# P17: Exact diagonalization for projected entangled pair states

## Project Description

Projected Entangled Pair States (PEPS) form a successful variational ansatz for strongly correlated quantum systems in two and higher dimensions ^{(1)}. At the same time, they can also be used to construct trial wave functions which appear as exact ground states of local Hamiltonians ^{(2)}, such as the AKLT model ^{(3)} or Resonating Valence Bond (RVB) states ^{(4),(5)}. In this way, solvable models can be constructed for which a variety of properties can be extracted from the leading eigenvalues and eigenvectors of the PEPS transfer operator, including the scaling of arbitrary correlation functions, the presence of symmetry breaking or of topological order, or the braiding statistics of excitations ^{(4),(6)-(8)}. Up to a certain system size, the properties of the transfer operator can be exactly computed numerically, which in combination with a finite size scaling analysis allows to make precise predictions about the behavior of the system in the thermodynamic limit. Yet, both memory and computation time required grow exponentially with the linear size of the system, with memory requirements typically forming the more severe bottleneck; moreover, the basis of the exponential strongly depends on the complexity of the PEPS wavefunction, given by the so-called* bond dimension. *

The aim of this project is to develop optimized methods for the exact diagonalization of PEPS transfer operators with the help of supercomputers ^{(9),(10)}. We will in particular focus on the possibilities for optimization which arise from the specific structure of the transfer operator. PEPS transfer operators are hermitian matrix product operators which exhibit a rich structure, in particular with respect to symmetries ^{(2)}. For instance, physical symmetries of the system, such as e.g. SU(2) symmetry, induce corresponding symmetries for the transfer operator and subsequently for its eigenvectors ^{(11),(12)} which allow for significant optimizations; similarly, the presence of topological order leads to strong symmetry constraints on the transfer operator^{ (6)}. At the same time, we will investigate how the structure of the transfer operator as a multi-index tensor network can be used to implement these methods on supercomputers with distributed memory architectures as efficiently as possible.

These efforts will be complemented by studying further properties which can be extracted from the structure of the transfer operator. As an example, we will investigate how the fact whether a system exhibits topological order, conventional symmetry breaking order, or both, can be extracted from the spectral properties of the transfer operator and its behavior under perturbations. We will also study whether there are alternative, more efficient ways to extract certain properties; for instance, the braiding of excitations a priori requires the study of the full transfer operator, and it would be a major step to understand how this can be reduced to properties of the leading eigenvalues.

Altogether, the results of this project will allow us to study new and increasingly complex PEPS ansatz wave functions using exact numerical methods and thereby help to improve our understanding of strongly correlated quantum systems.

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(11) N. Schuch, J. I. Cirac and D. Pérez-García, Ann. Phys.

**325**, 2153 (2010)

(12) D. Pérez-García, M. Sanz, C. E. Gonzalez-Guillen, M. M. Wolf and J. I. Cirac, New J. Phys.

**12**, 025010 (2010)