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Gradient methods for variational optimization of projected entangled-pair states

Vanderstraeten, L.; Haegeman, J.; Corboz, P.; Verstraete, F.DOI10.1103/PhysRevB.94.155123Publication date2016Document VersionFinal published versionPublished inPhysical Review B

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Citation for published version (APA):Vanderstraeten, L., Haegeman, J., Corboz, P., & Verstraete, F. (2016). Gradient methods forvariational optimization of projected entangled-pair states. Physical Review B, 94(15),[155123]. https://doi.org/10.1103/PhysRevB.94.155123

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Download date:19 Aug 2021

PHYSICAL REVIEW B 94, 155123 (2016)

Gradient methods for variational optimization of projected entangled-pair states

Laurens Vanderstraeten,1 Jutho Haegeman,1 Philippe Corboz,2 and Frank Verstraete1,3

1Ghent University, Department of Physics and Astronomy, Krijgslaan 281-S9, B-9000 Gent, Belgium2Institute for Theoretical Physics, University of Amsterdam, Science Park 904 Postbus 94485, 1090 GL Amsterdam, The Netherlands

3Vienna Center for Quantum Science, Universitat Wien, Boltzmanngasse 5, A-1090 Wien, Austria(Received 4 July 2016; published 14 October 2016)

We present a conjugate-gradient method for the ground-state optimization of projected entangled-pair states(PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPSmanifold. Our optimization is based on an efficient and accurate evaluation of the gradient of the global energyfunctional by using effective corner environments, and is robust with respect to the initial starting points. It has theadditional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide thetools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian.We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energiesand order parameters as compared to algorithms based on imaginary-time evolution.

DOI: 10.1103/PhysRevB.94.155123

I. INTRODUCTION

Ever since the birth of quantum mechanics, the quantummany-body problem has been at the center of theoreticaland computational physics. Despite the simplicity of thefundamental equations, it has been notoriously difficult tosimulate the quantum behavior of many-body systems. Thisis especially true for low-dimensional systems: becausequantum correlations are stronger, perturbation theory oftenfails and more sophisticated methods are needed. This cryfor better methods is loudest with respect to two-dimensionalsystems, because of the large range of unexplored quantumphenomena—quantum spin liquids [1], topological order [2],and quasiparticle fractionalization are only three examples.

Because Monte Carlo sampling is often plagued by thesign problem and exact diagonalization is necessarily limitedto small system sizes, it seems that variational methods arethe way to go for exploring the two-dimensional quantumworld. There are essentially two prerequisites for a successfulvariational approach: (i) an adequate variational ansatz thatcaptures the physics for the problem at hand, and (ii) anefficient way of computing observables and optimizing thevariational parameters. Examples such as the density-matrixrenormalization group [3,4] and Gutzwiller-projected wavefunctions [5,6] seem to meet the latter, but it is unclearto what extent they are the natural choice for simulatingtwo-dimensional quantum systems.

In recent years projected entangled-pair states (PEPS) [7,8]have emerged as a viable candidate for capturing the physicsof ground states of strongly correlated quantum lattice modelsin two dimensions. It is by explicitly modeling the distributionof entanglement in low-energy states of local Hamiltoniansthat PEPS parametrize the “physical corner of Hilbert space.”Indeed, PEPS have a built-in area law for the entanglemententropy [9], they provide a natural characterization of topo-logical order [10–14], and they can realize bulk-boundarycorrespondences explicitly [15–17]. Moreover, PEPS can beformulated directly in the thermodynamic limit [18] whichallows us to focus on bulk physics without any finite-size orboundary effects.

An efficient optimization of the parameters in a PEPS hasproven to be more challenging. According to the variationalprinciple, finding the best approximation to the ground statefor a given Hamiltonian H reduces to the minimization ofthe energy expectation value. For infinite PEPS this amountsto a highly nonlinear optimization problem for which theevaluation of, e.g., the gradient of the energy functional isa hard problem. For that reason, the state-of-the-art PEPSalgorithms have taken recourse to imaginary-time evolution[18–20]: a trial PEPS state is evolved with the operator e−τH ,which should result in a ground-state projection for verylong times τ . This imaginary-time evolution is integratedby applying small time steps δτ with a Trotter-Suzukidecomposition and, after each time step, truncating the PEPSbond dimension in an approximate way. This truncation canbe done by a purely local singular-value decomposition—theso-called simple-update [19] algorithm—or by taking the fullPEPS wave function into account—the full-update [18] or fastfull-update [20] algorithm.

These imaginary-time algorithms have allowed very ac-curate simulations of frustrated spin systems [21–30] andstrongly correlated electrons [30–33], but it remains unclearwhether they succeed in finding the optimal state in agiven variational class of PEPS. Although computationallyvery cheap, ignoring the environment in the simple-updatescheme is often a bad approximation for systems with largecorrelations. The full-update scheme takes the full wavefunction into account for the truncation, but requires theinversion of the effective environment which is potentiallybadly conditioned. This problem was solved by regularizingthe environment appropriately and fixing the gauge of thePEPS tensor [20,34]. Nonetheless, the truncation procedurein the full-update scheme is not guaranteed to providethe globally optimal truncated tensors in the sense thatthe global overlap of the truncated and the original PEPS ismaximized. Indeed, the truncated tensor is optimized locallyand afterwards put in at every site in the lattice to give anupdated (global) PEPS wave function.

Similar issues have been at the center of attention in the con-text of matrix product states (MPS) [35], the one-dimensional

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counterparts of PEPS, where a number of different strategieshave been around for optimizing ground-state approximationsdirectly in the thermodynamic limit [36–38]. Recently, theproblem of finding an optimal matrix product state has beenreinterpreted by (i) identifying the class of matrix productstates as a nonlinear manifold embedded in physical Hilbertspace [39,40], and (ii) formulating a minimization problemof the global energy functional on this manifold. A globallyoptimal state can then be recognized as a point on themanifold for which the gradient of the energy functionalis zero. Moreover, approximating time evolution within themanifold is optimized, as dictated by the time-dependentvariational principle [38], by projecting the time evolutiononto the tangent space of the manifold. In the case ofimaginary time, this tangent vector is exactly the gradient,which shows that different optimization algorithms can becompared within this unifying manifold interpretation [41].Moreover, whereas imaginary-time evolution more or lesscorresponds to a steepest-descent method [39], more advancedoptimization methods such as conjugate-gradient or quasi-Newton algorithms can find an optimal matrix product statemuch more efficiently [42,43].

In Ref. [44] it was shown how to implement these tangentspace methods for PEPS by introducing a contraction schemebased on the concept of a “corner environment.” Buildingon that work, this paper presents a PEPS algorithm thatoptimizes the global energy functional using a conjugate-gradient optimization method. In contrast to other methods,this algorithm has a clear convergence criterion, which canguarantee that an optimal state has been reached. Moreover,it allows us to more easily impose physical symmetries onthe PEPS. On the fly, the contraction scheme also allowsus to compute the energy variance of the variational groundstate—an unbiased measure of the accuracy of the variationalansatz and a tool for better energy extrapolations—as wellas general two-point correlation functions and static structurefactors.

In the next section (Sec. II) we review the “cornerenvironment” in considerable detail and show how to computestatic structure factors of a PEPS. Next (Sec. III) we discussour conjugate-gradient scheme for the PEPS optimization, andexplain how to evaluate the energy gradient and the energyvariance. We benchmark (Sec. IV) our method by applying itto the transverse Ising model, the XY model, and the isotropicHeisenberg model. In the last section (Sec. V) we discuss thepossible extensions and applications.

II. EFFECTIVE ENVIRONMENTS AND TWO-POINTCORRELATION FUNCTIONS

Consider an infinite square lattice with every site hostinga quantum degree of freedom with dimension d. For thisquantum spin system, a PEPS can be introduced formally as

|�(A)〉 =∑

{s}C2(A) |{s}〉 , (1)

where C2(. . . ) is the contraction of an infinite tensor network.This contraction is most easily represented graphically as

C2(A) = ,

with the red circle always representing the same five-leggedtensor A,

Asu,r,d,l = .

In order to obtain a physical state, a tensor A is associatedwith every site in the lattice and all virtual indices (u,r,d,l) arecontracted in the network. The physical indices s are left open,such that a coefficient is obtained for every spin configurationin the superposition in Eq. (1). The graphical representationis then obtained by connecting links that are contracted andleaving the physical links open. The virtual degrees of freedomin the PEPS carry the quantum correlations and mimic theentanglement structure of low-energy states. The dimensionof the virtual indices is called the bond dimension D and canbe tuned in order to enlarge the variational class; as such,it acts as a refinement parameter for the variational PEPSansatz.

The norm of an infinite PEPS can be pictorially representedas

〈Ψ(A)|Ψ(A)〉 = ,

where every block represents the tensor a obtained bycontracting the tensor A with its conjugate A over the physicalindex, i.e.,

a = =

As in the rest of this paper, the virtual indices of the ketand bra level are grouped into one index, so that these “topview” representations of double-layer tensor contractions aresimplified.

The norm of the PEPS is thus obtained by the contractionof an infinite tensor network and can, in general, only bedone approximately. Different numerical methods have beendeveloped to contract these infinite networks efficiently, whichallows the evaluation of the norm of a PEPS, as well asexpectation values and correlation functions.

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A. The linear transfer matrix

The first and most straightforward strategy is based on thelinear transfer matrix T , graphically represented as

T = .

This object carries all the correlations in the PEPS from onerow in the network to the next. One can of course definea similar transfer matrix in the vertical direction, and evendiagonal transfer matrices can be considered. Naturally thetransfer matrix is interpreted as an operator from the topto the bottom indices, so that the full contraction of thetwo-dimensional network reduces to successively multiplyingcopies of T . In the thermodynamic limit, the norm of a PEPSis thus given by

〈�(A)|�(A)〉 = limN→∞

T N = λN

with λ the leading eigenvalue of the transfer matrix. Theassociated leading eigenvector or fixed point contains all the in-formation on the correlations of a half-infinite part of thelattice.

An exact representation of the fixed point is only possiblein a number of special cases and approximate methods have tobe devised in general. Given the versatility of matrix productstates (MPS) for approximating the ground state of localgapped Hamiltonians [35], one expects that this class of statesmight provide a good variational ansatz for the case of gappedtransfer matrices as well. Moreover, the bond dimension of thematrix product state representation of the fixed point, denotedwith χ , can be tuned systematically, such that the errors can bekept under control perfectly. Whereas MPS approximationsfor fixed points go way back [45], a variety of efficienttensor-network methods have been developed [7,46] recently.Here we use an algorithm [43] in the spirit of Ref. [41], whichtreats the linear transfer matrix and corner transfer matrix(Sec. II B) on a similar footing.

The fixed-point equation can be stated graphically as

≈ λ .(2)

For this equation to hold, a relation of the form

≈

should hold to a very high precision [43]. Indeed, if this tensor(rectangle) exists, it maps the action of the transfer matrixback to the same MPS fixed point. The virtual dimension ofthe MPS fixed point will be denoted as χ and can be tuned toimprove the accuracy of the PEPS contraction.

Given that the fixed-point equation can be solved efficiently,the PEPS can now be normalized to one by rescaling the A

tensor such that the largest eigenvalue λ of the transfer matrixequals unity. With the MPS fixed point, the expectation value

of a local operator at an arbitrary site i,

〈Oi〉 = 〈�(A)| Oi |�(A)〉〈�(A)|�(A)〉 ,

can be easily computed. First the upper and lower halves ofthe network are replaced by the fixed points,

≈ ,

where a colored block tensor always indicates the presence ofa physical operator at that site. The resulting effective one-dimensional network can be evaluated exactly by finding theleading left and right eigenvectors (fixed points) of the channeloperator,

= μ

and

= μ .

The eigenvalue μ depends on the normalization of the MPStensors in the upper and lower fixed points of the linear transfermatrix, and its value can be put to one. The fixed points aredetermined up to a factor, which can be fixed by imposing thatthe norm of the PEPS

〈Ψ(A)|Ψ(A)〉 =

equals unity such that

〈Oi〉 = 〈Ψ(A)|Oi |Ψ(A)〉 = .

B. The corner transfer matrix

Another set of methods for contracting two-dimensionaltensor networks relies on the concept of the corner transfermatrix, which was first applied to classical lattice systems[45,47–49] and recently used extensively in tensor networksimulations [32,50,51]. The strategy now is to break up theinfinite tensor network in different regions, and represent these

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as tensors with a fixed dimension. Graphically, the setup is

≈ .

A red tensor represents the compression of one of the cornersof the network, whereas the blue tensors capture the effect ofan infinite row of a tensors. Together, they provide an effectiveone-site environment for the computation of the norm of thePEPS or local expectation values.

This scheme can now be extended [44] in order toevaluate nonlocal expectation values such as general two-pointcorrelation functions. Indeed, by not compressing the blueregion above, one can construct an environment that looks like

≈ ,

so that one could evaluate operators that have an arbitrarylocation in the lattice.

Finding this effective “corner environment” can again bedone by solving a fixed-point equation. Indeed, the greencorner-shaped environment should be the result of an infinitenumber of iterations of an equally corner-shaped transfermatrix; the fixed-point equation is

∝ .

Very far from the corner this equation reduces to the one forthe linear transfer matrix. This implies that, asymptotically,the fixed point can be well approximated by an MPS. Letus therefore make the ansatz that the full fixed point can beapproximated as an MPS, where we put an extra tensor on thevirtual level to account for the corner. With this ansatz, thefixed-point equation is given by

∝ . (3)

We expect [44] that this fixed point can be modeled using theMPS tensors from the fixed points of the linear transfer matrix,up to the corner matrix, which captures the effect of the cornershape. With this ansatz, we obtain a linear fixed-point equationfor the corner matrix, which corresponds to a simple eigenvalueequation and can be solved efficiently.

C. Channel environments

Once we have found (i) the fixed points of the linear transfermatrix in all directions [Eq. (2)], and (ii) the four cornertensors [Eq. (3)], we can contract the network correspondingto the norm, a local expectation value, or a correlation functionof the PEPS. Let us assume that the tensor A is normalizedsuch that the largest eigenvalue of the linear transfer matrixis unity. Computing the norm 〈�(A)|�(A)〉 with a channelenvironment then reduces to the contraction of

.

An infinitely long channel can be contracted by computingthe fixed point ρL of the “channel operator.” Therefore theeigenvector corresponding to the largest eigenvalue should befound, i.e.,

= λ ×

for the top channel. The boundary MPS tensors have to berescaled such that the largest eigenvalue λ is put to one.Similarly, the fixed point in the other direction ρR is definedas

= λ ×.

The inner product of the left and right fixed points is put toone. For further use, we note that, by subtracting the projectoron the largest eigenvector, an operator is constructed that hasspectral radius strictly smaller than one,

ρ

(−

)< 1.

The norm of the PEPS is then reduced to

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〈Ψ(A)|Ψ(A)〉 = ,

which can be scaled to 1 by rescaling the corner tensors bythe appropriate scalar. With these conventions, the norm ofthe state is well defined and expectation values can be safelycomputed. For a local one-site operator O we have

〈Ψ(A)|O |Ψ(A)〉 = ,

and, similarly, the expectation value of a two-site operator is

〈Ψ(A)|O |Ψ(A)〉 = ,

where the two-site operator can of course be oriented in theother channels as well.

The real power of the channel environment is now thatarbitrary two-point correlation functions can be computedstraightforwardly. Indeed, the expectation value of two op-erators at generic locations in the lattice is computed as

.

In fact, even three-point correlation functions can be evaluatedby orienting the corners in the right way, as in, e.g., thecontraction

.

From a computational point of view, the hardest step indetermining this corner environment is finding the fixed pointof the linear transfer matrix; state-of-the-art algorithms [43]scale as O(χ3D4 + χ2D6), with D the PEPS bond dimensionand χ the bond dimension of the fixed point. In the case ofstrongly correlated PEPS, finding the fixed point might takea lot of iterations. Determining the corner tensors and thechannel fixed points has similar scalings, but this has to bedone only once.

D. Static structure factor

As an example of the power of the corner environment,we will explicitly show how to compute a static correlationfunction directly in momentum space, i.e., the static structurefactor s(�q),

s(�q) = 1

|L|∑

i,j∈Lei �q·(�ni−�nj ) 〈�(A)| O†

i Oj |�(A)〉c ,

where only the connected part is taken up in the correlator, or,equivalently, the operators have been redefined such that theirground-state expectation value is zero.

The momentum superposition of all relative positions ofthe operators can be evaluated explicitly by moving theoperators independently through the channels and summingall contributions. This infinite number of contributions canbe resummed by realizing that one obtains a geometric seriesinside the channels. Summing all different contributions froman operator moving in the top channel can be done byintroducing a new momentum-resolved operator that capturesthe momentum superposition:

=∑

n

eiqyn( )n

=[1 − eiqy

(−

)]−1

+ 2πδ(qy) ×( )

,

(4)

where we have separated the projector onto the largesteigenvector. As we will see, the diverging δ contributionwill always drop out, such that the inverse is well defined.The momentum superposition inside the channel can berepresented as

eiqy + e2iqy + e3iqy + . . .

= eiqy ,

where the component along the channel fixed point is indeedalways zero—this component would correspond to the discon-nected part of the correlation function. The geometric seriesconverges for every value of the momentum and the inversecan be taken without problem.

By independently letting the two operators travel throughthe channels all relative positions can be taken into account.In addition, we also need the contribution where the twooperators act on the same site. The full expression is

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given by

s(�q) = + e−iqx + e+iqx

+ e+iqy + e−iqy + e+iqxe−iqy

+ e−iqxe−iqy + e+iqxe+iqy + e−iqxe+iqy ,

(5)

where the green tensor represents the action of the twooperators at the same site, and the blue and red tensorsrepresent actions of the operators on the ket and bra level. Thecomputational complexity for evaluating the structure factorscales as O(χ3D4 + χ2D6) in the PEPS bond dimension D

and the bond dimension of the environment χ , where thehardest step is computing the infinite sum inside a channelby an iterative linear solver.

III. VARIATIONAL CONJUGATE-GRADIENT METHOD

The PEPS ansatz defines a variational class of statesthat should approximate the ground state of two-dimensionalquantum lattice systems in the thermodynamic limit. Thesystem is described by its Hamiltonian, which we assume toconsist of nearest-neighbor interactions, i.e.,

H =∑

〈ij 〉hij ,

and the lattice structure, for which we will confine ourselvesto the square lattice. The following can be straightforwardlyextended to different lattices, larger unit cells, or longer-rangeHamiltonians.

As dictated by the variational principle, finding the bestapproximation to the ground state of H now amounts to solvingthe highly nonlinear minimization problem

minA

〈�(A)| H |�(A)〉〈�(A)|�(A)〉 . (6)

As we have seen, the evaluation of this energy functionalfor a certain tensor A is already nontrivial, but can be doneefficiently using a variety of numerical methods. Yet theevaluation of the energy is not enough, as efficient numericaloptimization algorithms also rely on the evaluation of thegradient or higher-order derivatives of the energy functional.For a translation-invariant PEPS, the gradient is a highlynontrivial object; it requires the evaluation of the change inenergy from a variation in the tensor A, for which the effectof local and nonlocal contributions should be added. In thisrespect, it is quite similar to a zero-momentum structure factor,

and can be evaluated using the channel environment that weintroduced in Sec. II. In Sec. III A we run through the differentdiagrams for the gradient’s explicit evaluation, and show thatit can be computed efficiently.

With the gradient, the easiest algorithm is the steepest-descent method, where in each iteration one minimizes theenergy in the direction of the gradient. One iteration i

corresponds to an update of the A tensor as

Ai+1 → Ai + αAi

with Ai = −gi (gi is the gradient at iteration i). The valueof α > 0 is determined with a line-search algorithm; we haveused a simple bisection algorithm with an Armijo condition onthe step size [52]. The performance can be greatly enhanced byimplementing a nonlinear conjugate-gradient method, wherethe search direction is a linear combination of the gradient andthe direction of the previous iteration:

Ai = −gi + βiAi−1.

For each nonlinear optimization problem, the parameter βi canbe chosen from a set of different prescriptions [52–54]. Herewe have exclusively used the Fletcher-Reeves scheme [55],according to which

βi = ‖gi‖2

‖gi−1‖2.

Crucially, these algorithms have a clear convergence crite-rion: when the norm of the gradient is sufficiently small, theenergy cannot be further optimized and an optimal solutionhas been found.

Note that these direct optimization methods allow us tocontrol the number of variational parameters in and/or imposecertain symmetries on the PEPS tensor A: the iterative searchcan be easily confined to a certain subspace of the PEPS varia-tional class by, e.g., projecting the gradient onto this subspacein each iteration. Moreover, this direct optimization strategyallows to start from a random input tensor A and systematicallyconverge to an optimal solution—all the results in Sec. IV wereobtained by starting from a random initial tensor.

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A. Computing the gradient

The objective function f that we want to minimize [seeEq. (6)] is a real function of the complex-valued A, or,equivalently, the independent variables A and A. The gradientis then obtained by differentiating f (A,A) with respect to A,

grad = 2 × ∂f (A,A)

∂A

= 2 × ∂A 〈�(A)| H |�(A)〉〈�(A)|�(A)〉

− 2 × 〈�(A)| H |�(A)〉〈�(A)|�(A)〉2 ∂A〈�(A)|�(A)〉,

where we have clearly indicated A and A as independentvariables. In the implementation we will always make surethe PEPS is properly normalized, such that the numeratorsdrop out. By subtracting from every term in the Hamiltonianits expectation value, the full Hamiltonian can be redefined as

H → H − 〈�(A)|H |�(A)〉, (7)

such that the gradient takes on the simple form

grad = 2 × ∂A〈�(A)|H |�(A)〉.The gradient is thus obtained by differentiating the energyexpectation value 〈�(A)| H |�(A)〉 with respect to every A

tensor in the bra level and taking the sum of all contributions.Every term in this infinite sum is obtained by omitting oneA tensor and leaving the indices open. The full infinite

summation is then obtained by letting the Hamiltonian operatorand this open spot in the network travel through the channelsseparately, just as in the case of the structure factor in Sec. II D.

Let us first define a new tensor that captures the infinite sumof Hamiltonian operators acting inside a channel,

= + + + . . .

= ,

where the big tensor is again the inverted channel operator ofEq. (4) with momentum zero. Because we have redefined theHamiltonian in Eq. (7), the inversion of the channel operatoris well defined, because the vector on which the inverse actshas a zero component along the channel fixed point ρL.

With this blue tensor all different relative positions of theHamiltonian terms and the tensor A that is being differentiated(the open spot) can be explicitly summed, similarly to theexpression for the structure factor [Eq. (5)]. There are a fewmore terms because every Hamiltonian term corresponds to atwo-site operator and has different orientations.

The full expression is

grad = + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + + ,

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where the red tensor indicates where the open spot in thebra level of the diagram is. Note that the diagrams on thesame line are always related by a rotation; in the case that thePEPS tensor A is rotationally invariant, these diagrams giveexactly the same contribution. This implies that the gradientcorresponding to a rotationally invariant tensor A is itselfrotationally invariant. The computational complexity for eval-uating the gradient scales similarly to the structure factor, i.e.,O(χ3D4 + χ2D6), where again the hardest step is computingthe infinite sum inside a channel by an iterative linear solver.

B. The energy variance

Like any variational method, the PEPS ansatz is a priori notguaranteed to provide an accurate parametrization of a groundstate. It is expected that increasing the PEPS bond dimensionprovides a good test for the reliability of the simulation: an ex-trapolation in D should provide the correct results. One prob-lem is that it is unclear how the energy or order parameter be-have as a function of D [30]. A better and completely unbiasedextrapolation quantity is the energy variance [56], defined as

v = 〈�(A)| (H − e)2|�(A)〉,with e = 〈�(A)| H |�(A)〉 the energy expectation value. Itmeasures to what extent a variational wave function approx-imates the ground state (or more generally, an eigenstate) ofthe Hamiltonian.

Because the variance can be interpreted as a zero-momentum structure factor of the Hamiltonian operator, thecomputation of the energy variance is again similar. In additionto the green tensor above, we will also need the followinggeometric series

= + + . . .

=

where

=∑

n

( )n

=[1 −

(−

)]−1

+ 2πδ(0) ×( )

with the fixed points of the two-site channels properlynormalized. We again renormalize the Hamiltonian as

H → H − 〈�(A)| H |�(A)〉,

such that disconnected contributions always drop out and theinverse of the operator above is well defined. The blue tensorhas χ2D4 elements, so its computation is by far the most costlystep for the variance evaluation. Approximating it by a tensordecomposition might reduce the cost considerably, but for ourpurposes this has not been necessary.

Let us now associate to each nearest-neighbor term 〈ij 〉 inthe Hamiltonian a variance term as

v〈ij 〉 = 〈�(A)| Hh〈ij〉|�(A)〉,

such that the energy variance per site is given by

v = 1

|L| 〈�(A)| H 2 |�(A)〉 = v〈ij〉,hor + v〈ij〉,ver,

the sum of the variances corresponding to the horizontal andvertical nearest-neighbor terms in the Hamiltonian.

The vertical contribution is given by

v〈ij〉,ver = + 2 × + 2 × + 2 ×

+ 2 × + 2 ×

+ 2 × + 2 ×

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+ 2 × + 2 × + 2 × + 2 ×

+ 2 × + 2 × + 2 × + 2 × .

The green tensors represent the double action of the Hamil-tonian operator: a two-site tensor if they fully overlap anda three-site tensor if the overlap is on one site only. In thisexpression, we have explicitly used the rotational invarianceof the PEPS tensor A, which can be easily imposed within ourframework. Under this symmetry, the horizontal and verticalcontributions to the variance are obviously equal, so theabove is the complete expression for the variance. If A is notrotationally invariant, all the other diagrams can be obtained byrotating the above ones. The complexity scaling of the varianceevaluation is larger than for the gradient, because of the extrageometric series in a two-site channel; the complexity scalesas O(χ3D6).

IV. BENCHMARKS

As a first check, we apply our PEPS algorithm to thetwo-dimensional transverse Ising model on the square lattice,defined by the Hamiltonian

HIsing =∑

〈ij〉Sz

i Szj + λ

∑

i

Sxi .

The model exhibits a phase transition at λc ≈ 3.044 [57] froma symmetry-broken phase to a polarized phase; the orderparameter is m = 〈Sz〉. The model has been extensively studiedwith the PEPS ansatz [18,20,50], and we use the model as abenchmark for our conjugate-gradient method.

In Fig. 1 we have plotted the magnetization curve of thetransverse Ising model for two different values of the bonddimension D, the parameter that controls the dimensions ofthe PEPS and can be tuned as a refinement parameter. Wesee that the phase transition is captured accurately alreadyfor D = 3; growing the bond dimension further will increasethe accuracy only slightly. Further on, we will observe that asystematic growing of the bond dimension is paramount forcapturing ground states with stronger correlations.

In Fig. 2 we have compared our variational search withimaginary-time evolution (full update), showing that we findlower energies and better order parameters, even as the Trottererror goes to zero. The plot clearly shows that, as the Trotterstep size goes to zero, the imaginary-time result does not

converge to the variational optimum that we obtain. Note thatthe variational freedom is slightly different: we optimize over arotationally symmetric PEPS with a one-site unit cell, whereasthe imaginary-time results break rotational symmetry and workwith a two-site unit cell. Although this larger rotationallyasymmetric unit cell might give lower energies, it appearsthat our optimization still gives better energies and orderparameters.

In Fig. 3 we provide some details on the convergenceof the conjugate-gradient algorithm. In particular, we havefound that rather high values of χ (the bond dimension ofthe corner environment) were needed to evaluate the gradientaccurately close to convergence. Indeed, in the case of astrongly correlated PEPS, a lot of different terms contributeto the expression for the gradient. Close to convergence thegradient becomes a vector of small magnitude, which can onlyhappen due to the subtle cancellations of a lot of differentterms; consequently, finding the gradient accurately is boundto require a large value of χ . Note that the large values of

λ

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

m

0

0.2

0.4

0.6

0.8

FIG. 1. The magnetization curve for the transverse Ising modelwith bond dimensions D = 2 (blue) and D = 3 (orange). We nicelycapture the phase transition, although the critical point has beenslightly shifted. The critical point can be estimated as the point wherethe slope of the curve is maximal; we arrive at λc ≈ 3.09 (D = 2)and λc ≈ 3.054 (D = 3).

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Trotter step size τ

0 0.05

e

−3.230

−3.229

Trotter step size τ

0 0.05m

0

0.1

0.2

0.3

0.4

FIG. 2. Our variational results compared to the results thatare obtained with imaginary-time evolution using the full-updatealgorithm; the comparison is done for the transverse Ising model atλ = 3.04 for bond dimensions D = 2 and D = 3. On the left we haveplotted the convergence for the energy (magnetization) as a functionof the Trotter step size of the full-update scheme (blue points), andour results (red line). For both plots, the upper (lower) lines are forD = 2 (D = 3).

χ are only necessary close to convergence, so we grow χ

throughout the optimization. We never impose the final valueof χ , because it is the correlations in the optimized PEPS thatdetermine the χ needed to reach a certain tolerance on thenorm of the gradient.

As a second application, we study two spin-1/2 Heisenbergmodels on the square lattice, defined by the Hamiltonian

HHeisenberg =∑

〈ij〉Sx

i Sxj + S

y

i Sy

j + JzSzi S

zj .

The model has been of great theoretical and experimentalinterest, because of its paradigmatic long-range antiferro-magnetic order [58]. In particular, Heisenberg models haveproven to be a hard case for the PEPS ansatz [59] because ofthe large quantum fluctuations around the antiferromagneticordering; as such, they provide a proper benchmark for ourconjugate-gradient method.

In contrast to most PEPS implementations, we prefer towork with a single-site unit cell, so we perform a sublatticerotation in order to capture the staggered magnetic order inthe ground state. Moreover, we impose rotational symmetryon the PEPS tensor A, so that our variational ground stateis automatically invariant under rotations of the lattice. InFigs. 4 and 5 we have plotted the energy expectation valueand staggered magnetization after convergence as a functionof the bond dimension, for the XY model (Jz = 0) andthe isotropic Heisenberg model (Jz = 1). Comparing withresults from imaginary-time evolution [20,59], we see thatour variational method reaches considerably lower energiesand order parameters at the same bond dimension.

In addition we also compute the variance of these PEPSvariational states, in order to get an idea of how well theyapproximate the true ground state. The result for the isotropicHeisenberg model is plotted in Fig. 6. We observe the expectedlinear behavior [56] to some extent, and a zero-varianceextrapolation based on the two last points (D = 4,5) improvesthe estimate of the energy by a factor of two. Better zero-variance extrapolations should be possible at higher bond

Iteration0 100 200 300 400

10−10

10−5

100

χ

0 50 100 150 200

‖g‖ 2

10−5

10−4

10−3

10−2

10−1

FIG. 3. Details on the convergence of the optimization algorithmfor the Ising model at λ = 3. Upper panel: The convergence of thenorm of the gradient ‖g‖ =

√g†g (blue), the error in the energy

(red), and the error in the magnetization (yellow), as a function of theiteration. The errors are computed as the relative error with respect tothe last iteration. In this D = 2 simulation the convergence criterionwas ‖g‖ � 10−5, a value for which the two plotted observableshave clearly converged. Lower panel: The convergence of the normof the gradient as a function of the bond dimension χ of thecorner environment, at a particular iteration of the conjugate-gradientscheme for D = 3 (close to convergence). This plot shows that largevalues of χ are needed to obtain a required tolerance on the norm ofthe gradient (in this case χ ≈ 100).

dimensions for which the linear behavior is expected to bestronger.

Another quantity that is within reach of our PEPSframework is the static structure factor, a central quantityfor detecting the order in the ground state, and of directexperimental relevance. It is defined as

s(�q) = 1

|L|∑

i,j∈Lei �q·(�ni−�nj ) 〈�Si · �Sj 〉c ,

where only the connected part of the correlation function istaken into account. The disconnected part will give a δ peakat �q = (π,π ) (the X point), corresponding to the staggered-

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bond dimension D

2 3 4 5

Δe

10−5

10−4

10−3

10−2

bond dimension D

2 3 4 5m

s0.43

0.44

0.45

0.46

FIG. 4. Results for the XY model (Jz = 0), compared to theMonte Carlo results in Ref. [60]. Left: The relative error �e =|(evar − eMC)/eMC| as a function of the bond dimension. Right: Thestaggered magnetization as a function of the bond dimension; the redline is the Monte Carlo result with error bars.

magnetization order parameter. The strong fluctuations aroundthis point will give an additional 1/q divergence, with q thedistance from the X point [63]. The structure factor becomeszero at �q = (0,0), because the ground state is in a singletstate. In Fig. 7 we observe that the regular parts of thestructure factor are perfectly reproduced, even at low bonddimensions, whereas the divergences can only be accuratelycaptured by observing the behavior as a function of the bonddimension.

V. CONCLUSIONS

In conclusion, we have presented an algorithm for numer-ically optimizing the PEPS ansatz for ground-state approx-imations. The algorithm is based on the efficient evaluationof the energy gradient, and is a direct implementation ofthe variational principle with a clear convergence criterion.Starting from a random PEPS tensor, it allows us to find avariational minimum for a given bond dimension.

As such, our approach is complementary to any otherPEPS algorithm. In fact, our variational search systematically

bond dimension D

2 3 4 5

Δe

10−4

10−3

10−2

10−1

bond dimension D

2 3 4 5

ms

0.30

0.32

0.34

0.36

0.38

0.40

FIG. 5. Results for the Heisenberg antiferromagnet (Jz = 1),compared to the Monte Carlo results in Refs. [61] and [62]. Left:The relative error �e = |(evar − eMC)/eMC| as a function of the bonddimension. Right: The staggered magnetization as a function of thebond dimension; the red line is the Monte Carlo result for which theerror bars are too small to plot.

v(D)0 0.005 0.010 0.015

e(D

)

−0.670

−0.668

−0.666

−0.664

−0.662

−0.660

FIG. 6. The energy expectation value as a function of the varianceper site for the isotropic (Jz = 1) Heisenberg antiferromagnet, for fourvalues of the bond dimension (D = 2 → 5). The red line representsthe Monte Carlo result for the ground-state energy [61]. A linearextrapolation with respect to the two best points (D = 4,5) gives anenergy with a relative error of �e ≈ 4.7 × 10−5. The striped line isdrawn between the exact MC result and the D = 5 point and servesonly as a guide to the eye.

finds lower energies than algorithms based on imaginary-timeevolution and local truncations. This observation is consistentwith the recent results in Ref. [65], where an alternativevariational algorithm was proposed. This confirms our beliefthat a variational approach will be crucial in the future forcapturing, e.g., phase transitions in two-dimensional latticesystems.

Our approach has the additional advantage that globalsymmetries can be exploited easily, which should lead to moreefficient simulations [66]. Moreover, the implementation ofsymmetries will prove crucial for simulating systems withtopological order, which can be imposed as a matrix productoperator symmetry on the virtual level of the PEPS tensor [14].Finally, our approach straightforwardly allows us to consider

M X S Γ M S

s(� k)

10−2

10−1

100

101

FIG. 7. Structure factor s(�q) of the isotropic (Jz = 1) Heisenbergantiferromagnet along a path through the Brillouin zone for optimizedPEPS states with bond dimensions D = 2 (blue), D = 3 (red), D =4 (orange), and D = 5 (purple), in agreement with the results inRefs. [63] and [64]. The divergence around the X point and the zeroaround the point are better reproduced as D increases, although theimprovement as a function of D seems not to be smooth.

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reduced PEPS parametrizations by confining our optimizationscheme to a certain PEPS subclass [67,68].

In addition, some of the methods that we have presented inthis paper could be applicable to the variational optimization ofPEPS on finite lattices [7,34,69–73] as well. With finite PEPSsimulations, the straightforward approach of optimizing thedifferent PEPS tensors sequentially is severely hampered bythe bad conditioning of the normalization matrix. In particular,because the energy and normalization matrix require differenteffective environments, the regularization of this bad conditionnumber is not well defined. With a finite-lattice version of thecorner environments, however, we could use the same effectiveenvironment for computing the energy and normalization,allowing a consistent regularization of both the energy andnormalization matrix. This should lead to efficient variationaloptimization methods for finite PEPS as well.

Our framework has allowed us to compute the structurefactor, which is of direct experimental relevance, and theenergy variance, which provides an unbiased measure of thevariational error of the PEPS ansatz. Although the variance ex-trapolations seem to be not straightforwardly implementable,this should contribute to better energy bond dimensionextrapolations in the future. For systems with a number ofcompeting ground states such as the Hubbard model [30],this extrapolation will be of crucial importance.

From the perspective of numerical optimization, aconjugate-gradient search is only a first step to more advancedschemes such as Newton or quasi-Newton methods. The Hes-sian of the energy functional is crucial in these optimizationschemes, the evaluation of which is straightforward with oureffective environment. Alternatively, the nontrivial geometricstructure of the PEPS manifold can be taken into account inthe optimization [74]. Also, imposing a certain gauge fixing onthe PEPS tensor might render the optimization more efficient.

Finally, it seems that tangent-space methods that haveproven successful in the context of matrix product states[39] are now within reach for PEPS simulations for generictwo-dimensional quantum spin models. In particular, thispaper opens up the prospect of simulating real-time evolutionaccording to the time-dependent variational principle [38]and/or computing the low-energy spectrum on top of a genericPEPS with the quasiparticle excitation ansatz [75,76].

ACKNOWLEDGMENTS

This research was supported by the Research FoundationFlanders (L.V. and J.H.), by the Delta-ITP (an NWO programfunded by the Dutch OCW) (P.C.), and by the Austrian FWFSFB through Grants FoQuS and ViCoM and the EuropeanGrants SIQS and QUTE (F.V.).

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