[Paper Review] Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks

Henrik Dreyer, Laurens Vanderstraeten+3 more|arXiv (Cornell University)|Aug 11, 2020

Quantum many-body systems44 references4 citations

TL;DR

This paper investigates the robustness of U(1) spin liquids and emergent symmetries in tensor networks using field theory and numerical simulations. It shows that doping critical U(1) spin liquids with long-range singlets is a relevant perturbation that gaps the system, but the gap opens extremely slowly, explaining prior numerical observations of long correlation lengths. Crucially, it presents the first numerical evidence for emergent U(1) symmetry in a PEPS wavefunction under U(1)-breaking perturbations, when the Luttinger parameter K < 1/2.

ABSTRACT

We study the response of critical Resonating Valence Bond (RVB) spin liquids to doping with longer-range singlets, and more generally of U(1)-symmetric tensor networks to non-symmetric perturbations. Using a field theory description, we find that in the RVB, doping constitutes a relevant perturbation which immediately opens up a gap, contrary to previous observations. Our analysis predicts a very large correlation length even at significant doping, which we verify using high-accuracy numerical simulations. This emphasizes the need for careful analysis, but also justifies the use of such states as a variational ansatz for critical systems. Finally, we give an example of a PEPS where non-symmetric perturbations do not open up a gap and the U(1) symmetry re-emerges.

Motivation & Objective

• To determine whether U(1) entanglement symmetry is essential for criticality in PEPS or if it can emerge under perturbations.
• To resolve the apparent contradiction between field theory predictions (gap opening) and numerical observations (long correlation lengths) in doped RVB states.
• To identify conditions under which U(1) symmetry can re-emerge in PEPS after being broken by perturbations.
• To provide a field-theoretic framework for analyzing the stability of critical phases in tensor network states.
• To present and validate a PEPS model where U(1) symmetry emerges dynamically despite explicit breaking of the symmetry in the Hamiltonian.

Proposed method

• The study employs an effective field theory description of the transfer matrix as a Luttinger liquid with Luttinger parameter K.
• Perturbations breaking U(1) symmetry are analyzed as relevant operators in the field theory, with scaling dimensions determined by 1/K.
• High-accuracy numerical simulations using the corner transfer matrix (CTM) method are performed to extract correlation lengths up to 10^4 sites.
• The sine-Gordon model is used to quantitatively predict the scaling of the correlation length with doping strength.
• A 6-vertex model with magnetic field h is used to construct a PEPS with K < 1/2, enabling the emergence of U(1) symmetry.
• Correlation functions of operators related by U(1) symmetry are measured numerically to detect emergent symmetry through identical power-law decay.

Figure 1: (a) RVB and dimer model. Numbers in the plaquettes are the height potential $h(\vec{x})$ obtained from the $\mathrm{U}(1)$ Gauss law (red). (b) PEPS tensor for the RVB state. The green line is the identity on the $\{\left|0\right\rangle,\left|1\right\rangle\}$ space, and all rotations are

Experimental results

Research questions

• RQ1 Is the U(1) symmetry in critical PEPS wavefunctions essential for maintaining criticality, or can it emerge dynamically under perturbations?
• RQ2 Why do numerical simulations of doped RVB states show long correlation lengths despite field theory predicting an immediate gap?
• RQ3 Under what conditions can U(1) symmetry re-emerge in a PEPS after being explicitly broken by perturbations?
• RQ4 Can the Luttinger parameter K < 1/2 lead to the emergence of U(1) symmetry in the infrared limit of a PEPS?
• RQ5 What numerical signatures can be used to detect emergent U(1) symmetry in tensor network states?

Key findings

• Doping the RVB state with long-range singlets is a relevant perturbation in the field theory, which should immediately open a gap, contrary to earlier numerical observations.
• The gap opens extremely slowly due to the small scaling dimension of the perturbation, explaining the long correlation lengths (up to 10^4 sites) observed in simulations.
• High-precision CTM simulations confirm correlation lengths on the order of 10^4, matching predictions from the sine-Gordon model.
• For K < 1/2, U(1)-breaking perturbations do not gap the system, and the correlation functions of operators related by U(1) symmetry decay with the same power law.
• Numerical evidence shows identical decay scaling for U(1)-related correlation functions at h = 0.48 (K = 0.435), indicating emergent U(1) symmetry in the infrared.
• This work presents the first observation of emergent U(1) symmetry in a PEPS, achieved in a 6-vertex model PEPS with K < 1/2 under U(1)-breaking perturbations.

Figure 2: Luttinger parameter $K$ obtained via different methods: $\mathbb{E}_{Q}$ and vison correlators ( $K_{\mathbb{E}_{Q}}$ , $K_{\mathrm{vison}}$ ), and finite size extrapolation ( $L=6,8$ ) of the transfer matrix spectrum ( $K_{\mathrm{TM}}$ ). The good agreement up to $\lambda\approx 0.5$ is