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 Netherlands3Vienna Center for Quantum Science, Universitat Wien, Boltzmanngasse 5, A-1090 Wien, Austria

(Dated: November 8, 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 principlewithin the PEPS manifold. Our optimization is based on an efficient and accurate evaluation ofthe gradient of the global energy functional by using effective corner environments, and is robustwith respect to the initial starting points. It has the additional advantage that physical and virtualsymmetries can be straightforwardly implemented. We provide the tools to compute static structurefactors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark ourmethod on Ising and Heisenberg models, and show a significant improvement on the energies andorder parameters as compared to algorithms based on imaginary-time evolution.

I. INTRODUCTION

Ever since the birth of quantum mechanics, the quan-tum many-body problem has been at the center of theo-retical and computational physics. Despite the simplicityof the fundamental equations, it has been notoriouslydifficult to simulate the quantum behavior of many-bodysystems. This is especially true for low-dimensional sys-tems: because quantum correlations are stronger, pertur-bation theory often fails and more sophisticated methodsare needed. This cry for better methods is loudest withrespect to two-dimensional systems, because of the largerange of unexplored quantum phenomena—quantum spinliquids1, topological order2 and quasi-particle fractional-ization are only three examples.

Because Monte Carlo sampling is often plagued by thesign problem and exact diagonalization is necessarily lim-ited to small system sizes, it seems that variational meth-ods are the way to go for exploring the two-dimensionalquantum world. There are essentially two prerequisitesfor a successful variational approach: (i) an adequate vari-ational ansatz that captures the physics for the problem athand, and (ii) an efficient way of computing observablesand optimizing the variational parameters. Examplessuch as the density-matrix renormalization group3,4 andGutzwiller-projected wave functions5,6 seem to meet thelatter, but it is unclear to what extent they are the naturalchoice for simulating two-dimensional quantum systems.

In recent years projected entangled-pair states(PEPS)7,8 have emerged as a viable candidate for captur-ing the physics of ground states of strongly-correlatedquantum lattice models in two dimensions. It is byexplicitly modeling the distribution of entanglement inlow-energy states of local Hamiltonians, that PEPSparametrize the “physical corner of Hilbert space”. In-deed, PEPS have a built-in area law for the entanglemententropy9, they provide a natural characterization of topo-logical order10–14, and they can realize bulk-boundarycorrespondences explicitly15–17. Moreover, PEPS can beformulated directly in the thermodynamic limit18 which

allows us to focus on bulk physics without any finite-sizeor boundary effects.

An efficient optimization of the parameters in a PEPShas proven to be more challenging. According to thevariational principle, finding the best approximation tothe ground state for a given Hamiltonian H reduces to theminimization of the energy expectation value. For infinitePEPS this amounts to a highly non-linear optimizationproblem for which the evaluation of, e.g., the gradientof the energy functional is a hard problem. For thatreason, the state-of-the-art PEPS algorithms have takenrecourse to imaginary-time evolution18–20: a trial PEPSstate is evolved with the operator e−τH , which shouldresult in a ground-state projection for very long times τ .This imaginary-time evolution is integrated by applyingsmall time steps δτ with a Trotter-Suzuki decompositionand, after each time step, truncating the PEPS bonddimension in an approximate way. This truncation can bedone by a purely local singular-value decomposition—theso-called simple-update19 algorithm—or by taking thefull PEPS wave function into account—the full-update18

or fast full-update20 algorithm.

These imaginary-time algorithms have allowed very ac-curate simulations of frustrated spin systems21–30 andstrongly correlated electrons30–33, but it remains unclearwhether they succeed in finding the optimal state in agiven variational class of PEPS. Although computation-ally very cheap, ignoring the environment in the simple-update scheme is often a bad approximation for systemswith large correlations. The full-update scheme takesthe full wave function into account for the truncation,but requires the inversion of the effective environmentwhich is potentially badly conditioned. This problem wassolved by regularizing the environment appropriately andfixing the gauge of the PEPS tensor20,34. Nonetheless,the truncation procedure in the full-update scheme isnot guaranteed to provide the globally optimal truncatedtensors in the sense that the global overlap of the trun-cated and the original PEPS is maximized. Indeed, thetruncated tensor is optimized locally and afterwards put

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in at every site in the lattice to give an updated (global)PEPS wave function.

Similar issues have been at the center of attentionin the context of matrix product states (MPS)35, theone-dimensional counterparts of PEPS, where a numberof different strategies have been around for optimizingground-state approximations directly in the thermody-namic limit36–38. Recently, the problem of finding anoptimal matrix product state has been reinterpreted by(i) identifying the class of matrix product states as a non-linear manifold embedded in physical Hilbert space39,40,and (ii) formulating a minimization problem of the globalenergy functional on this manifold. A globally optimalstate can then be recognized as a point on the mani-fold for which the gradient of the energy functional iszero. Moreover, approximating time evolution within themanifold is optimized, as dictated by the time-dependentvariational principle38, 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 interpretation41.Moreover, whereas imaginary-time evolution more or lesscorresponds to a steepest-descent method39, more ad-vanced optimization methods such as conjugate-gradientor quasi-Newton algorithms can find an optimal matrixproduct state much more efficiently42,43.

In Ref. 44 it was shown how to implement these tan-gent space methods for PEPS by introducing a contractionscheme based on the concept of a “corner environment”.Building on that work, this paper presents a PEPS algo-rithm that optimizes the global energy functional usinga conjugate-gradient optimization method. In contrastto other methods, this algorithm has a clear convergencecriterion, which can guarantee that an optimal state hasbeen reached. Moreover, it allows us to more easily im-pose physical symmetries on the PEPS. On the fly, thecontraction scheme also allows us to compute the energyvariance of the variational ground state—an unbiasedmeasure of the accuracy of the variational ansatz anda tool for better energy extrapolations—as well as gen-eral two-point correlation functions and static structurefactors.

In the next section [Sec. II] we review the “cornerenvironment” in considerable detail and show how tocompute static structure factors of a PEPS. Next [Sec. III]we discuss our conjugate-gradient scheme for the PEPSoptimization, and explain how to evaluate the energygradient and the energy variance. We benchmark [Sec. IV]our method by applying it to the transverse Ising model,the XY model and the isotropic Heisenberg model. In thelast section [Sec. V] we discuss the possible extensionsand applications.

II. EFFECTIVE ENVIRONMENTS ANDTWO-POINT CORRELATION FUNCTIONS

Consider an infinite square lattice with every site host-ing a quantum degree of freedom with dimension d. Forthis quantum spin system, a PEPS can be introducedformally as

|Ψ(A)〉 =∑s

C2(A) |s〉 (1)

where C2(. . . ) is the contraction of an infinite tensor net-work. This contraction is most easily represented graphi-cally 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 associ-ated with every site in the lattice and all virtual indices(u, r, d, l) are contracted in the network. The physical in-dices s are left open, such that a coefficient is obtained forevery spin configuration in the superposition in Eq. (1).The graphical representation is then obtained by connect-ing links that are contracted and leaving the physicallinks open. The virtual degrees of freedom in the PEPScarry the quantum correlations and mimic the entangle-ment structure of low-energy states. The dimension ofthe virtual indices is called the bond dimension D andcan be tuned in order to enlarge the variational class; assuch, it acts as a refinement parameter for the variationalPEPS ansatz.

The norm of an infinite PEPS can be pictorially repre-sented as

〈Ψ(A)|Ψ(A)〉 = ,

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

a = =

3

As in the rest of this paper, the virtual indices of theket and bra level are grouped into one index, so thatthese “top view” representations of double-layer tensorcontractions are simplified.

The norm of the PEPS is thus obtained by the con-traction of an infinite tensor network and can, in general,only be done approximately. Different numerical methodshave been developed to contract these infinite networksefficiently, which allows the evaluation of the norm ofa PEPS, as well as expectation values and correlationfunctions.

A. The linear transfer matrix

The first and most straightforward strategy is based onthe linear transfer matrix T , graphically represented as

T = .

This object carries all the correlations in the PEPS fromone row in the network to the next. One can of coursedefine a similar transfer matrix in the vertical direction,and even diagonal transfer matrices can be considered.Naturally the transfer matrix is interpreted as an opera-tor from the top to the bottom indices, so that the fullcontraction of the two-dimensional network reduces to suc-cessively multiplying copies of T . In the thermodynamiclimit, the norm of a PEPS is thus given by

〈Ψ(A)|Ψ(A)〉 = limN→∞

T N = λN

with λ the leading eigenvalue of the transfer matrix. Theassociated leading eigenvector or fixed point contains allthe information on the correlations of a half-infinite partof the lattice.

An exact representation of the fixed point is only possi-ble in a number of special cases and approximate methodshave to be devised in general. Given the versatility of ma-trix product states (MPS) for approximating the groundstate of local gapped Hamiltonians35, one expects thatthis class of states might provide a good variational ansatzfor the case of gapped transfer matrices as well. Moreover,the bond dimension of the matrix product state represen-tation of the fixed point, denoted with χ, can be tunedsystematically, such that the errors can be kept undercontrol perfectly. Whereas MPS approximations for fixedpoints go way back45, a variety of efficient tensor-networkmethods have been developed7,46 recently. Here we usean algorithm43 in the spirit of Ref. 41, which treats thelinear and corner transfer matrices [Sec. II B] on a similarfooting.

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 precision43. Indeed, if thistensor (rectangle) exists, it maps the action of the transfermatrix back to the same MPS fixed point. The virtualdimension of the MPS fixed point will be denoted as χand can be tuned to improve the accuracy of the PEPScontraction.

Given that the fixed-point equation can be solved ef-ficiently, the PEPS can now be normalized to one byrescaling the A tensor such that the largest eigenvalueλ of the transfer matrix equals unity. With the MPSfixed point, the expectation value of a local operator atan arbitrary site i,

〈Oi〉 =〈Ψ(A)|Oi |Ψ(A)〉〈Ψ(A)|Ψ(A)〉

,

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

≈ ,

where a colored block tensor always indicates the presenceof a physical operator at that site. The resulting effectiveone-dimensional network can be evaluated exactly byfinding the leading left and right eigenvectors (fixed points)of the channel operator,

= µ

and

= µ .

The eigenvalue µ depends on the normalization of theMPS tensors in the upper and lower fixed points of thelinear transfer matrix, and its value can be put to one.The fixed points are determined up to a factor, which can

4

be fixed by imposing that the 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 trans-fer matrix, which was first applied to classical latticesystems45,47–49 and recently used extensively in tensornetwork simulations32,50,51. The strategy now is to breakup the infinite tensor network in different regions, andrepresent these as tensors with a fixed dimension. Graph-ically, the set up is

≈ .

A red tensor represents the compression of one of thecorners of the network, whereas the blue tensors capturethe effect of an infinite row of a tensors. Together, theyprovide an effective one-site environment for the com-putation of the norm of the PEPS or local expectationvalues.

This scheme can now be extended44 in order to evaluatenon local expectation values such as general two-pointcorrelation functions. Indeed, by not compressing theblue region above, one can construct an environment thatlooks like

≈ ,

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

Finding this effective “corner environment” can againbe done by solving a fixed-point equation. Indeed, the

green corner-shaped environment should be the resultof an infinite number of iterations of an equally corner-shaped transfer matrix; the fixed-point equation is

∝ .

Very far from the corner this equation reduces to theone for the linear transfer matrix. This implies that,asymptotically, the fixed point can be well approximatedby an MPS. Let us therefore make the ansatz that thefull fixed point can be approximated as an MPS, wherewe put an extra tensor on the virtual level to account forthe corner. With this ansatz, the fixed point equation isgiven by

∝ . (3)

We expect44 that this fixed point can be modelled usingthe MPS tensors from the fixed points of the linear trans-fer matrix, up to the corner matrix, which captures theeffect of the corner shape. With this ansatz, we obtain alinear fixed point equation for the corner matrix, whichcorresponds to a simple eigenvalue equation and can besolved efficiently.

C. Channel environments

Once we have found (i) the fixed points of the lineartransfer matrix in all directions [Eq. (2)], and (ii) thefour corner tensors [Eq. (3)], we can contract the networkcorresponding to the norm, a local expectation value, or acorrelation function of the PEPS. Let us assume that thetensor A is normalized such that the largest eigenvalue ofthe linear transfer matrix is unity. Computing the norm〈Ψ(A)|Ψ(A)〉 with a channel environment then reduces

5

to the contraction of

.

An infinitely long channel can be contracted by computingthe fixed point ρL of the “channel operator”. Thereforethe eigenvector corresponding to the largest eigenvalueshould be found, i.e.,

= λ×

for the top channel. The boundary MPS tensors have tobe rescaled such that the largest eigenvalue λ is put toone. Similarly, the fixed point in the other direction ρRis defined as

= λ×

The inner product of the left and right fixed points isput to one. For further use, we note that, by subtractingthe projector on the largest eigenvector, an operator isconstructed that has spectral radius strictly smaller thanone,

ρ

(−

)< 1.

The norm of the PEPS is then reduced to

〈Ψ(A)|Ψ(A)〉 = ,

which can be scaled to 1 by rescaling the corner tensorsby the appropriate scalar. With these conventions, thenorm of the state is well defined and expectation valuescan be safely computed. For a local one-site operator Owe have

〈Ψ(A)|O |Ψ(A)〉 = ,

and, similarly, the expectation value of a two-site operatoris

〈Ψ(A)|O |Ψ(A)〉 = ,

where the two-site operator can of course be oriented inthe other 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 twooperators at generic locations in the lattice is computedas

.

In fact, even three-point correlation functions can beevaluated by orienting the corners in the right way, as in,e.g., the contraction

.

From a computational point of view, the hardest stepin determining this corner environment is finding thefixed point of the linear transfer matrix; state-of-the-artalgorithms43 scale as O(χ3D4 +χ2D6), with D the PEPSbond dimension and χ the bond dimension of the fixedpoint. In the case of strongly-correlated PEPS, findingthe fixed point might take a lot of iterations. Determiningthe corner tensors and the channel fixed points has similarscalings, but this has to be done 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 correla-tion function directly in momentum space, i.e., the staticstructure factor s(~q),

s(~q) =1

|L|∑i,j∈L

ei~q·(~ni−~nj) 〈Ψ(A)|O†iOj |Ψ(A)〉c

where only the connected part is taken up in the correlator,or, equivalently, the operators have been redefined suchthat their ground-state expectation value is zero.

The momentum superposition of all relative positionsof the operators can be evaluated explicitly by movingthe operators independently through the channels andsumming all contributions. This infinite number of contri-butions can be resummed by realizing that one obtains a

6

geometric series inside the channels. Summing all differentcontributions from an operator moving in the top channelcan be done by introducing a new momentum-resolvedoperator that captures the momentum superposition,

=∑n

eiqyn( )n

=

[1− eiqy

(−

)]−1

(4)

+ 2πδ(qy)×( )

,

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 be

represented as

eiqy + e2iqy + e3iqy + . . .

= eiqy ,

where the component along the channel fixed point isindeed always zero—this component would correspond tothe disconnected part of the correlation function. The ge-ometric series converges for every value of the momentumand the inverse can be taken without problem.

By independently letting the two operators travel through the channels all relative positions can be taken intoaccount. In addition, we also need the contribution where the two operators act on the same site. The full expressionis 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 two operators at the same site, and the blue and red tensorsrepresent actions of the operators on the ket and bra level. The computational complexity for evaluating the structurefactor scales as O(χ3D4 +χ2D6) in the PEPS bond dimension D and the bond dimension of the environment χ, wherethe hardest step is computing the infinite sum inside a channel by an iterative linear solver.

III. VARIATIONAL CONJUGATE-GRADIENTMETHOD

The PEPS ansatz defines a variational class of statesthat should approximate the ground state of two-dimensional quantum lattice systems in the thermody-namic limit. The system is described by its Hamiltonian,which we assume to consist of nearest-neighbor interac-

tions, i.e.,

H =∑〈ij〉

hij ,

and the lattice structure, for which we will confine our-selves to the square lattice. The following can be straight-forwardly extended to different lattices, larger unit cells,or longer-range Hamiltonians.

7

As dictated by the variational principle, finding the bestapproximation to the ground state of H now amounts tosolving the highly non-linear minimization problem

minA

〈Ψ(A)|H |Ψ(A)〉〈Ψ(A)|Ψ(A)〉

. (6)

As we have seen, the evaluation of this energy functionalfor a certain tensor A is already non-trivial, but can bedone efficiently using a variety of numerical methods. Yetthe evaluation of the energy is not enough, as efficientnumerical optimization algorithms also rely on the eval-uation of the gradient or higher-order derivatives of theenergy functional. For a translation-invariant PEPS, thegradient is a highly non trivial object; it requires theevaluation of the change in energy from a variation inthe tensor A, for which the effect of local and non-localcontributions should be added. In this respect, it is quitesimilar to a zero-momentum structure factor, and can beevaluated using the channel environment that we intro-duced in Sec. II. In Sec. III A we run through the differentdiagrams for the gradient’s explicit evaluation, and showthat it can be computed efficiently.

With the gradient, the easiest algorithm is the steepest-descent method, where in each iteration one minimizesthe energy in the direction of the gradient. One iterationi corresponds to an update of the A tensor as

Ai+1 → Ai + αAi

with Ai = −gi (gi is the gradient at iteration i). Thevalue of α > 0 is determined with a line-search algo-rithm; we have used a simple bisection algorithm withan Armijo condition on the step size52. The performancecan be greatly enhanced by implementing a non-linearconjugate-gradient method, where the search direction isa linear combination of the gradient and the direction ofthe previous iteration:

Ai = −gi + βiAi−1.

For each non linear optimization problem, the parameterβi can be chosen from a set of different prescriptions52–54.Here we have exclusively used the Fletcher-Reevesscheme55, according to which

βi =‖gi‖2

‖gi−1‖2.

Crucially, these algorithms have a clear convergencecriterion: when the norm of the gradient is sufficientlysmall, the energy cannot be further optimized and anoptimal solution has been found.

Note that these direct optimization methods allow usto control the number of variational parameters in and/orimpose certain symmetries on the PEPS tensor A: theiterative search can be easily confined to a certain sub-space of the PEPS variational class by, e.g., projecting thegradient onto this subspace in each iteration. Moreover,this direct optimization strategy allows to start from a

random input tensor A and systematically converge to anoptimal solution—all the results in Sec. IV were obtainedby starting from a random initial tensor.

A. Computing the gradient

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

grad = 2× ∂f(A, A)

∂A

= 2× ∂A 〈Ψ(A)|H |Ψ(A)〉〈Ψ(A)|Ψ(A)〉

− 2× 〈Ψ(A)|H |Ψ(A)〉〈Ψ(A)|Ψ(A)〉2

∂A 〈Ψ(A)|Ψ(A)〉 ,

where we have clearly indicated A and A as independentvariables. In the implementation we will always makesure the PEPS is properly normalized, such that the nu-merators drop out. By subtracting from every term in theHamiltonian its expectation value, the full Hamiltoniancan be redefined as

H → H − 〈Ψ(A)|H |Ψ(A)〉 , (7)

such that the gradient takes on the simple form

grad = 2× ∂A 〈Ψ(A)|H |Ψ(A)〉 .

The gradient is thus obtained by differentiating the en-ergy expectation value 〈Ψ(A)|H |Ψ(A)〉 with respect toevery A tensor in the bra level and taking the sum of allcontributions. Every term in this infinite sum is obtainedby omitting one A tensor and leaving the indices open.The full infinite summation is then obtained by letting theHamiltonian operator and this open spot in the networktravel through the channels separately, just as in the caseof the structure factor in Sec. II D.

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

= + + + . . .

= ,

where the big tensor is again the inverted channel operatorof Eq.(4) with momentum zero. Because we have redefinedthe Hamiltonian in Eq. (7), the inversion of the channel

8

operator is well defined, because the vector on which theinverse acts has a zero component along the channel fixedpoint ρL.

With this blue tensor all different relative positions ofthe Hamiltonian terms and the tensor A that is being

differentiated (the open spot) can be explicitly summed,similarly to the expression for the structure factor [Eq. 5].There are a few more terms because every Hamiltonianterm corresponds to a two-site operator and has differentorientations.

The full expression is

grad = + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + + ,

where the red tensor indicates where the open spot in the bra level of the diagram is. Note that the diagrams on thesame line are always related by a rotation; in the case that the PEPS tensor A is rotationally invariant, these diagramsgive exactly the same contribution. This implies that the gradient corresponding to a rotationally invariant tensorA is itself rotationally invariant. The computational complexity for evaluating the gradient scales similarly to thestructure factor, i.e. O(χ3D4 + χ2D6), where again the hardest step is computing the infinite sum inside a channel byan iterative linear solver.

B. The energy variance

Like any variational method, the PEPS ansatz is a priorinot guaranteed to provide an accurate parametrization of

a ground state. It is expected that increasing the PEPS

9

bond dimension provides a good test for the reliability ofthe simulation: an extrapolation in D should provide thecorrect results. One problem is that it is unclear how theenergy or order parameter behave as a function of D30.A better and completely unbiased extrapolation quantityis the energy variance56, defined as

v = 〈Ψ(A)| (H − e)2 |Ψ(A)〉 ,

with e = 〈Ψ(A)|H |Ψ(A)〉 the energy expectation value.It measures to what extent a variational wave functionapproximates the ground state (or more generally, aneigenstate) of the Hamiltonian.

Because the variance can be interpreted as a zero-momentum structure factor of the Hamiltonian operator,the computation of the energy variance is again similar.In addition to the green tensor above, we will also needthe following geometric 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 − 〈Ψ(A)|H |Ψ(A)〉

such that disconnected contributions always drop out andthe inverse of the operator above is well defined. Theblue tensor has χ2D4 elements, so its computation isby far the most costly step for the variance evaluation.Approximating it by a tensor decomposition might reducethe cost considerably, but for our purposes this has notbeen necessary.

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

v〈ij〉 = 〈Ψ(A)|Hh〈ij〉 |Ψ(A)〉 ,

such that the energy variance per site is given by

v =1

|L|〈Ψ(A)|H2 |Ψ(A)〉 = v〈ij〉,hor + v〈ij〉,ver,

the sum of the variances corresponding to the horizontaland vertical nearest-neighbour terms in the Hamiltonian.

The vertical contribution is given by

v〈ij〉,ver = + 2× + 2× + 2×

+ 2× + 2×

+ 2× + 2×

10

+ 2× + 2× + 2× + 2×

+ 2× + 2× + 2× + 2× .

The green tensors represent the double action of the Hamiltonian operator: a two-site tensor if they fully overlap and athree-site tensor if the overlap is on one site only. In this expression, we have explicitly used the rotational invarianceof the PEPS tensor A, which can be easily imposed within our framework. Under this symmetry, the horizontal andvertical contributions to the variance are obviously equal, so the above is the complete expression for the variance. IfA is not rotationally invariant, all the other diagrams can be obtained by rotating the above ones. The complexityscaling of the variance evaluation is larger than for the gradient, because of the extra geometric series in a two-sitechannel; the complexity scales as O(χ3D6).

IV. BENCHMARKS

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

HIsing =∑〈ij〉

Szi Szj + λ

∑i

Sxi .

The model exhibits a phase transition at λc ≈ 3.04457

from a symmetry broken phase to a polarized phase;the order parameter is m = 〈Sz〉. The model has beenextensively studied with the PEPS ansatz18,20,50, and weuse the model as a benchmark for our conjugate-gradientmethod.

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 dimensionsof the PEPS and can be tuned as a refinement parameter.We see that the phase transition is captured accuratelyalready for D = 3; growing the bond dimension furtherwill increase the accuracy only slightly. Further on, we willobserve that a systematic growing of the bond dimensionis paramount for capturing ground states with strongercorrelations.

In Fig. 2 we have compared our variational search withimaginary-time evolution (full update), showing that wefind lower energies and better order parameters, even asthe Trotter error goes to zero. The plot clearly shows that,as the Trotter step size goes to zero, the imaginary-timeresult does not converge to the variational optimum thatwe obtain. Note that the variational freedom is slightlydifferent: we optimize over a rotationally symmetric PEPS

6

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). Wenicely capture the phase transition, although the critical pointhas been slightly shifted. The critical point can be estimatedas the point where the slope of the curve is maximal; we arriveat λc ≈ 3.09 (D = 2) and λc ≈ 3.054 (D = 3).

with a one-site unit cell, whereas the imaginary-time re-sults break rotational symmetry and work with a two-siteunit cell. Although this larger rotationally asymmetricunit-cell might give lower energies, it appears that our op-timization still gives better energies and order parameters.

In Fig. 3 we provide some details on the convergence ofthe conjugate-gradient algorithm. In particular, we havefound that rather high values of χ (the bond dimension

11

Trotter step size =

0 0.05

e

!3:230

!3:229

Trotter step size =

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-update algorithm; the comparison is done for the transverseIsing model at λ = 3.04 for bond dimensions D = 2 and D = 3.On the left we have plotted the convergence for the energy(magnetization) as a function of the Trotter step size of thefull-update scheme (blue points), and our results (red line).For both plots, the upper (lower) lines are for D = 2 (D = 3).

of the corner environment) were needed to evaluate thegradient accurately close to convergence. Indeed, in thecase of a strongly correlated PEPS, a lot of differentterms contribute to the expression for the gradient. Closeto convergence the gradient becomes a vector of smallmagnitude, which can only happen due to the subtlecancellations of a lot of different terms; consequently,finding the gradient accurately is bound to require a largevalue of χ. Note that the large values of χ are onlynecessary close to convergence, so we grow χ throughoutthe optimization. We never impose the final value of χ,because it is the correlations in the optimized PEPS thatdetermine the χ needed to reach a certain tolerance onthe norm of the gradient.

As a second application, we study two spin-1/2 Heisen-berg models on the square lattice, defined by the Hamil-tonian

HHeisenberg =∑〈ij〉

Sxi Sxj + Syi S

yj + JzS

zi S

zj .

The model has been of great theoretical and experimentalinterest, because of its paradigmatic long-range antiferro-magnetic order58. In particular, Heisenberg models haveproven to be a hard case for the PEPS ansatz59 because ofthe large quantum fluctuations around the antiferromag-netic ordering; as such, they provide a proper benchmarkfor our conjugate-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 orderin the ground state. Moreover, we impose rotationalsymmetry on the PEPS tensor A, so that our variationalground state is automatically invariant under rotationsof the lattice. In Figs. 4 and 5 we have plotted theenergy expectation value and staggered magnetization

Iteration

0 100 200 300 40010!10

10!5

100

@

0 50 100 150 200

kgk 2

10!5

10!4

10!3

10!2

10!1

FIG. 3. Details on the convergence of the optimization algo-rithm for the Ising model at λ = 3. (Upper) The convergence

of the norm of the gradient ‖g‖ =√g†g (blue), the error in

the energy (red) and the error in the magnetization (yellow),as a function of the iteration. The errors are computed as therelative error with respect to the last iteration. In this D = 2simulation the convergence criterion was ‖g‖ ≤ 10−5, a valuefor which the two plotted observables have clearly converged.(Lower) The convergence of the norm of the gradient as afunction of the bond dimension χ of the corner environment,at a particular iteration of the conjugate-gradient scheme forD = 3 (close to convergence). This plot shows that largevalues of χ are needed to obtain a required tolerance on thenorm of the gradient (in this case χ ≈ 100).

after convergence as a function of the bond dimension,for the XY model (Jz = 0) and the isotropic Heisenbergmodel (Jz = 1). Comparing with results from imaginary-time evolution20,59, we see that our variational methodreaches considerably lower energies and order parametersat the same bond dimension.

In addition we also compute the variance of these PEPSvariational states, in order to get an idea of how wellthey approximate the true ground state. The result forthe isotropic Heisenberg model is plotted in Fig. 6. We

12

bond dimension D

2 3 4 5

"e

10!5

10!4

10!3

10!2

bond dimension D

2 3 4 5m

s

0:43

0:44

0:45

0:46

FIG. 4. Results for the XY model (Jz = 0), compared tothe Monte Carlo results in Ref. 60. (Left) The relative error∆e = |(evar − eMC)/eMC| as a function of the bond dimension.(Right) The staggered magnetization as a function of the bonddimension; the red line is the Monte Carlo result with errorbars.

bond dimension D

2 3 4 5

"e

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 ofthe bond dimension. (Right) The staggered magnetization asa function of the bond dimension; the red line is the MonteCarlo result for which the error bars are too small to plot.

observe the expected linear behavior56 to some extent,and a zero-variance extrapolation based on the two lastpoints (D = 4, 5) improves the estimate of the energyby a factor of two. Better zero-variance extrapolationsshould be possible at higher bond dimensions for whichthe linear behavior is expected to be stronger.

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∈L

ei~q·(~ni−~nj) 〈~Si · ~Sj〉c ,

where only the connected part of the correlation functionis taken into account. The disconnected part will givea δ-peak at ~q = (π, π) (the X point), corresponding tothe staggered-magnetization order parameter. The strong

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 thevariance per site for the isotropic (Jz = 1) Heisenberg antifer-romagnet, for four values of the bond dimension (D = 2→ 5).The red line represents the Monte Carlo result for the ground-state energy61. A linear extrapolation with respect to the twobest points (D = 4, 5) gives an energy with a relative error of∆e ≈ 4.7× 10−5. The striped line is drawn between the exactMC result and the D = 5 point and serves only as a guide tothe eye.

M X S ! M S

s(~ k)

10!2

10!1

100

101

FIG. 7. Structure factor s(~q) of the isotropic (Jz = 1) Heisen-berg antiferromagnet along a path through the Brillouin zonefor optimized PEPS states with bond dimensions D = 2 (blue),D = 3 (red), D = 4 (orange), and D = 5 (purple), in agree-ment with the results in Refs. 63 and 64. The divergencearound the X point and the zero around the Γ point are betterreproduced as D increases, although the improvement as afunction of D seems not to be smooth.

fluctuations around this point will give an additional 1/qdivergence, with q the distance from the X point63. Thestructure factor becomes zero at ~q = (0, 0), because theground state is in a singlet state. In Fig. 7 we observethat the regular parts of the structure factor are perfectlyreproduced, even at low bond dimensions, whereas thedivergences can only be accurately captured by observingthe behavior as a function of the bond dimension.

13

V. CONCLUSIONS

In conclusion, we have presented an algorithm for nu-merically optimizing the PEPS ansatz for ground-stateapproximations. The algorithm is based on the efficientevaluation of the energy gradient, and is a direct im-plementation of the variational principle with a clearconvergence criterion. Starting from a random PEPStensor, it allows us to find a variational minimum for agiven bond dimension.

As such, our approach is complementary to any otherPEPS algorithm. In fact, our variational search system-atically finds lower energies than algorithms based onimaginary-time evolution and local truncations. This ob-servation is consistent with the recent results in Ref. 65,where an alternative variational algorithm was proposed.This confirms our belief that a variational approach will becrucial in the future for capturing, e.g., phase transitionsin two-dimensional lattice systems.

Our approach has the additional advantage that globalsymmetries can be exploited easily, which should leadto more efficient simulations66. Moreover, the implemen-tation of symmetries will prove crucial for simulatingsystems with topological order, which can be imposed asa matrix product operator symmetry on the virtual levelof the PEPS tensor14. Finally, our approach straightfor-wardly allows us to consider reduced PEPS parametriza-tions by confining our optimization scheme to a certainPEPS subclass67,68.

In addition, some of the methods that we have presentedin this paper could be applicable to the variational opti-mization of PEPS on finite lattices7,34,69–73 as well. Withfinite PEPS simulations, the straightforward approachof optimizing the different PEPS tensors sequentially isseverely hampered by the bad conditioning of the nor-malization matrix. In particular, because the energy andnormalization matrix require different effective environ-ments, the regularization of this bad condition number isnot well defined. With a finite-lattice version of the corner

environments, however, we could use the same effectiveenvironment for computing the energy and normalization,allowing a consistent regularization of both the energyand normalization matrix. This should lead to efficientvariational optimization 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 ofthe variational error of the PEPS ansatz. Although thevariance extrapolations seem to be not straightforwardlyimplementable, this should contribute to better energybond dimension extrapolations in the future. For systemswith a number of competing ground states such as theHubbard model30, this extrapolation will be of crucialimportance.

From the perspective of numerical optimization, aconjugate-gradient search is only a first step to moreadvanced schemes such as Newton or quasi-Newton meth-ods. The Hessian of the energy functional is crucial inthese optimization schemes, the evaluation of which isstraightforward with our effective environment. Alterna-tively, the non trivial geometric structure of the PEPSmanifold can be taken into account in the optimization74.Also, imposing a certain gauge fixing on the PEPS tensormight render the optimization more efficient.

Finally, it seems that tangent-space methods that haveproven successful in the context of matrix product states39

are now within reach for PEPS simulations for generictwo-dimensional quantum spin models. In particular,this paper opens up the prospect of simulating real-timeevolution according to the time-dependent variationalprinciple38 and/or computing the low-energy spectrumon top of a generic PEPS with the quasiparticle excitationansatz75,76.

This research was supported by the Research Founda-tion Flanders (L.V. & J.H.), by the Delta-ITP [an NWOprogram funded by the Dutch OCW] (P.C.), and by theAustrian FWF SFB through Grants FoQuS and ViCoMand the European Grants SIQS and QUTE (F.V.).

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