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Correlation length in random MPS and PEPS

Cécilia Lancien, David Pérez-García

TL;DR

This work analyzes random translation-invariant MPS and PEPS built from Gaussian 1-site tensors to determine which features are generic. It develops two complementary routes to explain typical exponential decay of correlations: (i) injective MPS/PEPS via the gap of the parent Hamiltonian, and (ii) the transfer operator gap, which in 2D (PEPS) depends on system size. The authors provide explicit, dimension-aware bounds showing that, in 1D, the parent Hamiltonian gap is typically large for sufficiently large physical dimension relative to the bond dimension, while the transfer operator gap is typically large for both MPS and PEPS under precise scaling of $d$ and $D$ with system size. In 2D, they demonstrate that random PEPS typically exhibit exponential decay of correlations in a polynomial-scaling regime, and they obtain quantitative correlation-length bounds via refined Lieb–Robinson arguments (scaling as $1/\log D$ in favorable regimes). Overall, the results establish that random tensor-network states are generically gapped and exhibit strong exponential decorrelation, with implications for random quantum expanders and holographic contexts, underpinned by Gaussian concentration and Wishart-identity techniques.

Abstract

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS.

Correlation length in random MPS and PEPS

TL;DR

This work analyzes random translation-invariant MPS and PEPS built from Gaussian 1-site tensors to determine which features are generic. It develops two complementary routes to explain typical exponential decay of correlations: (i) injective MPS/PEPS via the gap of the parent Hamiltonian, and (ii) the transfer operator gap, which in 2D (PEPS) depends on system size. The authors provide explicit, dimension-aware bounds showing that, in 1D, the parent Hamiltonian gap is typically large for sufficiently large physical dimension relative to the bond dimension, while the transfer operator gap is typically large for both MPS and PEPS under precise scaling of and with system size. In 2D, they demonstrate that random PEPS typically exhibit exponential decay of correlations in a polynomial-scaling regime, and they obtain quantitative correlation-length bounds via refined Lieb–Robinson arguments (scaling as in favorable regimes). Overall, the results establish that random tensor-network states are generically gapped and exhibit strong exponential decorrelation, with implications for random quantum expanders and holographic contexts, underpinned by Gaussian concentration and Wishart-identity techniques.

Abstract

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS.

Paper Structure

This paper contains 35 sections, 69 theorems, 386 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivations
  3. The model
  4. Summary of our main results and connections with previous works
  5. A few key results in Gaussian concentration
  6. Typical spectral gap of the parent Hamiltonian of random MPS and PEPS
  7. The case of MPS
  8. Approximating the local ground space projectors
  9. Conclusions
  10. The case of PEPS
  11. Approximating the local ground space projectors
  12. Conclusions
  13. Consequence: Typical correlation length in random MPS and PEPS
  14. Rough upper bound on the typical correlation length via the detectability lemma
  15. Tighter upper bound on the typical correlation length via a refined Lieb--Robinson bound
  16. ...and 20 more sections

Key Result

Theorem 1.3

Let $d,D,N\in\mathbf{N}$. Denote by $H_{MPS}$, resp. $H_{PEPS}$, the parent Hamiltonian of our random MPS, resp. PEPS. If $d\geqslant D^{10+\epsilon}$ for some $\epsilon>0$, then and if $d\geqslant D^{26+\epsilon}$ for some $\epsilon>0$, then where $C,c>0$ are universal constants.

Figures (7)

  • Figure 1: Tensor network state construction
  • Figure 2: Graphical representation of a Gaussian vector in $\mathbf{C}^d\otimes(\mathbf{C}^D)^{\otimes 2p}$, for $p=1,2$
  • Figure 3: Composition and decoration of Gaussian diagrams
  • Figure 4: MPS: $1$-site tensor and transfer operator
  • Figure 5: Random translation-invariant MPS with periodic boundary conditions
  • ...and 2 more figures

Theorems & Definitions (124)

  • Definition 1.1: Injectivity and normality
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Gaussian concentration inequality, global version BorST
  • Theorem 1.6: Gaussian concentration inequality, local version ASW
  • Theorem 1.7: Strong convergence of Wishart matrices AS
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • ...and 114 more