Reconciling Translational Invariance and Hierarchy

TL;DR

The paper addresses the challenge of describing gapless, translationally invariant quantum GSs within tensor-network frameworks by introducing a translationally invariant, pentagon-based hierarchical TN for Motzkin/ Fredkin states and coupling it with a uniform MPS transfer-matrix approach. This combination yields exact, analytic access to critical properties: at the critical point q = 1, the TM analysis gives a power-law decay with η = 3/2 and, under q-deformation, a finite correlation length with ν = 2/3, via RG arguments and TM spectra. The work unifies translational symmetry with hierarchical TNs, clarifies how MERA-like structures can be compatible with translational invariance, and provides a framework for extracting critical exponents from TNs using transfer matrices and RG. These results deepen the connection between TN representations and exact solvable GSs, with potential implications for improving MERA-based simulations of translationally invariant gapless systems.

Abstract

Tensor networks are not only numerical tools for describing ground states of quantum many-body systems, but also conceptual aids for understanding their entanglement structures. The proper way to understand tensor networks themselves is through explicit examples of solvable ground states that they represent exactly. In fact, this has historically been how tensor networks for gapped ground states, such as the matrix product state (MPS) and the projected entangled paired state (PEPS), emerged as an elegant analytical framework from numerical techniques like the density matrix renormalization group. However, for gapless ground states, generically described by the multiscale entanglement renormalization ansatz (MERA), a corresponding exactly solvable model has so far been missing. This is because the hierarchical structure of MERA intrinsically breaks the translational invariance. We identify a condition for MERA to be compatible with translational invariance by examining equivalent networks of rank-3 tensors. The condition is satisfied by the previously constructed hierarchical tensor network for the Motzkin and Fredkin chains, which can be considered a non-unitary generalization to the MERA. The hierarchical TN description is complemented by a translationally invariant MPS alternative, which is used to derive the power-law decay of the correlation function and critical exponents of a $q$-deformation phase transition.

Figures (12)

• Figure 1: (a) Pentagonal TN for the GS of a Motzkin chain of length 16. (b) The 32 non-vanishing entries of the rank-5 tensor in the bulk. The tensors on the boundary of the network has rank two or four, depending on its location, and is only non-vanishing for the corresponding compatible configurations.
• Figure 2: (a) Decomposition of a rank-5 tensor into the product of three rank-3 tensors, and recombination of two of the rank-3 tensors into a rank-4 tensor. (b) Equivalent representation of the pentagonal TN in Fig. \ref{['fig:penTN']} (a) using rank-3 and 4 tensors. (c)-(e) List of tensor configurations for the four types of tensors with value 1. Tensor entries for the rest of the configurations all have value 0.
• Figure 3: Pentagonal identity for the $F$ symbols of transformations among different decompositions of the same 5-leg tensor.
• Figure 4: Local $F$ moves that shifts a slice of a hierarchical TN structure by one lattice spacing.
• Figure 5: (a) Diagrammatic depiction of the Fredkin TM (left) and its nonzero elements (right), where $0\le h, h' \le L-1$. (b) The block diagonal Fredkin TM for $L = 2$, and the corresponding disjoint path graphs. The two numbers marking the rows (resp. columns) denote the values of the $h_{j}h'_{j}$ (resp. $h_{j+1}h'_{j+1}$) indices and the nodes in the graph are labeled by the values of the indices of the TM.