely, there is at least a loop wrapping around theith direction. This is indicated as Ri = 1; otherwise, Ri = 0.The average wrapping probability R = (1/d)

∑i〈Ri〉 is then

calculated, with 〈·〉 representing the ensemble average. In thesub-percolating phase, the loops are too small to percolate,and the R value quickly drops to 0 as L becomes larger. Inthe super-percolating phase, there is at least one giant loopoccupying a finite fraction of the whole system, and the Rvalue rapidly converges to 1. At the percolation threshold, theR values for different system sizes L have an asymptoticallycommon intersection with a non-trivial value between 0 and 1.In short, the wrapping probability R is a dimensionless quan-tity characterizing the topological feature of loops on torus.

5

0.48

0.49

0.50

0.51

0.52

0.9998 0.9999 1.0000 1.0001 1.0002

1D

-0.02

0.00

0.02

-0.10 0.00 0.10

R↓

h

64256512

0.4995

fit

R↓vs.

(h− hc)Lyt

FIG. 4. (Color online) Wrapping probability R↓ in 1D. The grayband indicates an interval of 2σ above and below the estimate hc =

1.000 001(5). The yellow band indicates an interval of 2σ above andbelow the estimate Rc = 0.4995(3). The inset shows R↓ = R↓ − Rc −

biLyi versus (h − hc)Lyt , with a1 = −0.1625(9), a2 = −0.34(13), bi =

4.3(1.5), yi = −2 and yt = 1.

In many cases, such wrapping probabilities are found to ex-hibit small finite-size corrections, and have been widely usedto critical points24–28.

For the QTFI, two types of non-intersecting loops, spin-up(↑) or -down (↓), can be constructed. The Monte Carlo (MC)results in Fig. 3 show that irrespective of the spatial dimen-sion (1D or 2D), both the spin-down and -up loops displaycritical behaviors near the quantum critical point hc. In 1D,particularly rich behaviors are observed. In the whole disor-dered phase (0 ≤ h < hc), both R↑ and R↓ have non-trivialvalues, 0 < R↑(R↓) < 1, indicating the fractal structures ofthese loops. At the transition point h = hc, the R↑ and R↓values have a sharp drop, which becomes infinitely sharp forL → ∞. For h > hc, the R↓ value drops to 0, meaning that thespin-down loops are too small to percolate, while the R↑ valueconverges to non-trivial value if h is not too large, suggestingthat the spin-up loops still exhibit fractal properties. Never-theless, as h is further increased, the R↑ value also graduallyapproaches to 0. This is understandable because, while theimaginary-time lines are dominated by the spin-up state, thenumber of kinks decreases when h increases, and thus the spinloops are less likely to percolate. In 2D, the R↑ and R↓ valuesconverge to 1 in the disordered phase 0 ≤ h < hc, suggesting asuper-percolating phase both for the spin-up and -down loops.For h > hc, the R↓ value quickly reaches 0, but R↑ convergesto 1 for L→ ∞. At h = hc, the R↓ value has a sharp drop, andthe derivative of R↑ with respect to h probably also develops asingularity as L increases.

Extensive simulations are then carried out at h = hc = 1 in1D and h = 3.04435 in 2D, and the data nearby are obtainedby the standard reweighting technique29. The system size istaken as L = 16, 64, 256, 512 in 1D, with at least 4× 108 sam-ples for each L, and is L = 8, 16, 32, 64, 128 in 2D, with atleast 1 × 108 samples for each L.

According to the least-squares criterion, the R↓ data, partly

0.51

0.52

0.53

0.54

3.04415 3.04425 3.04435 3.04445 3.04455

2D

-0.01

0.00

0.01

-0.1 0.0 0.1

R↓

h

163264

128

0.5281

fit

R↓vs.

(h− hc)Lyt

FIG. 5. (Color online) Wrapping probability R↓ in 2D. The grayband indicates an interval of 2σ above and below the estimatehc = 3.044 330(6). The yellow band indicates an interval of 2σabove and below the estimaten Rc = 0.528 1(14). The inset displaysR↓ = R↓ − Rc − biLyi versus (h − hc)Lyt , with a1 = −0.0855(8), bi =

−0.031(8), yi = −0.821 and yt = 1.568.

shown in Figs. 4 and 5, are fitted by

R↓ = R↓,c +

2∑k=1

ak(h − hc)kLkyt + biLyi + b2Ly2 . (15)

The thermal renormalization exponent is fixed at the knownvalues in the classical (d+1) Ising universality–i.e., yt(1d) = 1and yt(2d) = 1.586830. The term with bi comes from the lead-ing irrelevant thermal field, which has the exponent yi(1d) =

−2 and yi(2d) = −0.82128,30. The subleading correction expo-nents are set as y2(1d) = −3 and y2(2d) = −2. As a precaution,we gradually increase Lmin and exclude the L < Lmin data fromthe fit to see how the ratio of the residual χ2 to the degree offreedom changes as Lmin.

In 1D, it is found that the MC data for Lmin = 64 can bewell described by Eq. (15) without the correction-to-scalingterm (b2 = 0). The fit yields hc = 1.000 001(5), in excellentagreement with the exact quantum critical point hc = 1. Also,we have R↓,c = 0.4995(3), which suggests that it might exactlybe 1/2; see the inset of Fig. 3(b).

In 2D, an eye-view fitting of the R↓ data in Fig. 5 alreadygives the critical point approximately as hc ≈ 3.044 33, withuncertainty at the 5th place. We find that it is sufficient to de-scribe these data by Eq. (15) with a2 = 0 and for Lmin = 16,b2 can also be set to zero. The fit gives R↓,c = 0.528 1(14) andhc = 3.044 330(6). To test the reliability of the value and theerror bar of hc, we plot in Fig. 6 the R↓ data against L at somefixed h near hc. It can be seen that at h = hc = 3.044 330, thewrapping probability R↓ converges to a constant value withinthe 2-sigma shadow area in Fig. 6. In contrast, as L increases,the R↓ data for h = 3.044 30 and 3.044 36 bend upward anddownward, respectively, suggesting that they are clearly awayfrom the thermodynamic critical point. For h = 3.044 38,which is the estimated critical point in Ref. 5, the down-ward bending is stronger, meaning that it cannot be the critical

6

0.520

0.525

0.530

0.535

20 40 60 80 100 120

2D

R↓

L

3.0443003.0443303.0443603.044380

0.5281

FIG. 6. (Color online) Plot of R↓ versus L for various values of hfor the 2D QTFI. The shaded yellow strip indicate an interval of 2σabove and below the estimate Rc = 0.528 1(14). The solid lines areplotted according to the fitting result.

point. Table I gives a (incomplete) list of the existing resultsfor hc in 2D.

We now briefly discuss the efficiency of the current wormalgorithm, which is already reflected by the precision of theestimated critical point hc. For a quantitative evaluation, wecalculate at criticality the integrated autocorrelation times τint

for the energy E, magnetization M and kink number Nk, inthe unit of MC sweeps. A unit of MC sweep is defined suchthat on average, each imaginary-time spin line is updated byβ times. From the least-squares fitting τ ∝ Lz, we obtainthe dynamical exponent as z

E= 0.38(3), z

M= 0.35(3) and

zNk

= 0.41(3) for 1D, and zE

= 0.28(3), zM

= 0.23(4) andzNk

= 0.30(3) for 2D. The efficiency of the worm algorithm ismuch better than that of the Metropolis algorithm for Hamil-tonian (1), and is comparable to that of the Wolff-like clustermethod. Note that as the spatial dimension increases, the val-

TABLE I. Estimated critical point hc on the square lattice.

Method hc

This work 3.044 330(6)CMC5 3.044 38(2)SSE8 3.044 2(4)S-W11 3.044(1)HOSVD(D=14)16 3.043 9iPEPS31 3.04MERA32 3.075CTM33 3.14

1 CMC: cluster Monte Carlo methodSSE: stochastic series expansionS-W: Swendsen-Wang in continuous timeHOSVD: Tensor renormalization group method based on the higher-order singular value decompositioniPEPS: infinity projected entangled-pair stateMERA: multiscale entanglement renormalization ansatzCTM: corner transfer matrix

102

103

104

16 32 64 128 256

1D

16 32 64 128 256101

102

103

104

1D

S1↑

L

0.40.60.81.01.2

S1↓

L

0.40.60.81.0

FIG. 7. (Color online) The largest-loop size S 1↑ and S 1↓ versus Lat h = 0.4, 0.6, 0.8, 1.0, 1.2 for 1D. For S 1↑, the straight lines have aslope 7/4, irrespective of the h value. For S 1↓, the lines have slope7/4 for h < hc and 11/8 for h = hc.

ues of z decreases. From the numerical results of the wormalgorithm for the classical Ising model21,34, we expect z = 0(without critical slowing down) for d ≥ dc, where dc = 3 isthe upper critical dimensionality for the QTFI.

B. Geometric properties of loops

We have determined the quantum critical point hc with ahigh precision by locating the percolation threshold of theloop configurations. Hereby, we shall further explore othergeometric properties of the spin-up and -down loops at andaway from hc. In the Z configuration space, we measurethe average length S 1 of the largest loop and the probabil-ity distribution that a randomly chosen loop is of size s, i.e.,P(s, L) ≡ (1/N`(L))dN`(s, L)/ds, where N`(L) ∼ βLd is thetotal number of loops and N`(s, L) is the number of loops ofsize in range (s, s + ds).

In 1D, one knows from Fig. 3 that for the whole region0 ≤ h ≤ hc, both the spin-up and -down loops exhibit criticalscaling behaviors. For the spin-up loops, such fractal proper-ties further survive in the ordered phase h > hc. In these cases,we expect that the largest-loop size scales as S 1 ∝ Ld` , whered` < 2 is the loop fractal dimension, and that the loop-sizedistribution behaves as35,36

P(s, L) ∼ s−τ f (s/Ld` ), (16)

where τ is called the Fisher exponent. Moreover, the ex-ponents, τ and d`, are related by the hyperscaling relationτ = 1 + (d + 1)/d`. The function f (x ≡ s/Ld` ) is universaland describes the finite-size cut-off of s near S 1 ∼ Ld` .

We simulate at h = 0.4, 0.6, 0.8, 1.0, 1.2 and the results areshown in Fig. 7. The straight lines in the log-log plot sug-gest that indeed, the largest loop has a fractal structure. Forthe spin-up loops, irrespective of the h value, the straight lineshave the same slope approximately as 7/4. Further, the am-plitude of the power law S 1↑ ∼ L7/4 increases as a functionof h, at least in the range 0.4 ≤ h ≤ 1.2. In contrast, as hincreases, the largest-loop size S 1↓ decreases and then dropsto a significantly smaller value at h = hc. Further, while thelines for h < hc still have a slope near 7/4, the line for h = hchas a smaller slope which is about 11/8. This suggests thatthe spin-down loops start with a dense and critical phase forh < hc, experiences a critical state at h = hc, and then enters

7

10−9

10−7

10−5

10−3

10−1

(a)

spin-up loops

10−9

10−7

10−5

10−3

10−1

(b)

10−9

10−7

10−5

10−3

10−1

100 101 102 103 104

(c)

(a’)

spin-down loops

(b’)

100 101 102 103 104

(c’)

64128256

P(s,L

)

s

64128256

s

FIG. 8. (Color online) Loop-size distribution P(s, L) for different hvalues in 1D. The figures in the left (right) panel are for the spin-up (-down) loops, and the 1st, 2nd and 3rd rows correspond to h =

0.8, 1.0, 1.2, respectively. The straight lines, with slope −15/7 or−27/11, are a guide for the eye.

into a sparse phase containing enormous small loops. The S 1data for both the spin-up and -down loops are fitted by

S 1 = Ld` (a0 + b1Ly1 ) , (17)

with different choices of the correction exponent y1 =

−0.5,−1.0 or −1.5. We find that the fits are rather stable, andthe results are shown in Tab. II.

We notice that the configuration of the classical O(n)loop model on the honeycomb lattice also consists of non-intersecting loops37–39. Moreover, the O(n) loop modelwith n = 1 corresponds to the 2D Ising model, and hasa hull/loop dimension as dhull = 11/8 at the critical point

xc = 1/√

2 +√

2 − n = 1/√

3 and dhull = 7/4 in the densephase x > xc, where x is the statistical weight for each loopunit37–39. These behaviors are very similar to those of thespin-down loops for the 1D QTFI. Accordingly, we conjecturethat in 1D, the fractal dimensions d`↓(h = hc) = 1.37(1) andd`↓(h < hc) = 1.750(6) are exactly identical to 11/8 and 7/4,respectively. We also conjecture that the fractal dimensiond`↑ = 1.747(5), which is independent of the h value, is also

TABLE II. Estimates of d`↑ and d`↓ at different h in 1D.

h 0.4 0.6 0.8 1.0(hc) 1.2

d`↑ 1.754(6) 1.750(5) 1.747(5) 1.750(6) 1.751(3)d`↓ 1.751(5) 1.749(7) 1.750(7) 1.37(1)

0

4

8

12

16(a)

spin-up loops

0

4

8

12

16 (b)

0

4

8

12

0 0.4 0.8 1.2 1.6

(c)

0.0

1.0

2.0

(a’)

spin-down loops

0.0

0.4

0.8

1.2(b’)

0 0.4 0.8 1.20.0

0.3

0.6

(c’)

64128256

P(s,L

)sτ

s/Ld`↑

64128256

P(s,L

)sτ

P(s,L

)sτ

s/Ld`↓

FIG. 9. (Color online) P(s, L)sτ versus s/Ld` in 1D. The plots in theleft (right) panel are for the spin-up (-down) loops, and the 1st, 2ndand 3rd rows correspond to h = 0.8, 1.0, 1.2, respectively. The valueof d`↑ and d`↓ is listed in Tab. II, and the τ value is calculated fromthe hyperscaling relation τ = 1 + (d + 1)/d`.

exactly equivalent to 7/4. Further, it is noted that by the dual-ity relation, the loops on the honeycomb lattice can be mappedonto the boundaries of the spin domains for the Ising model onthe triangular lattice. In the dense phase x > xc, these domainsare simply critical site percolation clusters. Therefore, we ex-pect that the domains, enclosed by the spin-up or -down loops,are also fractal and have a fractal dimension corresponding tothat for critical Ising spin domains or percolation clusters in2D.

To further demonstrate the fractal structure of the spin-upand -down loops in 1D, we display in Fig. 8 the MC data forthe loop-size distribution P(s, L). Indeed, one observes alge-braically decaying behaviors, s−τ, for the spin-up loops withh = 0.8, 1.0, 1.2 and for the spin-down loops with h = 0.8, 1.0.The cut-off size of s for the power-law scaling, due to finite-size effects, increases as the system size L. The hyperscalingrelation τ = 1 + (d + 1)/d` is well confirmed by the fact thatthe data for different L collapse onto the straight lines withslope −15/7 or −27/11. For the spin-down loops in the or-dered phase h = 1.2, the P(s, L) data for different L dropquickly, illustrating that the loop sizes are finite even in thethermodynamic limit L → ∞. We further plot sτP(s, L) ver-sus s/Ld` in Fig. 9. With the values of (d`, τ) as (7/4, 15/7)or (11/8, 27/11), the data for different L collapse well ontoa single curve, illustrating the universal feature of the cut-off

function f (x). It is interesting to see that for the spin-downloops at h = hc, function f (x) displays a two-peak structure(Fig. 9 (b’)). We regard that the first peak at the smaller valueof x reflects the residual effect of the spin-down loops in thedisordered phase h < hc. Meanwhile, it is observed that for the

8

102

103

104

24 25 26 27 28

1D

24 25 26102

103

104

2D

Tω

L L

FIG. 10. (Color online) Worm-return time at criticality hc. Thestraight lines, with slope 2yh − (d + 1), are a guide for the eye.

spin-up loops with h = 1.2, function f (x) exhibits a shoulderfeature on the smaller-x side. We expect that as h increases,such a shoulder feature would become more pronounced andits location would move toward the value x = 0. This isbecause that as h is enhanced, the number of kinks will begradually suppressed and the sizes of the spin-up loops willeventually start to decrease. In the limiting case h → ∞, allthe spin-up loops will become individual imaginary-time lineswith length β.

In 2D, the spin-down loops are fractal only at h = hc, andthe spin-up loops are always in a super-percolating phase. Thefit of the S 1↓ data by Eq. (17) gives d`↓ = 1.75(3). Again,this is in excellent agreement with the loop dimension dhull =

1.734(4) for the classical O(n = 1) loop model on the 3Dhydrogen-peroxide lattice40, on which the loops are also non-intersecting. As expected, for the spin-down loops at h = hc =

3.044 330, the loop-size distribution P(s, L) follows Eq. (16).

C. Worm-return time

The worm-return time Tw is the average update times be-tween two subsequent Z configurations in the markov chainMonte Carlo simulation. Mathematically, it can be expressedas the integral of spin-spin correlation function (8) over thelattice and the imaginary time as

Tw =ωG

ZTr

Tτ∫ β

0

∫ β

0dτIdτM

∑i, j

σxi (τ

I)σx

j(τM )e−βH

=ωG

ZTr

∫ β

0dτ

∑i

σxi (τ)

2

e−βH

=ωG

ZTr

(∫ β

0dτ Mx(τ)

)2

e−βH . (18)

With the choice of ωG = 1/βN, the worm-return Tw is pre-cisely equal to the dynamic magnetic susceptibility χxx =

〈(∫ β

0 dτ Mx(τ))2〉/βN. Fig. 10 shows the Tw data at h = hcfor both 1D and 2D, which are fitted by

Tw = L2yh−(d+1)(a0 + biLyi ) , (19)

In 1D, the fit with yi = −2 gives yh = 1.876(2), in excel-lent agreement with the exact value 15/8. In 2D, we setyi = −0.82130 and obtain yh = 2.484(4), which is again well

0

1

2

3

0.90 0.95 1.00 1.05 1.10

1D

0.31 0.32 0.33 0.34 0.350

10

20

30

402D

0.00

0.06

−6 −3 0 3 6

2D

−1.0 0.0 1.00.0

1.02D

χF/Ld

t

1624324864

t

8121624

χF/L2yt

(t−tc)Lyt

χF/L2yt

(t−tc)Lyt

FIG. 11. (Color online) Fidelity susceptibility χF (t) versus t, with(a) for 1D and (b) for 2D. The solid lines indicate the critical pointstc(1d) = 1 and tc(2d) = 1/3.044 330.

consistent with the result yh = 2.4816(1) for the classical 3DIsing model30.

D. Fidelity susceptibility

It is well known that many systems can undergo quantumphase transitions without spontaneous symmetry breaking andthus without good definition of local order parameter. Thesephase transitions are beyond the Ginzburg-Landau paradigm,and difficult to be detected by conventional thermodynamicobservables. Fidelity susceptibility, a quantity proposed in thequantum information science41, has been shown to be usefulfor such a purpose8,42–45. Consider a quantum phase transitiondriven by some given parameter λ and let |φ(λ)〉 represent thecorresponding wave function, the fidelity F(λ, ε) of the sys-tem is defined as the overlap between the wave functions withdifferent values of λ–i.e., F(λ, ε) = |〈φ(λ)|φ(λ + ε)〉|. Accord-ingly, the fidelity susceptibility χF (λ) is calculated as:

χF (λ) = −∂2 ln F(λ, ε)

∂ε2

∣∣∣∣∣∣ε=0

. (20)

For the QTFI, we hereby choose the driving parameter λ to bethe pairwise interaction t, which is conjugate to the numberof kinks Nk. Given a Z configuration, let Nk,1 and Nk,2 de-note the total number of kinks in the 1st-half imaginary-timedomain 0 < τ ≤ β/2 and the 2nd-half one β/2 < τ ≤ β,respectively. It can be shown by following Ref. 43 that the fi-delity susceptibility χF (t) is proportional to the covariance ofNk,1 and Nk,2, and can be written as

χF (t) =〈Nk,1Nk,2〉 − 〈Nk,1〉〈Nk,2〉

2t2 , (21)

where the external field h is now set to be 1. The MC dataof χF (t) for the 1D and 2D QTFIs are shown in Fig. 11. Asexpected, the χF (t) data for each L display a peak near thecritical point tc. As system size L increases, the peak locationtL , called the pseudo-critical point, moves toward the thermo-dynamic critical point tc, and the peak itself becomes sharperwith a smaller width. Following the standard finite-size scal-ing analysis, we expect that near the critical point tc, the fi-delity susceptibility χF scales as

χF (t, L) = L2yt fχF(Lyt (t − tc)) . (22)

9

Indeed, making use of yt(1d) = 1 and yt(2d) = 1.5868, weobtain a good collapse when plotting χF/L

2yt versus (t− tc)Lyt ,as shown in the insets of Fig. 11.

For the fidelity F(λ, ε), we can also choose the driving pa-rameter to be the external field h for the QTFI, which is con-jugate to the σz-component magnetization M. In this case,we should consider the magnetization M1 for 0 < τ ≤ β/2andM2 for β/2 < τ ≤ β, and the fidelity susceptibility χF (h)would be proportional to the covariance of M1 and M2. Atthe critical point, we expect χF ∝ L2yh .

While Fig. 11 illustrates the applicability of the fidelitysusceptibility χF as a tool for studying the quantum phasetransition, it is worth mentioning that by calculating the co-variance of two quantities of the same kind but in separatedspatial/imaginary-time domains, χF normally has large fluctu-ations. Thus, to achieve a good statistics for χF would requestextensive simulations.

IV. DISCUSSION

We formulate a worm-type algorithm and study the QTFIin a path-integral representation in which configurations aresets of non-intersecting loops. By locating the percolationthreshold of loop configurations via the so-called wrappingprobability, we obtain a high-precision quantum critical pointhc = 3.044 330(6) for the QTFI on the square lattice. Thesenon-intersecting loops are further observed to exhibit rich ge-ometric properties, particularly in 1D, where both the spin-down and -up loops have fractal structures over a wide pa-rameter range. By examining the similarity of the scalingbehaviors for the d-dimensional QTFI and for the (d + 1)-dimensional classical O(n = 1) model, we conjecture thatin 1D the two fractal dimensions are d`↓(hc) = 11/8 andd`↓(h < hc) = 7/4, and that in 2D, d`↓(hc) = 1.75(3) is equal tothe hull dimension dhull = 1.734(4) for the classical 3D loopmodel. The finite-size scaling of magnetic and fidelity sus-ceptibilities are also examined. It is confirmed that the fidelitysusceptibility can be used to probe quantum phase transitions.

Motivated by the fact that the classical O(1) loop model isa specific case of the O(n) loop model with n = 1, we cangeneralize the loop path-integral representation of the QTFIby giving each spin-down loop a statistical weight n. As a

consequence, the partition function (5) is generalized to be

Z(t, t′, h, n) =∑{α0}

∞∑N=0

∫ β

0

∫ β

τ1

· · ·

∫ β

τN−1

N∏k=1

dτi

nN`↓ tNh t′Np e−∫ β

0 U(τ)dτ

=C∑{α0}

∞∑N=0

∫ β

0

∫ β

τ1

· · ·

∫ β

τN−1

N∏k=1

dτi

nN`↓ tNh t′Np (e−2h)S`↓ , (23)

where C = ehβN ,N`↓ specifies the number of spin-down loopsand S`↓ is the total length of spin-down loops. We expect thatfor t = t′, the phase transition of such a “quantum O(n) loop”model in d dimensions will belong to the same universalityclass as that for the classical O(n) loop model in (d + 1) di-mensions. In 1D, we further expect that the exact value of thequantum critical point hc(n) can be obtained for the“quantumO(n) loop” model, and that the spin-down loops would ex-hibit rich geometric properties both at criticality hc(n) and inthe disordered phase h < hc. In particular, for (d = 1, n = 2),the phase transition would be of the celebrated Berenzinskii-Kosterlize-Thouless topological transition. All these expec-tations can be explored by the current worm-type algorithm,and remain to be a future work.

The efficiency of the current worm algorithm implies itsbroad applications in a variety of spin and hard-core systems.A straightforward application is to simulate the QTFI on other2D lattices and in 3D; for the latter, one expects very minor orabsent critical slowing down, and thus interesting logarithmiccorrections can be examined. It can be also of significant rele-vance in solid-state experiments, since the pairing termsσ+

i σ+j

and σ−i σ−j are found to occur in frustrated quantum materials

due to the dipolar-octupolar doublets46–60. Further, in additionto the external field h, one can introduce pairing interactionσz

iσzj along the σz direction, which can be either ferromag-

netic or anti-ferromagnetic. This allows the worm-type studyof quantum spin systems with geometric frustration with re-spect to the σz component. In combination with the so-calledclock Monte Carlo method61, one can even study spin sys-tems with long-range σz

iσzj interaction without heavy compu-

tational overhead. Finally, we mention that a similar wormalgorithm has recently been used in the SSE representation ofthe hard-core bosonic Hubbard model with pairing terms62.

ACKNOWLEDGMENTS

We acknowledge Nikolay Prokofiev for initializing theproject and Xu-Ping Yao for technical help in preparingFig. 1(c). This work is supported by the National ScienceFoundation for Distinguished Young Scholars (NSFDYS) un-der Grant No. 11625522.

∗ jiangyi@ustc.edu.cn † yjdeng@ustc.edu.cn

10

1 Sachdev Subir, Quantum Phase Transitions (Cambridge Univer-sity Press, 2011).

2 A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T. F. Rosen-baum, and D. Sen, “Quantum phase transitions in transverse fieldspin models: from statistical physics to quantum information,”arXiv:1012.0653 (2010).

3 P. Pfeuty, “One-dimensional ising model with a transverse field,”Ann. Phys. 57, 79 (1970).

4 G. Vidal, “Classical simulation of infinite-size quantum latticesystems in one spatial dimension,” Phys. Rev. Lett. 98, 070201(2007).

5 Henk WJ Blote and Youjin Deng, “Cluster monte carlo simulationof the transverse ising model,” Phys. Rev. E 66, 066110 (2002).

6 Alexei Kitaev and John Preskill, “Topological entanglement en-tropy,” Phys. Rev. Lett. 96, 110404 (2006).

7 Michael Levin and Xiao-Gang Wen, “Detecting topological or-der in a ground state wave function,” Phys. Rev. Lett. 96, 110405(2006).

8 A. Fabricio Albuquerque, Fabien Alet, Clement Sire, and Syl-vain Capponi, “Quantum critical scaling of fidelity susceptibility,”Phys. Rev. B 81, 064418 (2010).

9 Tuncer Kaya, “One-dimensional quantum transverse-field isingmodel: A semiclassical transfer matrix approach,” Int. J. Mod.Phys. B 24, 5457–5468 (2010).

10 Olav F. Syljuø asen and Anders W. Sandvik, “Quantum montecarlo with directed loops,” Phys. Rev. E 66, 046701 (2002).

11 Heiko Rieger and Naoki Kawashima, “Application of a continu-ous time cluster algorithm to the two-dimensional random quan-tum ising ferromagnet,” Eur. Phys. J. B 9, 233–236 (1999).

12 N.V Prokof’ev, B.V Svistunov, and I.S Tupitsyn, “worm algo-rithm in quantum monte carlo simulations,” Phys. Lett. A 238,253 – 257 (1998).

13 Tohru Ikegami, Seiji Miyashita, and Heiko Rieger, “Griffiths-mccoy singularities in the transverse field ising model on the ran-domly diluted square lattice,” J. Phys. Soc. Jpn. 67, 2671–2677(1998).

14 S. V. Isakov and R. Moessner, “Interplay of quantum and ther-mal fluctuations in a frustrated magnet,” Phys. Rev. B 68, 104409(2003).

15 V. I. Iglovikov, R. T. Scalettar, R. R. P. Singh, and J. Oitmaa,“Disorder line and incommensurate floating phases in the quan-tum ising model on an anisotropic triangular lattice,” Phys. Rev.B 87, 214415 (2013).

16 Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang,“Coarse-graining renormalization by higher-order singular valuedecomposition,” Phys. Rev. B 86, 045139 (2012).

17 Steven R. White, “Density matrix formulation for quantum renor-malization groups,” Phys. Rev. Lett. 69, 2863 (1992); Phys. Rev.B 48, 10345(1993).

18 F. Verstraete and J. I. Cirac, “Renormalization algorithms forquantum-many body systems in two and higher dimensions,”arXiv:cond-mat/0407066 (2004).

19 Giuseppe Carleo and Matthias Troyer, “Solving the quantummany-body problem with artificial neural networks,” Science 355,602–606 (2017).

20 Juan Carrasquilla and Roger G. Melko, “Machine learning phasesof matter,” Nat. Phys. 13, 431– (2017).

21 Nikolay Prokof’ev and Boris Svistunov, “Worm algorithms forclassical statistical models,” Phys. Rev. Lett. 87, 160601 (2001).

22 Youjin Deng, Timothy M. Garoni, and Alan D. Sokal, “Dynamiccritical behavior of the worm algorithm for the ising model,” Phys.Rev. Lett. 99, 110601 (2007).

23 Under this condition, τa is almost constant for different βLd. Be-sides, the acceptance ratio will not be too small or too large.

24 M. E. J. Newman and R. M. Ziff, “Efficient monte carlo algorithmand high-precision results for percolation,” Phys. Rev. Lett. 85,4104–4107 (2000).

25 P. H. L. Martins and J. A. Plascak, “Percolation on two- and three-dimensional lattices,” Phys. Rev. E 67, 046119 (2003).

26 Xiaomei Feng, Youjin Deng, and Henk W. J. Blote, “Percolationtransitions in two dimensions,” Phys. Rev. E 78, 031136 (2008).

27 Junfeng Wang, Zongzheng Zhou, Wei Zhang, Timothy M. Garoni,and Youjin Deng, “Bond and site percolation in three dimensions,”Phys. Rev. E 87, 052107 (2013).

28 Pengcheng Hou, Sheng Fang, Junfeng Wang, Hao Hu, and YoujinDeng, “Geometric properties of the fortuin-kasteleyn representa-tion of the ising model,” Phys. Rev. E 99, 042150 (2019).

29 D. P. Landau and K. Binder, A guide to Monte Carlo simula-tions in statistical physics, chap 7, 4th. ed. (Cambridge UniversityPress, 2014).

30 Youjin Deng and Henk W. J. Blote, “Simultaneous analysis ofseveral models in the three-dimensional ising universality class,”Phys. Rev. E 68, 036125 (2003).

31 Roman Orus and Guifre Vidal, “Simulation of two-dimensionalquantum systems on an infinite lattice revisited: Corner transfermatrix for tensor contraction,” Phys. Rev. B 80, 094403 (2009).

32 G. Evenbly and G. Vidal, “Entanglement renormalization in twospatial dimensions,” Phys. Rev. Lett. 102, 180406 (2009).

33 Roman Orus, “Exploring corner transfer matrices and corner ten-sors for the classical simulation of quantum lattice systems,” Phys.Rev. B 85, 205117 (2012).

34 Eren Metin Elci, Jens Grimm, Lijie Ding, Abrahim Nasrawi, Tim-othy M. Garoni, and Youjin Deng, “Lifted worm algorithm for theising model,” Phys. Rev. E 97, 042126 (2018).

35 K Binder, “clusters in the ising model, metastable states and es-sential singularity,” Ann. Phys. 98, 390 – 417 (1976).

36 Michael E. Fisher, “The theory of condensation and the criticalpoint,” Phys. Phys. Fiz. 3, 255–283 (1967).

37 Qingquan Liu, Youjin Deng, and Timothy M. Garoni, “Wormmonte carlo study of the honeycomb-lattice loop model,” NuclearPhysics B 846, 283 – 315 (2011).

38 H. Saleur and B. Duplantier, “Exact determination of the percola-tion hull exponent in two dimensions,” Phys. Rev. Lett. 58, 2325–2328 (1987).

39 Antonio Coniglio, “Fractal structure of ising and potts clusters:Exact results,” Phys. Rev. Lett. 62, 3054–3057 (1989).

40 Qingquan Liu, Youjin Deng, Timothy M. Garoni, and Henk W.J.Blte, “The o(n) loop model on a three-dimensional lattice,” Nu-clear Physics B 859, 107 – 128 (2012).

41 M. Nielsen and I. Chuang, Quantum Computation and QuantumInformation (Cambridge University Press, 2010).

42 Paolo Zanardi and Nikola Paunkovic, “Ground state overlap andquantum phase transitions,” Phys. Rev. E 74, 031123 (2006).

43 Lei Wang, Ye-Hua Liu, Jakub Imriska, Ping Nang Ma, andMatthias Troyer, “Fidelity susceptibility made simple: A unifiedquantum monte carlo approach,” Phys. Rev. X 5, 031007 (2015).

44 Shi-Jian Gu and Hai-Qing Lin, “Scaling dimension of fidelity sus-ceptibility in quantum phase transitions,” EPL 87, 10003 (2009).

45 Lorenzo Campos Venuti and Paolo Zanardi, “Quantum criticalscaling of the geometric tensors,” Phys. Rev. Lett. 99, 095701(2007).

46 Yi-Ping Huang, Gang Chen, and Michael Hermele, “Quantumspin ices and topological phases from dipolar-octupolar doubletson the pyrochlore lattice,” Phys. Rev. Lett. 112, 167203 (2014).

47 M. Ciomaga Hatnean, M. R. Lees, O. A. Petrenko, D. S. Kee-ble, G. Balakrishnan, M. J. Gutmann, V. V. Klekovkina, andB. Z. Malkin, “Structural and magnetic investigations of single-crystalline neodymium zirconate pyrochlore Nd2Zr2O7,” Phys.

11

Rev. B 91, 174416 (2015).48 A. Bertin, P. Dalmas de Reotier, B. Fåk, C. Marin, A. Yaouanc,

A. Forget, D. Sheptyakov, B. Frick, C. Ritter, A. Amato,C. Baines, and P. J. C. King, “Nd2Sn2O7: An all-in–all-out py-rochlore magnet with no divergence-free field and anomalouslyslow paramagnetic spin dynamics,” Phys. Rev. B 92, 144423(2015).

49 E. Lhotel, S. Petit, S. Guitteny, O. Florea, M. Ciomaga Hatnean,C. Colin, E. Ressouche, M. R. Lees, and G. Balakrishnan, “Fluc-tuations and all-in–all-out ordering in dipole-octupole Nd2Zr2O7,”Phys. Rev. Lett. 115, 197202 (2015).

50 J. Xu, V. K. Anand, A. K. Bera, M. Frontzek, D. L. Aber-nathy, N. Casati, K. Siemensmeyer, and B. Lake, “Magneticstructure and crystal-field states of the pyrochlore antiferromag-net Nd2Zr2O7,” Phys. Rev. B 92, 224430 (2015).

51 V. K. Anand, A. K. Bera, J. Xu, T. Herrmannsdorfer, C. Ritter,and B. Lake, “Observation of long-range magnetic ordering in py-rohafnate Nd2Hf2O7: A neutron diffraction study,” Phys. Rev. B92, 184418 (2015).

52 Owen Benton, “Quantum origins of moment fragmentation inNd2Zr2O7,” Phys. Rev. B 94, 104430 (2016).

53 J. Xu, C. Balz, C. Baines, H. Luetkens, and B. Lake, “Spin dy-namics of the ordered dipolar-octupolar pseudospin- 1

2 pyrochloreNd2Zr2O7 probed by muon spin relaxation,” Phys. Rev. B 94,064425 (2016).

54 V. K. Anand, D. L. Abernathy, D. T. Adroja, A. D. Hillier, P. K.Biswas, and B. Lake, “Muon spin relaxation and inelastic neu-tron scattering investigations of the all-in/all-out antiferromagnetNd2Hf2O7,” Phys. Rev. B 95, 224420 (2017).

55 P. Dalmas de Reotier, A. Yaouanc, A. Maisuradze, A. Bertin, P. J.Baker, A. D. Hillier, and A. Forget, “Slow spin tunneling in theparamagnetic phase of the pyrochlore Nd2Zr2O7,” Phys. Rev. B

95, 134420 (2017).56 Romain Sibille, Elsa Lhotel, Vladimir Pomjakushin, Chris

Baines, Tom Fennell, and Michel Kenzelmann, “Candidate quan-tum spin liquid in the Ce3+ pyrochlore stannate Ce2Sn2O7,” Phys.Rev. Lett. 115, 097202 (2015).

57 Yao-Dong Li and Gang Chen, “Symmetry enriched U(1) topolog-ical orders for dipole-octupole doublets on a pyrochlore lattice,”Phys. Rev. B 95, 041106 (2017).

58 J. Gaudet, E. M. Smith, J. Dudemaine, J. Beare, C. R. C. Buhari-walla, N. P. Butch, M. B. Stone, A. I. Kolesnikov, GuangyongXu, D. R. Yahne, K. A. Ross, C. A. Marjerrison, J. D. Garrett,G. M. Luke, A. D. Bianchi, and B. D. Gaulin, “Quantum spin icedynamics in the dipole-octupole pyrochlore magnet Ce2Zr2O7,”Phys. Rev. Lett. 122, 187201 (2019).

59 Bin Gao, Tong Chen, David W Tam, Chien-Lung Huang,Kalyan Sasmal, Devashibhai T Adroja, Feng Ye, Huibo Cao,Gabriele Sala, Matthew B Stone, Christopher Baines, Joel A TVerezhak, Haoyu Hu, Jae-Ho Chung, Xianghan Xu, Sang-WookCheong, Manivannan Nallaiyan, Stefano Spagna, M Brian Maple,Andriy H Nevidomskyy, Emilia Morosan, Gang Chen, andPengcheng Dai, “Experimental signatures of a three-dimensionalquantum spin liquid in effective spin-1/2 Ce2Zr2O7 pyrochlore,”Nat. Phys. (2019), 10.1038/s41567-019-0577-6.

60 Yao Dong Li and Gang Chen, “Non-spin-ice pyrochlore U(1)quantum spin liquid: Manifesting mixed symmetry enrichments,”arXiv:1902.07075 (2019).

61 Manon Michel, Xiaojun Tan, and Youjin Deng, “Clock montecarlo methods,” Phys. Rev. E 99, 010105 (2019).

62 Alvin J. R. Heng, Wenan Guo, Anders W. Sandvik, and PinakiSengupta, “Pair hopping in systems of strongly interacting hard-core bosons,” Phys. Rev. B 100, 104433 (2019).