Simulating 2+1d Z 3 lattice gauge theory with iPEPS
SSimulating 2+1d Z lattice gauge theory with iPEPS Daniel Robaina, Mari Carmen Ba˜nuls,
1, 2 and J. Ignacio Cirac
1, 2 Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen (Dated: July 24, 2020)We simulate a zero-temperature pure Z Lattice Gauge Theory in 2+1 dimensions by usingan iPEPS (Infinite Projected Entangled-Pair State) ansatz for the ground state. Our results aretherefore directly valid in the thermodynamic limit. They clearly show two distinct phases separatedby a phase transition. We introduce an update strategy that enables plaquette terms and Gauss-law constraints to be applied as sequences of two-body operators. This allows the use of the mostup-to-date iPEPS algorithms. From the calculation of spatial Wilson loops we are able to prove theexistence of a confined phase. We show that with relatively low computational cost it is possible toreproduce crucial features of gauge theories. We expect that the strategy allows the extension ofiPEPS studies to more general LGTs.
Introduction.–
For years, Tensor Networks (TN) havebeen exploited to study quantum many-body problems,especially in the context of condensed matter physics,since they provide efficient ans¨atze for ground states, lowlying excitations and thermal equilibrium states of lo-cal hamiltonians [1–5]. The application of TN to Lat-tice Gauge Theories (LGT) constitutes a much newer,but also fast growing field. Their suitability for 1+1dimensional problems has already been widely demon-strated using the matrix product state (MPS) ansatz. Innumerous studies, MPS have been shown to efficientlyand accurately describe the relevant equilibrium physicsof abelian and non-abelian LGTs, even at finite densitywhere the infamous sign-problem would turn traditionalMonte Carlo approaches infeasible, TN enable contin-uum limit extrapolations, as well as simulations in out-of-equilibrium scenarios (see [6, 7] for recent reviews).The one-dimensional success strongly motivates an ex-tension of the TN study to LGT in higher spatial dimen-sions, where the natural generalization of the MPS ansatzis provided by projected entangled pair states (PEPS) [8],or its infinite version defined directly in the thermody-namic limit, iPEPS [9]. More restricted TN have allowedsome first encouraging steps for two dimensional models.Early on, the phase diagram of a Z LGT was stud-ied with MERA[10, 11], and, more recently, tree tensornetworks [12] were applied to explore the U (1) quantumlink model on a finite lattice [13]. But a fully variationalPEPS calculation for a LGT does not yet exist.Although the fast progress in iPEPS algorithms has al-lowed reaching some of the most competitive results forcertain condensed matter problems [14–20] and there isno conceptual limitation to apply them to LGTs [21], un-til the date the only numerical results of (i)PEPS simula-tions of LGTs have been limited to toy models without anactual optimization of the most general tensors [22–26].Apart from the obvious increase in computational cost,another more limiting factor is the presence of plaquetteterms in the LGT Hamiltonian. While it is possible todirectly apply a plaquette term to PEPS [27, 28], this in- volves a considerably higher computational cost than thetwo-body interactions for which the most efficient PEPSalgorithms are optimized, and ultimately limits the bonddimension that can be explored to only very small values,not enough to approach convergence.In this work we develop a new update strategy that al-lows the standard plaquette term of a LGT to be appliedas a sequence of purely two-body operations. This allowsus to use an iPEPS ansatz to study the phase diagram ofa Z -invariant LGT in two spatial dimensions. In agree-ment to predictions in the literature [29–31], we observea confining and a non-confining phase. We are able toquantitatively locate the transition at a value of the cou-pling constant g c = 1 . Model.–
We consider a Z invariant Lattice GaugeTheory given by the following Hamiltonian in 2+1 space-time dimensions H = H E + H (cid:3) , (1)where H E = g (cid:88) x E ( x + i /
2) + E ( x + j / H (cid:3) = − g (cid:88) x U P ( x ) + U † P ( x ) . The plaquette operator is written as U P ( x ) = U † ( x + j / U † ( x + i / j ) U ( x + i + j / U ( x + i / x is the position of a vertex and i , j are unit-vectorsin both space directions connecting two adjacent vertices.The physical degrees of freedom are the link variableswhich have a local Hilbert space of dimension d = 3 andconsequently E takes values in {− , , } . The unitary a r X i v : . [ h e p - l a t ] J u l operators U and U † , lower and raise respectively the elec-tric field at the corresponding link by one unit U | e (cid:105) = | e − (cid:105) U † | e (cid:105) = | e + 1 (cid:105) and Z -symmetry implies U = ( U † ) = .In the limit of d → ∞ this Z d Hamiltonian yields a U (1) Lattice Gauge Theory where H E corresponds to theelectric field and the plaquette terms in H (cid:3) reproduce themagnetic parts [32].The Hamiltonian in (1) commutes with the Gauss-lawoperator G ( x ) at every point in space giving rise to alocal Z gauge symmetry where G ( x ) is given by G ( x ) = e πi ( E l ( x )+ E d ( x ) − E r ( x ) − E u ( x )) (2)where the subscripts l, d, r, u correspond to the linkswhich are to the left, down, right, up of the vertex atposition x . Notice that G ( x ) is defined at the vertices ofthe lattice while the links live inbetween vertices. Giventhat [ G ( x ) , H ] = 0, the hamiltonian is block diagonaland physical states that satisfy the Gauss-law obey G ( x ) | ψ (cid:105) = e πi q ( x ) | ψ (cid:105) , (3)where q ( x ) ∈ {− , , } can be thought of as the staticcharge at vertex x . Although the ground state of thesystem lives in the charge sector with q ( x ) = 0, ∀ x , it isalso interesting to study different charge patterns, as wewill do. Method.–
An iPEPS ansatz consists of a unit-cell ofrank-5 tensors arranged in a 2D-grid which is repeated inboth space directions infinitely many times. Those ten-sors have a physical index of dimension d equal to thatof the local Hilbert space of each degree of freedom (3 inour case) and 4 additional virtual indices of bond dimen-sion D that allow for the interactions with neighbouringtensors. As D increases the ansatz becomes more gen-eral and, consequently, a better description of the truequantum state is expected.There are several ways of optimizing the tensors withinthe unit-cell in order to find the ground state. One pos-sibility relies on a variational approach in which onlyone tensor is varied at a time by keeping the rest fixed.The optimal tensor is then found by solving a General-ized Eigenvalue Problem before moving to the next one[2]. While the variational method has been able to ob-tain very accurate energies [15, 16], the most widely usedstrategy for iPEPS, which we also adopt here, is still animaginary time evolution, very much in the spirit of thepopular Time Evolving Block Decimation (TEBD) algo-rithm [33]. In the most efficient version, a simple update(SU) [34] strategy is used to find the optimized tensors.We use a second order Suzuki-Trotter [35, 36] expan-sion of the Hamiltonian exponential e − β ( H E + H (cid:3) ) = lim n →∞ (cid:16) e − δτ H E e − δ τ H (cid:3) e − δτ H E (cid:17) n (4) FIG. 1. iPEPS unit-cell. with δ τ = β/n and β the total imaginary time evolveduntil convergence.Traditional iPEPS algorithms have been optimized forHamiltonians with nearest neighbor interactions. Longerrange or higher-order terms considerably increase thecomputational cost. Therefore, in order to apply thesemethods to our problem, we need a simple and efficientupdate strategy that takes into account 4-body plaquetteoperators like the ones that appear in LGTs.In order to apply the plaquette operator in its expo-nential form we import an idea originally envisioned fordigital quantum simulations of LGTs [37–40]. The keyaspect consists in including an auxiliary degree of free-dom with the same Hilbert space as the links themselvesat the center of each plaquette. This ancilla is prepared ina state which is an equal weight symmetric superpositionof all basis states. Following the notation of [38] we callit | (cid:101) in (cid:105) = √ (cid:80) m = − , , | ˜ m (cid:105) . The derivation presented inthe above mentioned papers allows us to write the ac-tion of the four-body operator e − δ τ H (cid:3) as a sequence oftwo-body gates (we call this the entangler ) followed bya local operation on the ancilla. The inverse of the en-tangler (the disentangler ) leaves the ancilla back in itsoriginal state | (cid:101) in (cid:105) , ready for the next update. The fullidentity reads U † (cid:3) e δτ g ( ˜ U + ˜ U † ) U (cid:3) | (cid:101) in (cid:105) = | (cid:101) in (cid:105) e − δτH (cid:3) (5)where the entangler U (cid:3) = U † l U † u U r U d is the product offour two-body gates between ancilla and the correspond-ing links. Each of these two-body gates is written as U i = U i ⊗ (cid:101) P + i ⊗ (cid:101) P + U † i ⊗ (cid:101) P − (6)where U i with i = l, u, r, d act on the links and (cid:101) P m are ordinary projectors in the ancilla Hilbert space thatproject onto state | ˜ m (cid:105) . The local operation on the ancilla e δτ g ( (cid:101) U + (cid:101) U † ) involves (cid:101) U and (cid:101) U † which are nothing but ordi-nary U (and U † )-operators acting on the ancilla degreesof freedom. Note, that (5) is a mathematical identity andthere is no approximation involved. We refer the inter-ested reader to the original papers for a clean derivationof (5).The electrical evolution corresponds to a sequential ac-tion of e − δτ g E -single-site operators onto the physicalindices of all links. Since we employ the simple updateprocedure (SU) this operation does not increase the bonddimensions and thus carries no truncation errors.In order to implement the update procedure describedabove, we choose a 4 × (cid:96) i , with i = 1 , ..., G ( x ), it is enough to ap-ply the projector at the beginning of the imaginary timeevolution. To cope with potential errors introduced bythe truncation, we subsequently monitor the expectationvalue of G ( x ) to be sure to stay in the sector of inter-est. We observe that the deviation (with respect to thedesired sector) is not larger than 10 − in any of our sim-ulations.Similarly to other Tensor Networks, iPEPS allow forthe calculation of local observables. This requires anaccurate approximation of the environment around agiven tensor. In this work we calculate the environmentwith the Corner Transfer Matrix (CTM)-method [46, 47],which introduces an additional bond dimension, control-ling the precision of such approximation [48].Altogether, this strategy allows us to simulate theimaginary time evolution of a LGT including the four-body plaquette operator by means of well-known tools tothe iPEPS practitioners like single and two-body gates. Phase Diagram.–
When g → ∞ , the electric fieldterm dominates and, in the case of vanishing staticcharges at all the vertices, the lowest energy is attainedwhen all links are in the zero electric flux state. Theground state thus becomes a product state with zero en-ergy. Similarly, in the weak coupling regime when g → − /g where the ground state is again a prod-uct state. It is well known that Z d gauge theories aredual to spin systems with nearest neighbour interactions[29]. For Z in 2 + 1 dimensions the system undergoes a first order phase transition [30, 31] around some criticalcoupling g c .We have performed calculations at D = 3, 4, 5 for thewhole range of couplings from g = 0 .
01 to g = 5 . D = 5was not able to provide a lower estimate than D = 4.We attribute this to a lack of full convergence of the SUon those points. Since for the rest of parameters therelative difference between the results for D = 4 and 5is extremely small (see SM), we take D = 4 as our bestdata-set and use D = 3 and 5 to estimate numericalerrors [49]. Our ground-state energy results are plottedin Fig. 2.First order phase transitions can be cleanly detectedby TN simulations [50] as cusps in the energy curve, cor-responding to a level crossing. This effect is apparent inFig. 2 at intermediate values of the coupling (the dashedlines are meant to guide the eye). A cleaner way of lo-cating the phase transition is by the discontinuity in thefirst derivative of the energy, which can be calculated as ∂E q ( x )=00 ∂g = (cid:104) ψ GS | ∂H∂g | ψ GS (cid:105) = 1 g (cid:104) ψ GS | H E − H (cid:3) | ψ GS (cid:105) (7)and is plotted in Fig. 3. A clear discontinuity between g c = 1 .
15 and g c = 1 .
175 can be identified.We also consider a different charge sector, in which weproject two adjacent vertices to static charges 1 and -1 re-spectively (as illustrated in Fig. 5). Below the phase tran-sition, both sectors are close to degenerate (see Fig. 2),and as soon the transition is crossed, they separate. Theenergy per plaquette of the static charges tends to g / g → ∞ since our unit cell contains 4plaquettes and in that limit there is a single link whose E -expectation value is 1, while the rest vanish. The factthat the energies of both sectors start to strongly deviatefrom each other exactly at the phase transition repre-sents a consistency check that we have correctly locatedthe transition region. We will attempt a more accuratedetermination of g c via Wilson loops in the following sec-tion. Wilson loops.–
The phase transition separates a non-confining (for small g ) from a confining (for large g )phase. We can characterize it by investigating the groundstate expectation value of several closed spatial Wilsonloops, the simplest of them being the plaquette whichenters the calculation of the energy. In the confiningphase, these values are expected to decay exponentiallywith the area of the loop. Due to the large computationalcost of these quantities, we restrict ourselves to loopsof width 1 and length n = 1 , . . .
6. The correspondingoperator can be written in closed form as − − g − . − . . . . g E q ( x )=(1 , − , D = 4 g E q ( x )=(1 , − , D = 5 g E q ( x )=00 , D = 4 g E q ( x )=00 , D = 5 . . . g − . − . − . . FIG. 2. Ground State Energies for the zero charge sector withbond dimensions D = 4, 5. We compare to the sector of twoadjacent vertices respectively projected to charges 1 and -1with bond dimension D = 4 ,
5. Inset: Transition region zoomin. . . . . . g − − D ∂H∂g E FIG. 3. Expectation value of ∂H∂g on the ground state for thezero charge sector. Bond dimension is D = 4. W × n = U † ( x + j / ⊗ (cid:32) n − (cid:79) α =0 U ( x + ( α + 1 / i ) (cid:33) ⊗ U ( x + n i + j / ⊗ n − (cid:79) β =0 U † ( x + ( n − β − / i + j ) . (8)We calculate (cid:104) ψ GS | W × n | ψ GS (cid:105) and show the results infigure 4. We perform a linear fit of the logarithm ofthe real part of (cid:104) W × n (cid:105) (the imaginary part is consistentwith zero) vs. the area n , and read off the slope σ . Thephase transition clearly manifests in a sudden increase of σ when the coupling approaches a critical value g c . Inorder to extract this critical value, we perform several fitsof the data to a form A ( g − g c ) α and estimate the errors g . . . . . . . A ( g − g c ) α σ ( g ) 1 2 3 4 5 6 n − − l og ( R e h W × n i ) . . . . . . . . FIG. 4. Area-law coefficient σ obtained from the fit of theexpectation value of the Wilson loops that is shown in theinset for D = 4 ground states. The colorbar represents thevalue of the coupling g . The blue band represents an errorestimation for the fitted curve. by varying the number of points included in the fit. Wefind A = 2 . , g c = 1 . , α = 0 . . (9) Electric field map.–
In order to illustrate clearly thevery different behavior of the electric field in both phases,in figure 5 we plot (cid:104) ψ GS | E (cid:96) | ψ GS (cid:105) for all 8 links in theunit-cell in different charge sectors. The zero charge sec-tor keeps translational symmetry for all values of the cou-pling and above the phase transition the electric field ispractically zero. For the case of two static charges, wesee that below the phase transition the behavior is verysimilar as in the zero charge sector, while as soon as thetransition is crossed, the electric field is confined to asingle link between two charges. Conclusions.–
We find that iPEPS are capable of ac-curately capturing the Physics of a gauge theory withdifferent phases in 2+1 space-time dimensions. Withmoderate bond dimension, the iPEPS ansatz allows usnot only to determine the ground state energy but alsoto explore the phenomenology of the model, includingthe location of a confinement phase transition.Key to this development is a special update strategythat employs additional ancillary degrees of freedom andreduces many-body terms to sequences of two-body oper-ations. This allows us to deal with plaquette terms in anefficient way, and also to correctly implement Gauss-lawconstraints at the vertices as a way to impose the localsymmetry.The strategy can be immediately applied to otherLGTs, but also to other hamiltonians that require theinclusion of a 4-body operator. Since the original con-struction [40] on which this update is based can be ap-plied to non-Abelian Lie groups and also to operators − − FIG. 5. Ground state expectation values of E -operators act-ing on the links of the unit-cell for g = 0 . , .
0. The upperrow corresponds to the zero static charge sector while thelower has two vertices (yellow circles) projected to 1 and − acting on a larger number of sites [39, 51], we expectthat the method can be further generalized. Dynamicalfermions can additionally be included in the approachwithout involving a sign-problem, and we leave this di-rection for future work. Altogether, this opens the doorto more ambitious iPEPS studies of LGTs.We thank Claudius Hubig for insightful discussions onthe SyTen toolkit used in this work [52, 53]. This workwas partly supported by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation)under Ger-many’s Excellence Strategy – EXC-2111 – 390814868,and EU-QUANTERA project QTFLAG (BMBF grantNo. 13N14780). [1] J. I. Cirac and F. Verstraete, Journal of Physics A: Math-ematical and Theoretical , 504004 (2009).[2] F. Verstraete, V. Murg, and J. Cirac, Adv. Phys. ,143 (2008).[3] U. Schollw¨ock, Ann. Phys. , 96 (2011), january 2011Special Issue.[4] R. Or´us, Annals Phys. , 117 (2014), arXiv:1306.2164[cond-mat.str-el].[5] P. Silvi, F. Tschirsich, M. Gerster, J. J¨unemann,D. Jaschke, M. Rizzi, and S. Montangero, SciPost Phys.Lect. Notes , 8 (2019).[6] M. C. Ba˜nuls and K. Cichy, Rept. Prog. Phys. , 024401(2020), arXiv:1910.00257 [hep-lat]. [7] M. C. Ba˜nuls, R. Blatt, J. Catani, A. Celi, J. I. Cirac,M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein,S. Montangero, C. A. Muschik, B. Reznik, E. Rico,L. Tagliacozzo, K. Van Acoleyen, F. Verstraete, U. J.Wiese, M. Wingate, J. Zakrzewski, and P. Zoller, arXive-prints , arXiv:1911.00003 (2019), arXiv:1911.00003[quant-ph].[8] F. Verstraete and J. Cirac, (2004), arXiv:cond-mat/0407066.[9] J. Jordan, R. Or´us, G. Vidal, F. Verstraete, and J. I.Cirac, Phys. Rev. Lett. , 250602 (2008).[10] G. Vidal, Phys. Rev. Lett. , 220405 (2007).[11] L. Tagliacozzo and G. Vidal, Phys. Rev. B , 115127(2011), arxiv:1007.4145.[12] Y.-Y. Shi, L.-M. Duan, and G. Vidal, Phys. Rev. A ,022320 (2006).[13] T. Felser, P. Silvi, M. Collura, and S. Montangero,“Two-dimensional quantum-link lattice quantum electro-dynamics at finite density,” (2019).[14] P. Corboz, Phys. Rev. B , 045116 (2016).[15] P. Corboz, Phys. Rev. B , 035133 (2016).[16] L. Vanderstraeten, J. Haegeman, P. Corboz, and F. Ver-straete, Phys. Rev. B , 155123 (2016).[17] P. Corboz, P. Czarnik, G. Kapteijns, and L. Tagliacozzo,Phys. Rev. X , 031031 (2018).[18] M. Rader and A. M. L¨auchli, Phys. Rev. X , 031030(2018).[19] L. Vanderstraeten, J. Haegeman, and F. Verstraete,Phys. Rev. B , 165121 (2019).[20] C. Hubig and J. I. Cirac, SciPost Phys. , 31 (2019).[21] K. Zapp and R. Or´us, Phys. Rev. D , 114508 (2017).[22] L. Tagliacozzo, A. Celi, and M. Lewenstein, Phys. Rev.X , 041024 (2014).[23] E. Zohar, M. Burrello, T. B. Wahl, and J. I. Cirac, Ann.Phys. (Amsterdam) , 385 (2015).[24] J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac,and F. Verstraete, Phys. Rev. X , 011024 (2015).[25] E. Zohar, T. B. Wahl, M. Burrello, and J. I. Cirac,Annals Phys. , 84 (2016), arXiv:1607.08115 [quant-ph].[26] E. Zohar and J. I. Cirac, Phys. Rev. D , 034510 (2018).[27] S. Dusuel, M. Kamfor, R. Or´us, K. P. Schmidt, andJ. Vidal, Phys. Rev. Lett. , 107203 (2011).[28] M. D. Schulz, S. Dusuel, R. Or´us, J. Vidal, and K. P.Schmidt, New Journal of Physics , 025005 (2012).[29] C. P. K. Altes, Nuclear Physics B , 315 (1978).[30] H. W. J. Bl¨ote and R. H. Swendsen, Phys. Rev. Lett. ,799 (1979).[31] G. Bhanot and M. Creutz, Phys. Rev. D , 2892 (1980).[32] The limit of U(1) is recovered when d → ∞ if the Hamil-tonian is written in the form of [ ? ] but for d = 3 ourformulation is equivalent except for a trivial rescaling of g and the U operator and a constant overall shift in theHamiltonian.[33] G. Vidal, Phys. Rev. Lett. , 147902 (2003).[34] H. C. Jiang, Z. Y. Weng, and T. Xiang, Phys. Rev. Lett. , 090603 (2008).[35] H. F. Trotter, Proc. Amer. Math. Soc. , 545 (1959).[36] M. Suzuki, Journal of Mathematical Physics , 601(1985).[37] E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Phys.Rev. Lett. , 070501 (2017), arXiv:1607.03656 [quant-ph].[38] E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Phys. Rev.
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Symmetry-Protected Tensor Networks , Ph.D.thesis, LMU M¨unchen (2017).
SUPPLEMENTARY MATERIALGauss-Law Constrains
In order to enforce the Gauss-Law at every vertex, we define the projector P q ( x ) = 13 (cid:88) n = − , , (cid:16) e πi ( E l ( x )+ E d ( x ) − E r ( x ) − E u ( x ) − q ( x ) (cid:17) n (10)which projects vertex x to charge q ( x ). Since E -field operators in the exponent commute with eachother, this projectorhas the same structure as H (cid:3) since it can be written as a product of four single-site operators. Taking q ( x ) = 0 asan example case, it is convenient to consider a slight modification of identity (5) G † (cid:16) ˜ + ˜ U + ˜ U † (cid:17) G| (cid:101) in (cid:105) = | (cid:101) in (cid:105) P (11)where now the entangler between vertex x and the links surrounding it can be again written as a sequence of fourtwo-body gates G = G l G d G † r G † u . Each of the two-body gates is written as G i = g i ⊗ ˜ P + i ⊗ ˜ P + g † i ⊗ ˜ P − (12)with g i = e πi E i ( x ) and i = l, u, r, d . Similarly to the case with the ancillas, (11) is only true if vertex tensors areinitialized in their | (cid:101) in (cid:105) states. In this way, enforcing the Gauss-law at every vertex is as simple as applying a sequenceof single and two-body gates. Only a minor modification to the local operation (cid:16) ˜ + ˜ U + ˜ U † (cid:17) on the vertex allowsus to also obtain P q ( x ) with q ( x ) = ± Errors
The left plot in Fig. 6 shows for different values of the couplings our results for the ground state energy for differentvalues of the bond dimension. It can be seen that at weak coupling the error is negligible. In fact, the differencebetween D = 4 and D = 5 is less than 10 − . This is not surprising, since the true ground state tends to a productstate for g →