Universal tensor-network algorithm for any infinite lattice

Abstract

We present a general graph-based Projected Entangled-Pair State (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural-matrix which codifies the details of tensor networks on any graphs in any dimension d, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., D = 100), and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the 3-state quantum Potts model in field on the kagome lattice and the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the 2d quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies. arXiv:1808.00680v3 [cond-mat.str-el]

Tensor Network Contractions

“…Besides the contractions of TN's, the concept of environment becomes more important for the TNS update algorithms, where the central task is to optimize the tensors for minimizing the cost function. According to the environment, the TNS up-date algorithms are categorized as the simple [141,143,210,221,231,232], cluster [141,231,233,234], and full update [221,228,230,235,236,237,238,239,240]. The simple update uses local environment, hence has the highest efficiency but limited accuracy.…”

“…The simulations of 3D quantum systems are much more consuming than the 2D cases, where the task become to contract the 4D TN. The 4D TN contraction is extremely consuming, one may consider to generalize the simple update [234,232], or to construct finite-size effective Hamiltonians that mimic the infinite 3D quantum models [234,258] Many technical details of the approaches can be flexibly modified according to the problems under consideration. For example, the iPEPO formulation is very useful when computing a 3D statistic model, which is to contract the corresponding 3D TN.…”

Topological Z2 resonating-valence-bond quantum spin liquid on the ruby lattice

“…The extension of MPS to 2D and higher dimensions has also been put forward in the form of projected entangled-pair state (PEPS) [64][65][66]. During past years, PEPS has been successfully used for representing the ground state of numerous quantum many-body systems and has played a key role in a better understanding of strongly correlated systems [5,43,[67][68][69][70][71][72][73].…”

Introduction

“…Besides the contractions of TNs, the concept of environment becomes more important for the TNS update algorithms, where the central task is to optimize the tensors for minimizing the cost function. According to the environment, the TNS update algorithms are categorized as the simple [141,143,210,221,231,232], cluster [141,231,233,234], and full update [221,228,230,[235][236][237][238][239][240]. The simple update uses local environment, hence has the highest efficiency but limited accuracy.…”