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arXiv:1903.09439 · Revista Matemática Complutense

Mathematical open problems in projected entangled pair states

J. I. Cirac, José Garre-Rubio, David Pérez-Garcı́a

Published Jul 29, 2019• 17 citations
arXivHTMLPDF
DOI
10.1007/s13163-019-00318-x

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Projected entangled pair states (PEPS) provide a powerful variational class for strongly correlated ground states in two dimensions, but how to apply them to clusters with periodic boundary conditions was an open question. To solve this problem, a novel algorithm is introduced to contract PEPS on a torus and variational ground state optimization is performed using gradient methods in combination with state-of-the-art automatic differentiation. The authors observe finite-size scaling of critical (2+1)d quantum models and use this to successfully extract scaling dimensions.
Matrix product states and projected entangled pair states (PEPS) are powerful analytical and numerical tools to assess quantum many-body systems in one and higher dimensions, respectively. While matrix product states are comprehensively understood, in PEPS fundamental questions, relevant analytically as well as numerically, remain open, such as how to encode symmetries in full generality, or how to stabilize numerical methods using canonical forms. Here, we show that these key problems, as well as a number of related questions, are algorithmically undecidable, that is, they cannot be fully resolved in a systematic way. Our work thereby exposes fundamental limitations to a full and unbiased understanding of quantum many-body systems using PEPS.