Fidelity Susceptibility Made Simple: A Unified Quantum Monte Carlo Approach

Lei Wang,1 Ye-Hua Liu,1 Jakub Imriška,1 Ping Nang Ma,2 and Matthias Troyer11Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland

2Yotcopi Technologies, National University of Singapore, 117576, Singapore(Received 10 March 2015; revised manuscript received 8 June 2015; published 15 July 2015)

The fidelity susceptibility is a general purpose probe of phase transitions. With its origin in quantuminformation and in the differential geometry perspective of quantum states, the fidelity susceptibility canindicate the presence of a phase transition without prior knowledge of the local order parameter, as well asreveal the universal properties of a critical point. The wide applicability of the fidelity susceptibility toquantum many-body systems is, however, hindered by the limited computational tools to evaluate it. Wepresent a generic, efficient, and elegant approach to compute the fidelity susceptibility of correlatedfermions, bosons, and quantum spin systems in a broad range of quantum Monte Carlo methods. It can beapplied to both the ground-state and nonzero-temperature cases. The Monte Carlo estimator has a simpleyet universal form, which can be efficiently evaluated in simulations. We demonstrate the power of thisapproach with applications to the Bose-Hubbard model, the spin-1=2 XXZ model, and use it to examine thehypothetical intermediate spin-liquid phase in the Hubbard model on the honeycomb lattice.

DOI: 10.1103/PhysRevX.5.031007 Subject Areas: Computational Physics,Condensed Matter Physics,Quantum Physics

I. INTRODUCTION

Phase transitions highlight the beauty of universality,despite the great diversity of nature. For example, one findsa unified description for systems ranging from ultracoldbosons [1,2] to magnetic insulators [3–5] on the verge of aphase transition. Phase transitions originate from thecompetition between different tendencies when a macro-scopic system tries to organize itself. Thermal fluctuationscan drive classical phase transitions at nonzero temper-atures, while quantum phase transitions can occur even atzero temperature because of the competition between non-commuting terms in the quantum-mechanical Hamiltonian[6,7]. At the phase transition point, physical observablesoften exhibit singular behavior. In this respect, phasetransitions are the most dramatic manifestation of the lawsof statistical and quantum mechanics.Traditional descriptions of phase transitions are based on

low-energy effective theories of local order parameters,which have had enormous success in explaining variousphase transitions of superfluids, superconductors [8], andquantum magnets [9,10]. However, in recent years, excep-tions to this Ginzburg-Landau-Wilson paradigm haveemerged [11]. In particular, topological phase transitions[12–14] do not have a local order parameter on either side of

the phase transition. Therefore, new theoretical tools areneeded to search for and characterize these new quantumphases and phase transitions. Many concepts in quantuminformation science [15], such as quantum fidelity andquantum entanglement, have proven to be useful [16,17].Having a point of view that is totally different from thetraditional condensed matter approach, they do not assumethe presence of a local order parameter and thus offer newperspectives of the phase transitions and their universalities.Specifically, we consider the following one-parameter

family of Hamiltonians with a driving parameter λ:

HðλÞ ¼ H0 þ λH1: ð1ÞAs λ changes, the system may go through one or severalphase transition(s) because of the competition between H0

and H1. The quantum fidelity measures the distance on themanifold of λ, which is defined as the overlap betweenthe ground-state wave functions at two different values ofthe driving parameter,

Fðλ; ϵÞ ¼ jhΨ0ðλÞjΨ0ðλþ ϵÞij; ð2Þwhere HðλÞjΨnðλÞi ¼ EnðλÞjΨnðλÞi, and n ¼ 0 corre-sponds to the ground state. Unless otherwise stated, weassume the wave functions are normalized and there is noground-state degeneracy. It is anticipated that the fidelitywill exhibit a dip when the two wave functions arequalitatively different, e.g., when they belong to differentphases [18]. This wave function overlap is also related tothe Anderson orthogonality catastrophe [19] and theLoschmidt echo in quantum dynamics [20].

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

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Since in general the quantum fidelity vanishes exponen-tially with the system size for a many-body system, itis more convenient to study the change of its logarithmwith respect to the driving parameter, called the fidelitysusceptibility [21]:

χFðλÞ ¼ −∂2 lnF∂ϵ2

����ϵ¼0

: ð3Þ

The first-order derivative vanishes because F is at itsmaximum when ϵ ¼ 0. In general, the fidelity susceptibil-ity is an extensive quantity away from the critical point, butit exhibits a maximum or even diverges at the critical point,thus indicating a quantum phase transition [22,23]. Similarto conventional thermodynamic quantities, it also follows ascaling law close to the critical point [22–25], which can beused to extract universal information about the phasetransition. An important feature of the fidelity susceptibilityis that it can reveal a phase transition without priorknowledge of the local order parameter. This makes itsuitable for detection of topological phase transitions[26–29] and Berezinsky-Kosterlitz-Thouless-type transi-tions [30–32] as well as for tackling challenging caseswhere an in-depth understanding of the underlying physicsis still lacking [33,34]. Interestingly, the fidelity suscep-tibility may also be accessible to experiments [35–37].Despite its appealing features, the difficulty in calculat-

ing the fidelity susceptibility has hindered its use innumerical simulations. Many previous studies were thuslimited to the cases where the ground-state wave functionoverlap could be calculated from the analytical solution,exact diagonalization, or density-matrix renormalization-group (DMRG) methods [16].There are several equivalent formulations of the fidelity

susceptibility Eq. (3), which reveal different aspects of thequantity. From a computational point of view, they offerdirect ways to calculate the fidelity susceptibility withoutthe need to perform numerical derivatives of the fidelity asin Eq. (2).(a) Expanding jΨ0ðλþ ϵÞi for small ϵ, one can cast the

definition Eq. (3) into an explicit form [22,38]:

χFðλÞ¼h∂λΨ0j∂λΨ0ihΨ0jΨ0i

−hΨ0j∂λΨ0ihΨ0jΨ0i

h∂λΨ0jΨ0ihΨ0jΨ0i

: ð4Þ

The above form does not assume properly normalizedwave functions jΨ0i. Equation (4) reveals the geo-metric content of the fidelity susceptibility [22,38],since this expression is the real part of the quantumgeometric tensor [39].

(b) Alternatively, one can calculate the first-order pertur-bation for jΨ0ðλþ ϵÞi and get

χFðλÞ ¼Xn≠0

jhΨnðλÞjH1jΨ0ðλÞij2½EnðλÞ − E0ðλÞ�2

: ð5Þ

Compared to Eq. (4), Eq. (5) does not containderivatives but involves all eigenstates and the fullspectrum. It explicitly shows that χF ≥ 0 and suggeststhe divergence of χF when the energy gap of thesystem closes.

(c) Reference [21] views Eq. (5) as the zero-frequencycomponent of a spectral representation; thus, a Fouriertransform is performed to obtain an alternativeexpression,

χFðλÞ ¼Z

∞

0

dτ½hΨ0jH1ðτÞH1jΨ0i − hΨ0jH1jΨ0i2�τ;

ð6Þ

where H1ðτÞ ¼ eHτH1e−Hτ. Equation (6) has the formof a linear-response formula and is computationallymore friendly than Eq. (4) or Eq. (5). Referen-ces [24,25] generalize it to nonzero temperature byreplacing the integration limit with β=2:

χFðλÞ ¼Z

β=2

0

dτ½hH1ðτÞH1i − hH1i2�τ; ð7Þ

where h� � �i denotes the thermal average at inversetemperature β. Besides reducing to Eq. (6) as β → ∞,Eq. (7) is nevertheless a well-defined quantity atnonzero temperatures. It bounds the divergence ofan alternative “mixed state” fidelity susceptibility[40,41], which is based on the Uhlmann fidelity[42,43], and both quantities follow the same scalinglaw close to a quantum critical point [24,25]. Ingeneral, the evaluation of Eq. (7) is still a formidablecomputational task, which requires ad hoc implemen-tation depending on the details of the Hamiltonian.For example, the fidelity susceptibility for two-dimensional quantum spin systems is calculated inRefs. [24,25] using a quantum Monte Carlo (QMC)method, while for a one-dimensional quantum spinsystem, Ref. [44] computed it using the transfer-matrix DMRGmethod. The nontrivial implementationof these specific approaches and the overhead in thecalculation still limits the wide applicability of thefidelity susceptibility approach to a broad range ofquantum many-body systems.

In this paper, we present a simple yet generic approach tocompute the fidelity susceptibility in a largevariety ofmodernquantum Monte Carlo methods, including the continuous-time worldline [45–48] and stochastic series expansion(SSE) [49] methods for bosons and quantum spins, andthe diagrammatic determinantal methods for quantumimpurity [50–53] and fermion lattice models [54–56].

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In all cases, the Monte Carlo estimator is generic and theimplementations are straightforward. As long as the quan-tumMonte Carlo simulation is feasible (not hindered by thesign problem), the fidelity susceptibility can be easilycalculated. Our finding can boost the investigation ofquantum phase transitions from a quantum informationperspective and becomes especially advantageous for theexploration of exotic phases beyond the Ginzburg-Landau-Wilson paradigm.The organization of the paper is as follows. In Sec. II,

we present our estimator for the fidelity susceptibilityand discuss its implementations in various quantumMonte Carlo methods. Section III presents derivations ofthe estimator. In Sec. IV, we demonstrate the power of thefidelity susceptibility approach with applications to variousmodels, including correlated bosons, fermions, andquantum spins, using a variety of quantum Monte Carlomethods. Section V discusses the relation between the zero-temperature and nonzero-temperature estimators for thefidelity susceptibility and compares them to the previousapproaches [24,25]. We conclude with future prospectsin Sec. VI.

II. RESULTS

We first present our results on the estimator of thefidelity susceptibility in a general setting, then discuss itsimplementations in various QMC methods, includingthe continuous-time worldline [45–48] and diagrammaticdeterminantal approaches [50–56], and the stochastic seriesexpansion method [49]. In all cases, the fidelity suscep-tibility can be measured with little effort.

A. Universal covariance estimator

Many modern QMC methods [45–56] share a unifiedconceptual framework, namely that the partition function iscalculated as a perturbative series expansion for the λH1

term,

Z ¼ Trðe−βHÞ ¼X∞k¼0

λkXCk

wðCkÞ; ð8Þ

where the second summation runs over all the Monte Carloconfigurations of a given expansion order k. The detailedmeaning of the configuration depends on the specific QMCalgorithm and will be explained in the next subsection.Figure 1(a) depicts a generic configuration, where the kobjects residing on the periodic imaginary-time axisrepresent the vertices λH1 in the expansion, with aMonte Carlo weight λkwðCkÞ for this configuration.QMC simulations [45–56] sample the summation over kand Ck on an equal footing. Specific algorithms differ bythe detailed form of wðCkÞ and by the sampling schemes.Nevertheless, these QMC methods share a unified frame-work provided by Eq. (8), which is the only requirement for

the estimator of the fidelity susceptibility Eq. (7) to possessan appealing universal form in nonzero-temperature QMCsimulations:

χT≠0F ¼ hkLkRi − hkLihkRi2λ2

; ð9Þ

where kL and kR are the number of vertices residing inthe range ½β=2; βÞ and ½0; β=2Þ of the imaginary-timeaxis, respectively, shown in Fig. 1(a). The Monte Carloaverage of an observable is defined as hOi ¼ 1

Z

P∞k¼0 λ

k×PCkwðCkÞOðCkÞ, where OðCkÞ denotes the value measured

for the configuration Ck. In practice, because of the periodicboundary condition on the imaginary-time axis, the divi-sion of the time axis to halves may be done at an arbitrarylocation. Moreover, it is even possible to perform multiplemeasurements on the same configuration by generatingseveral random divisions.QMC methods [45–56] can also be utilized at zero

temperature, where the unnormalized ground-state wavefunction is obtained from an imaginary-time projection:

jΨ0i ¼ limβ→∞

e−βH=2jΨTi: ð10Þ

Here, β is a projection parameter and the trial wave funcitonjΨTi shall not be orthogonal to the true ground state.A similar framework as Eq. (8) applies, except that onenow samples from the overlap hΨT je−βHjΨTi instead of thepartition function. In the projection scheme, the fidelitysusceptibility has the estimator

χT¼0F ¼ hkLkRi − hkLihkRi

λ2; ð11Þ

where kL and kR are the number of vertices for the bra andket states, which reside in the range ½β=2; βÞ and ½0; β=2Þ ofthe imaginary-time axis respectively, shown in Fig. 1(b).

FIG. 1. Measurement of the fidelity susceptibility in the(a) nonzero-temperature formalism and in the (b) ground-stateprojection scheme. Each red object represents a term in thedriving Hamiltonian λH1, denoted as a vertex. To measure thefidelity susceptibility Eqs. (9) and (11), we divide the imaginary-time axis into two halves and count the number of vertices kL andkR, respectively. The nonzero-temperature formalism allows anarbitrary division because of the periodic boundary condition inthe imaginary-time axis, while in the ground-state projectionscheme the division has to be at β=2.

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Since the fidelity susceptibility is non-negative, the covari-ance formula Eq. (11) reveals positive correlation of kL andkR in a Monte Carlo simulation.Equations (9) and (11) are the central results of this

paper. As is obvious from the discussions in this section,neither the details of the Hamiltonian nor the statistics ofthe system need to be specified. These estimators are thusgeneral and can be readily implemented in a varietyof QMC methods for correlated fermionic, bosonic, orquantum spin systems [45–56].

B. Implementations

We now discuss implementation of the estimatorsEqs. (9) and (11) in various concrete QMC methods.See Sec. II B 1 for discussions about continuous-timeworldline [45–48] and diagrammatic determinantal[50–56] approaches and Sec. II B 2 for discussions aboutthe stochastic series expansion approach [49].

1. Continuous-time worldline and diagrammaticdeterminantal approaches

Continuous-time worldline methods [45–48] are widelyused to simulate boson and quantum spin systems, whilethe diagrammatic determinantal approaches are the state-of-the-art methods for solving quantum impurity [50–53]and fermion lattice models [54–56]. A common feature ofthese methods is to split the Hamiltonian in the form ofEq. (1) and perform a time-dependent expansion in λH1,

Z ¼X∞k¼0

λkZ

β

0

dτ1…Z

β

τk−1dτk

× Tr½ð−1Þke−ðβ−τkÞH0H1…H1e−τ1H0 �; ð12Þ

which obviously fits in the general framework of Eq. (8).In the continuous-time worldline approach [45–48], theH1 term corresponds to hoppings of bosons or spin flips,depicted as kinks of the worldlines in Fig. 2(a). Incontinuous-time diagrammatic determinantal approaches

[50–56], λH1 contains the fermion interactions, drawn asinteraction vertices in Fig. 2(b). Equation (12) has the formof a grand canonical partition function for a classical gas,where λ plays the role of fugacity and k is the number ofcertain classical objects (kinks or vertices) residing on theimaginary-time axis. Typical updates of continuous-timediagrammatic determinantal approaches [50–56] consist ofrandomly inserting or removing vertices, which are iden-tical to the updates of the grand canonical Monte Carlomethod for molecular simulations [57,58]. For bosons andquantum spins there are more effective nonlocal updates,such as the worm and directed loop updates [45–49]. In anycase, the Monte Carlo estimators Eqs. (9) and (11) areindependent to the detailed sampling procedures. It sufficesto count kL and kR of Monte Carlo configurations tocalculate the fidelity susceptibility. Examples are presentedin Secs. IVA and IV C.

2. Stochastic series expansion

SSE is based on a Taylor expansion of the partitionfunction [49],

Z ¼X∞n¼0

ð−βÞnn!

Tr½Hn�; ð13Þ

which may seem to be different from the framework ofEq. (8). However, as is shown in Ref. [59], one can formallytreat the SSE as the time-dependent expansion Eq. (12)with respect to the full Hamiltonian H ¼ H0 þ λH1.In the implementation of SSE, one truncates the sum to a

large number M and pads M − n identity operators in thesquare bracket of Eq. (13). SSE then samples operators inthe fixed-length operator string. To map to a Monte Carloconfiguration in the continuous-time formalism, one canassign an imaginary time to each operator, as shown in thebottom of Fig. 3. As long as the mapping keeps the relativeorder in the original operator string, the Monte Carloweight remains unchanged [60]. In particular, the configu-ration is sampled with a weight proportional to λk if thereare k of λH1 operators in the operator string. In this way,although the sampling of SSE is carried out differently fromEq. (12), the general framework of Eq. (8) still applies. Thefidelity susceptibility is then measured easily by countingthe numbers kL and kR of operators associated with λH1 inthe two halves of the imaginary-time axis after themapping.From Fig. 3 it is clear that even though one performs an

equal bipartition in the imaginary-time axis, the corre-sponding location of division is not always in the centerof the operator string. In fact, it is easier to directly samplethe location of division in the operator string, as shown inthe upper part of Fig. 3. A division at the lth position(l ¼ 0; 1;…;M) means that there are l slots being mappedto one half of the imaginary-time axis and M − l slots to

FIG. 2. (a) Measurement of the fidelity susceptibility in acontinuous-time worldline QMC simulation of bosons andquantum spins, where one counts the number kinks kL and kRafter division of the imaginary-time axis. (b) In the diagrammaticdeterminantal QMC simulation of correlated fermions, thenumber of interaction vertices kL and kR are counted.

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the other half. Therefore, the division l itself follows abinomial distribution pðlÞ ¼ 1

2MðMlÞ that can be sampled

directly. In this way, the fidelity susceptibility can beefficiently calculated in SSE similar to the continuous-timeQMC approaches discussed in Sec. II B 1. As M → ∞, thebinomial distribution approaches a delta function peaked atthe center of the operator string. Only in this limit theposition in the operator string can be directly interpreted asthe imaginary time and a bipartition of the operator string inthe center will yield the correct result for the fidelitysusceptibility.

III. DERIVATIONS

In this section, we derive the estimators for the fidelitysusceptibility at nonzero temperature Eq. (9) and forground-state projector formalism Eq. (11). Readers mayskip this section and continue reading with the followingsections.

A. Ground state

Using the diagrammatic expansion for the projectionoperator, the unnormalized ground-state wave functionEq. (10) has the following form:

jΨ0i ¼ limβ→∞

X∞k¼0

λkZ

β=2

0

dτ1…Z

β=2

τk−1dτk

× ½ð−1Þke−ðβ=2−τkÞH0H1…H1e−τ1H0 �jΨTi: ð14Þ

Substituting it into Eq. (4) and noticing that the partialderivatives affect only the prefactor λk, one obtainsEq. (11). The estimator also holds for the continuous-time

auxiliary field expansion methods (CT-AUX [52] and LCT-AUX [56]), because one can cast the ground-state wavefunction to a similar form as Eq. (14), assuming the shiftparameter used in these methods [54] to be proportionalto λ.

B. Nonzero temperature

We present two derivations of the nonzero temperatureestimator Eq. (9).

1. Derivation based on the definition ofnonzero-temperature fidelity

The nonzero-temperature estimator Eq. (9) can beobtained directly from the definition of the nonzero-temperature fidelity [61]:

F ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTrðe−βHðλÞ=2e−βHðλþϵÞ=2Þ

½Trðe−βHðλÞÞTrðe−βHðλþϵÞÞ�1=2

s: ð15Þ

This is a nonzero-temperature generalization of Eq. (2)and leads to Eq. (7) by using the definition of thefidelity susceptibility Eq. (3) [44]. We expand the tracesof the density matrices around the partition functionZ ¼ Trðe−βHðλÞÞ to Oðϵ2Þ,

Trðe−βHðλþϵÞÞ ¼ Z þ ϵ∂λZ þ ϵ2

2∂2λZ;

Trðe−βHðλÞ=2e−βHðλþϵÞ=2Þ ¼ Z þ ϵ~∂λZ þ ϵ2

2~∂2λZ;

where the notation ~∂λ indicates that the partial derivativeacts only on operators in the imaginary-time interval0 ≤ τ < β=2. Substituting the above two expansions intoEq. (15) and keeping terms up to Oðϵ2Þ, one obtains

χT≠0F ¼ ð~∂λZÞ22Z2

− ~∂2λZ2Z

þ ∂2λZ4Z

− ð∂λZÞ24Z2

¼ hkRi22λ2

− hkRðkR − 1Þi2λ2

þ hkðk − 1Þi4λ2

− hki24λ2

¼ hkLkRi − hkLihkRi2λ2

: ð16Þ

To obtain the second line, we use the partition functionexpansion Eq. (8) and the fact that the partial derivatives actonly on the prefactor λk ¼ λkLþkR. For the third line, we usek ¼ kL þ kR and hkLi ¼ hkRi ¼ hki=2. This derivation isabstract and is independent of the details of a QMC scheme.Carrying out a similar procedure starting from Eq. (2), onecan also prove the ground-state estimator Eq. (11).

FIG. 3. Division of the operator string in SSE to measure thefidelity susceptibility. The slots represent the fixed-length oper-ator string where empty slots hold identity operators, the redcircles (blue squares) correspond to the operators in H1 (H0).These operators can be mapped to a continuous-time configu-ration indicated by the arrows. A division can then be made on theimaginary-time axis, for example, at β=2. An equivalent approachwithout explicit mapping to continuous time is to divide theoperator string at the location indicated by the vertical dashedline, where the integer l is drawn from a binomial distribution.For the estimators Eqs. (9) and (11), one counts the number of redcircles (operators in λH1) in both sides for kL and kR. In thisexample, M ¼ 12; n ¼ 6;l ¼ 7, and kL ¼ kR ¼ 2.

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2. Derivation based on theimaginary-time correlator Eq. (7)

This derivation starts from the definition of fidelitysusceptibility based on the imaginary-time correlatorEq. (7). We utilize its connection to the Monte Carloweight that appears in Eq. (12) to derive the nonzero-temperature estimator Eq. (9). First of all, the second termin the square bracket of Eq. (7) can be measured directlyfrom the average expansion order [49,50,53,54]:

hH1i ¼ − hkiβλ

: ð17Þ

Integrating over the imaginary-time and using hkLi ¼hkRi ¼ hki=2, one has

Zβ=2

0

dτ½−hH1i2�τ ¼ − hki28λ2

¼ − hkLihkRi2λ2

: ð18Þ

We then consider the QMC estimator for the first term ofEq. (7),

Gðτ1 − τ2Þ ≜ hT ½H1ðτ1ÞH1ðτ2Þ�i

¼ 1

λ2

�Xi≠j

δðτi − τ1Þδðτj − τ2Þ�; ð19Þ

where T is the time-ordering operator and τi and τj are theimaginary times of two vertices in the Monte Carloconfiguration. Integrating both sides of Eq. (19) and usingthe fact that Gðτ1 − τ2Þ depends only on jτ1 − τ2j, one hasZ

Λ

0

dτ1

ZΛ

0

dτ2Gðτ1 − τ2Þ ¼ 2

ZΛ

0

dτGðτÞðΛ − τÞ

¼ 1

λ2hkðΛÞ½kðΛÞ − 1�i; ð20Þ

where kðΛÞ is the number of vertices in the range of0 ≤ τ < Λ. For example, kðβÞ ¼ k and kðβ=2Þ ¼ kR. Whenchoosing Λ ¼ β and using GðτÞ ¼ Gðβ − τÞ, Eq. (20)becomes

β

Zβ

0

dτGðτÞ ¼ 1

λ2hkðk − 1Þi: ð21Þ

When setting Λ ¼ β=2, Eq. (20) reads

2

Zβ=2

0

dτGðτÞ�β

2− τ

�¼ 1

λ2hkRðkR − 1Þi: ð22Þ

Together with Eq. (21), it leads to

Zβ=2

0

dτGðτÞτ ¼ hkðk − 1Þi4λ2

− hkRðkR − 1Þi2λ2

¼ hkLkRi2λ2

: ð23Þ

In combination with Eqs. (7) and (18), we obtain Eq. (9).

IV. APPLICATIONS

We first demonstrate the power of the new approach byidentifying quantum and thermal phase transitions in theBose-Hubbard model and in the spin-1=2 XXZ model.Then, we use the fidelity susceptibility to address thepresence of the intermediate quantum spin-liquid state inthe Hubbard model on the honeycomb lattice. In all cases,this requires only minimal modifications to existing codes.We purposely chose a variety of QMC methods inthe following to demonstrate the wide applicability of thecovariance estimators. Because of the flexibility of thenonzero-temperature estimator, we use Eq. (9). In Sec. VA,we compare it to the zero-temperature scheme.

A. Quantum phase transition in theBose-Hubbard model

First, we use the fidelity susceptibility to probe thequantum phase transition in the Bose-Hubbard model,

H ¼Xi

�U2niðni − 1Þ − μni

�− λ

Xhi;ji

ðb†i bj þ b†j biÞ;

ð24Þ

where U is the on-site interaction and μ is the chemicalpotential. The driving parameter λ has the physical meaningof a tunneling amplitude. The Bose-Hubbard model has awell-known quantum phase transition between the Mottinsulating state and the superfluid state as λ=U increases[1,2]. In particular, for integer fillings, the system has anemergent Lorentz invariance at the critical point and thedynamical critical exponent is z ¼ 1 [7].The fidelity susceptibility has previously been calculated

using DMRG methods for the one-dimensional Bose-Hubbard model [62–64]. We now calculate the fidelitysusceptibility on a square lattice with N ¼ L2 sites at unitfilling by tuning μ. In accordance with the dynamicalcritical exponent z ¼ 1, we scale the inverse temperatureproportionally to the system length βU ¼ 4L. The simu-lation employs the directed worm algorithm [46,65,66]. Weutilize Eq. (9) to sample the fidelity susceptibility bycounting the number of kinks in the worldline configura-tion, as illustrated in Fig. 2(a). Figure 4 shows that, as thesystem size increases, the peak in the fidelity susceptibility(as a function of the driving parameter λ) is becomingmore pronounced around the previously determinedcritical point ðλ=UÞc ¼ 0.05974ð3Þ [67]. The inset of

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Fig. 4 shows the scaled fidelity susceptibility χFL−2=νversus ½ðλ=UÞ − ðλ=UÞc�L1=ν according to the scaling lawof Refs. [22,24,25]. The data collapses well under thecritical exponent ν ¼ 0.6715 of the 3D XY universalityclass [67]. The ability to calculate the fidelity susceptibilityusing the state-of-the-art directed worm algorithm[46,65,66] will greatly advance the study of quantum phasetransitions of ultracold bosons. It is worth pointing out thatthe fidelity susceptibility is related to the quantity (kinetic-energy correlator) previously calculated in the study ofHiggs mode in a two-dimensional superfluid [68].

B. Thermal phase transition in the XXZ model

Next, we consider the spin-1=2 antiferromagnetic XXZmodel on a square lattice with N ¼ L2 sites,

H ¼ JzXhi;ji

Szi Szj þ λ

Xhi;ji

ðSxi Sxj þ Syi SyjÞ; ð25Þ

where the driving parameter λ plays the role of the couplingstrength in the XY plane. When λ dominates, theHamiltonian favors Néel order in the XY plane, while ifJz dominates, the system has an antiferromagnetic Isingground state. The Heisenberg point λ ¼ Jz is a quantumcritical point, which separates the XY order and the Isingorder. This quantum critical point can be easily locatedfrom the peak of the fidelity susceptibility (not shown). Ourapproach makes it possible to obtain the fidelity suscep-tibility in much larger systems compared to the previousexact diagonalization study [69], and thus can enable amore accurate scaling analysis.At nonzero temperature, thermal fluctuations will

destroy the antiferromagnetic Ising phase at a second-orderphase transition. Since one can cross the phase boundary bychanging either λ or the temperature T, we see that the

fidelity susceptibility can also indicate thermal phasetransitions. As a demonstration, we fix λ ¼ 1, Jz ¼ 1.5and scan the temperature T to drive a phase transition fromthe low-temperature antiferromagnetic Ising phase to thehigh-temperature disordered phase. Figure 5 shows thefidelity susceptibility calculated using Eq. (9) via the SSEmethod [49,71]. The peak in the fidelity susceptibilitycorrectly singles out the previously determined criticaltemperature ðT=λÞc ≈ 0.75 [70].

C. Intermediate phase in the Hubbard modelon the honeycomb lattice

Finally, we apply the fidelity susceptibility estimator to amore challenging and controversial example—the Hubbardmodel on the honeycomb lattice,

H ¼ − tXhi;ji

Xσ¼f↑;↓g

ðc†iσ cjσ þ c†jσ ciσÞ

þ λXi

�ni↑ − 1

2

��ni↓ − 1

2

�; ð26Þ

where λ has the meaning of on-site Hubbard interactionstrength. The simulation employs the recently developedefficient continuous-time QMC method for lattice fermions(LCT-INT) [56,72]. We consider lattices with N ¼ 2L2

sites, with L ¼ 6; 9; 12, and scale the inverse temperatureβt ¼ L.The ground-state phase diagram of the Hubbard model

on the honeycomb lattice [74] has been controversial.It was suggested to possess an intermediate nonmagneticspin-liquid phase for λ=t ∈ ½3.5; 4.3� [75]. However, morerecent QMC studies on larger systems [76] and withimproved observables [77,78] suggest a single continuousphase transition at λ=t ≈ 3.8 belonging to the Gross-Neveuuniversality class [79]. Other less unbiased methods, such

FIG. 4. Fidelity susceptibility per site of a Bose-Hubbard modelon a square lattice at unit filling. The vertical line indicates thecritical point determined in Ref. [67]. The inset shows the datacollapse of the scaled fidelity susceptibility.

FIG. 5. Fidelity susceptibility per site of a XXZ model onsquare lattice versus temperature. The vertical line indicates thecritical temperature determined in Ref. [70].

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as quantum cluster approaches, give conflicting results onthe presence of the intermediate phase [80–85], dependingon implementation details.The fidelity susceptibility offers a new perspective on the

debate about the phase diagram. In the scenario with anintermediate phase, there shall be two features in χF whenλ=t approaches the two phase boundaries. This consider-ation is independent of the presence of a local-order-parameter description of the possible intermediate phase.Figure 6 shows the fidelity susceptibility per site for

various system sizes obtained using Eq. (9). The calculationof the fidelity susceptibility for fermions is more challeng-ing than that for bosons (Sec. IVA) and quantum spins(Sec. IV B), because of the unsatisfactory localMonte Carlo updates and the cubic (instead of linear)scaling with respect to the system size N. Given the limitedsystem sizes, in Fig. 6 we can identify only a single broadpeak for small systems (and high temperature). The peakbecomes sharper and shifts towards smaller interactionstrength as the system size increases; however, theexamined system sizes are insufficient for a reliabledetermination of the phase diagram. Further algorithmicimprovements (nonlocal updates and better scaling) on thefermionic QMC methods are needed to access accuratefidelity susceptibility data for larger systems and lowertemperatures to better locate phase transitions withoutknowing the local order parameter. Moreover, by usingthe histogram reweighting [86,87] or the quantum Wang-Landau approach [88], it is possible to obtain the fidelitysusceptibility in a continuous range of λ. Such data may beused to precisely determine the critical point and even thecritical exponent of the quantum phase transition.

V. DISCUSSION

To help the reader gain a better understanding of theestimators Eqs. (9) and (11), we first discuss their

relationship then compare them with the previous approachadopted in SSE calculations [24,25]. Finally, we comparethe fidelity susceptibility approach with other genericapproaches for detecting phase transitions.

A. Relation of the ground-state and nonzero-temperature estimators

The factor of 2 difference in Eqs. (9) and (11) is due to thedifferent boundary conditions of the imaginary-time axis inthe ground-state projection and nonzero-temperature QMCformalisms; see Fig. 1. We use a four-site Hubbard model[Eq. (26)] as an illustrative example. Consider the integrandof Eq. (7), the correlatorGðτÞ ¼ hH1ðτÞH1i is related to thedistribution of the vertices on the imaginary-time axis. For agiven configuration, the probability of finding two verticeswith a time difference τ is proportional to λ2GðτÞ. If weequally divide the imaginary-time axis into two halves andimpose the additional constraint that the two vertices residein different halves (denoted as a separable vertex pair),the joint probability changes to λ2GðτÞminfτ; β − τg.Figure 7(a) shows the histogram of separable vertex pairsaccumulated in the imaginary time, which indeed agreeswith the exact curve. Summing up the histogram gives thetotal number of separable vertex pairs, which equals thefollowing integration:

hkLkRi ¼ λ2Z

β=2

0

dτGðτÞτ þ λ2Z

β

β=2dτGðτÞðβ − τÞ:

ð27Þ

FIG. 6. Fidelity susceptibility per site of the Hubbard model onthe honeycomb lattice Eq. (26) with N ¼ 2L2 sites.

(a)

(b)

FIG. 7. (a) Histogram obtained by counting separable vertexpairs with distance τ compared with the exact result ofhH1ðτÞH1imaxfτ; β − τg. (b) Histogram obtained by countingvertices with distance τ compared with the exact results ofhH1ðτÞH1i. For τ close to β=2, the correlator approaches hH1i2,indicated by the dashed blue line. These simulations are per-formed for the Hubbard model Eq. (26) on a four-site open chainwith λ=t ¼ −2 and βt ¼ 8.

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Since GðτÞ is symmetric around τ ¼ β=2 in the nonzero-temperature simulation, the two terms of Eq. (27) are equal.Thus, Eq. (27) reduces to Eq. (23). Furthermore, Fig. 7(b)shows the correlator GðτÞ sampled by accumulating thehistograms of distances between vertices [51,53] togetherwith the exact results (solid black line). The correlationbetween vertices decays rapidly with imaginary-time dis-tance and approaches the uncorrelated value hH1i2 (dashedblue line).However, in the zero-temperature limit, the correlator

GðτÞ decays monotonically with τ and two vertices willdecorrelate for τ ≥ β=2, where β → ∞ in the projectionscheme. Therefore, the second term of Eq. (27) reduces toðλ2β2hH1i2Þ=8 ¼ hki2=8 and cancels half of the secondterm in the estimator Eq. (11), resolving the apparentdifference by the factor of 2. In practical calculations, it is,however, crucial to adopt the correct formula to obtainconsistent results, as illustrated in Fig. 8. The fidelitysusceptibility calculated using Eq. (9) in a nonzero-temperature LCT-INT [56] simulation agrees perfectly withexact diagonalization results. The blue square shows thevalue obtained using Eq. (11) in a projector LCT-INTcalculation [89], which correctly reproduces the exact valueof the ground-state fidelity susceptibility.Figure 7(b) also reveals the difficulty of computing the

fidelity susceptibility. If the decay of GðτÞ is faster than1=τ, the integrand of Eq. (7) has vanishing contributions atlarge τ. However, as the two terms in Eq. (7) are sampledindependently in the actual QMC simulations, uncorrelatedvertices at large imaginary-time distance will cause noisesin the fidelity susceptibility signal. For the applications inSec. IV, we thus perform the calculations at nonzerotemperature as it provides a natural cutoff.

B. Comparison to previous approaches

The present approach to sample the fidelity susceptibilityis more generic and efficient than those developed inRefs. [24,25] specifically for the SSE method. It is

nevertheless instructive to compare them in detail. Thekey difference lies in the sampling of the first term ofEq. (7). References [24,25] employ the SSE estimator[90,91]

GðτÞ ¼ M − 1

λ2β2

×XM−2

n¼0

�M − 2

n

��1 − τ

β

�M−n−2�τ

β

�nhGðnÞi;

where GðnÞ is the number of occurrences of two operatorsfrom H1 that are separated by n positions in the fixed-length operator string (n ¼ 0 if they are next to each other).Multiplying both sides with maxfτ; β − τg and integratingover the imaginary time, one finds

λ2Z

β

0

dτGðτÞmaxfτ; β − τg ¼XM−2

n¼0

WðnÞhGðnÞi; ð28Þ

where the weight function WðnÞ is written in terms of theregularized incomplete beta function Ixða; bÞ [92],

WðnÞ ¼ I1=2ðnþ 2;M − n − 1Þ nþ 1

M

þ I1=2ðM − n; nþ 1ÞM − n − 1

M: ð29Þ

References [24,25] explicitly go through kðk − 1Þ=2pairs of vertices to accumulate hGðnÞi and multiply it withthe weight function WðnÞ. In Eq. (27), however, themultiplication by the imaginary time τ is taken into accountimplicitly by the sampling procedure (which requiresseparable vertices). Besides being more generic, ourapproach reduces the computational cost from Oðk2Þ toOðkÞ, which is crucial for the simulation of bosonicand quantum spin systems. In this sense, the specificationof our general result Eq. (9) for the SSE method can beregarded as an improved estimator of Eq. (28), which byitself already improves the approach of Refs. [24,25] inseveral aspects [93]. The improved estimator Eq. (9) notonly unifies the SSE approach in a broader context ofcontinuous-time diagrammatic QMC methods, it also givesbetter statistics with less computational cost comparedto Eq. (28).Figure 9 shows the weight function WðnÞ for various

truncation lengths. As M increases, it approaches twostraight lines, and a division in the center of the operatorstring would yield increasingly accurate results for thefidelity susceptibility, consistent with the discussion inSec. II B 2 concerning the large M limit.

C. Relationship to other quantities

The fidelity susceptibility is related to the second-orderderivative of the free energy A ¼ −ð1=βÞ lnZ [25,94].

FIG. 8. QMC results for the fidelity susceptibility comparedwith exact results (solid line). The nonzero-temperature QMCdata (red dots) are obtained from Eq. (9), while the ground-statedata (blue square at β−1 ¼ 0) are obtained from Eq. (11) in aprojector LCT-INT calculation [89]. The system is the sameas in Fig. 7.

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Because of the Hellmann-Feynman theorem [95,96], hH1iequals to the first-order derivative of the free energy withrespect to λ. A further derivative following the Kuboformula gives

∂2A∂λ2 ¼ ∂hH1i

∂λ ¼ −Z

β

0

dτ½hH1ðτÞH1i − hH1i2�

¼ hk2i − hki2 − hki−βλ2 : ð30Þ

The third equality follows from Eqs. (17) and (21) [97].The quantity resembles the widely used SSE estimatorfor the specific heat [49,98], but can be used to probequantum phase transitions [25]. At zero temperatureð∂hH1iÞ=∂λ ¼ −ð1=λÞð∂hH0iÞ=∂λ, and the latter quantityis computed using numerical differentiation of the kineticenergy, so as to address the quantum phase transition in theHubbard model on the honeycomb lattice [75,85]. As ispointed out in Refs. [25,94], the fidelity susceptibility has astronger singularity compared to the second-order deriva-tive of the free energy and is thus a better indicator ofquantum phase transitions. In particular, notice that theintegrand of Eq. (30) differs from Eq. (7) by a multipli-cative factor τ. One can show that in the critical region thesingular part of χF diverges faster than Eq. (30) by a factorof Lz, where z is the dynamical critical exponent [22,25].There are concrete examples in a class of topological phasetransitions, which do not exhibit singularity in the second-order derivative of the ground-state energy [99], but canstill be detected using the fidelity susceptibility [100].The covariance that appears in the estimators Eqs. (9)

and (11) can also be written as

hkLkRi − hkLihkRi ¼1

2½VarðkÞ − VarðkLÞ − VarðkRÞ�;

where VarðxÞ ¼ hx2i − hxi2 is the variance of x. Thisexpression has an appealing meaning, i.e., the distributionsof the vertices residing on the whole and on the halves ofthe imaginary-time axis have different widths, and thedifference in these widths gives the estimate. This formresembles the bipartite fluctuation [101], which was

proposed to be a diagnostic tool for phase transitions[102] because of its relation to the entanglement entropy.However, there are important differences. First, thefidelity susceptibility estimator requires a division in theimaginary-time axis for vertices, not in the real space forthe physical particles. Second, the total number of verticesis fluctuating in the QMC simulations as opposed to beingconserved in the case of bipartition fluctuations. Third, it iseasier to locate the critical point using the fidelity suscep-tibility. As is shown in this paper and in many previousstudies [16], the fidelity susceptibility exhibits an increas-ingly sharp peak at a phase transition as the system sizeenlarges. On the other hand, to utilize bipartite fluctuationsand entanglement entropy for phase transition, one typi-cally needs to resolve the scaling or subleading behaviorwith the system size, which is often difficult in finite sizesimulations.

VI. OUTLOOK

We present a general approach to compute the fidelitysusceptibility of correlated fermions, bosons, and quantumspin systems in a broad class of quantum Monte Carlomethods [45–56]. The calculation of the fidelity suscep-tibility is surprisingly simple yet generic. It provides ageneral purpose indicator of quantum phase transitionswithout the need for a prior knowledge of the local orderparameter.Conceptually, our work shows it is rewarding to view the

modern QMC methods [45–56] in a unified frameworkprovided by Eq. (8), which deals with the same type ofclassical statistical problem irrespective of microscopicdetails of the original quantum system. In the QMCsimulations, a quantum phase transition manifests itselfas a particle condensation transition driven by changing ofthe fugacity of the corresponding classical model. Thisconnection suggests generic ways to detect and characterizequantum phase transition through studying classical par-ticle condensations. For example, Eq. (30) actually relatesthe second-order derivative of free energy of a quantumsystem to the particle compressibility of a virtual classicalsystem. In this respect, the significance of the covarianceestimators Eqs. (9) and (11) is evident because they capturethe key critical fluctuation upon a particle condensationtransition.It is straightforward to generalize our Eqs. (9) and (11) to

cases with multiple driving parameters, where one needs tocount the vertices of different types (as is already donein the SSE calculations in Secs. II B 2 and IV B). It isinteresting to find out whether this can lead to a generalapproach to measure the Berry curvature (the imaginarypart of the quantum geometric tensor) in quantumMonte Carlo simulations. Related to these efforts, thenonequilibrium QMC method has been developed in recentyears [103,104] to study the nonadiabatic response ofquantum systems in imaginary time. In particular, it also

FIG. 9. The weight function according to Eq. (29) for varioustruncation lengths M in a SSE calculation.

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allows the extraction of the fidelity susceptibility and theBerry curvature [103,105]. It would be interesting tocompare the nonequilibrium QMC approach [103,104]to the equilibrium one presented in this paper.Last but not least, the Hamiltonian Eq. (1) has further

implications beyond quantum phase transitions. The effi-cient estimators Eqs. (9) and (11) may also provide usefulinsights in the simulations of adiabatic quantum compu-tation [106–108] and nonadiabatic quantum dynamics[109,110].

ACKNOWLEDGMENTS

The authors thank Mauro Iazzi, Sergei Isakov, LodePollet, and Hiroshi Shinaoka for helpful discussions.Simulations were performed on the Mönch cluster ofPlatform for Advanced Scientific Computing (PASC)and on the Brutus cluster at ETH Zurich. We have usedALPS libraries [111] for Monte Carlo simulations and dataanalysis. The results of the Bose-Hubbard model wereobtained using the DWA (directed worm algorithm) appli-cation, and the results of the XXZ model were obtainedusing the DIRLOOP_SSE (stochastic series expansion withdirected loop updates) application of the ALPS project.This work was supported by ERC Advanced GrantSIMCOFE and by the Swiss National ScienceFoundation through the National Center of Competencein Research Quantum Science and Technology QSIT.

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