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arXiv:2009.11383

Emergent universality in critical quantum spin chains: entanglement Virasoro algebra

Hu, Qi, Adrián Franco-Rubio, Guifré Vidal

Published Sep 23, 2020• 5 citations
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Abstract

Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions $A$ and $B$, they can be obtained in terms of the $Schmidt~ values$, or eigenvalues $λ_α$ of the reduced density matrix $ρ_A$ for region $A$. In this paper we draw attention instead to the $Schmidt~ vectors$, or eigenvectors $|v_α\rangle$ of $ρ_A$. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors $|v_α\rangle$ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums $H_n$ of the lattice Hamiltonian density $h_{j,j+1}$ over region $A$ and show that the matrix elements $\langle v_αH_n |v_{α'}\rangle$ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for $H_n^{\tiny \text{CFT}} = \frac 1 2 (L_n + L_{-n})$ in the boundary CFT, where $L_n$'s are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.

DOI
10.48550/arxiv.2009.11383

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Entanglement entropy and entanglement spectrum have been widely used to characterize quantum entanglement in extended many-body systems. Given a pure state of the system and a division into regions AAA and BBB, they can be obtained in terms of the Schmidt valuesSchmidt~ valuesSchmidt values, or eigenvalues λαλ_αλα​ of the reduced density matrix ρAρ_AρA​ for region AAA. In this paper we draw attention instead to the Schmidt vectorsSchmidt~ vectorsSchmidt vectors, or eigenvectors ∣vα⟩|v_α\rangle∣vα​⟩ of ρAρ_AρA​. We consider the ground state of critical quantum spin chains whose low energy/long distance physics is described by an emergent conformal field theory (CFT). We show that the Schmidt vectors ∣vα⟩|v_α\rangle∣vα​⟩ display an emergent universal structure, corresponding to a realization of the Virasoro algebra of a boundary CFT (a chiral version of the original CFT). Indeed, we build weighted sums HnH_nHn​ of the lattice Hamiltonian density hj,j+1h_{j,j+1}hj,j+1​ over region AAA and show that the matrix elements ⟨vαHn∣vα′⟩\langle v_αH_n |v_{α'}\rangle⟨vα​Hn​∣vα′​⟩ are universal, up to finite-size corrections. More concretely, these matrix elements are given by an analogous expression for HnCFT=12(Ln+L−n)H_n^{\tiny \text{CFT}} = \frac 1 2 (L_n + L_{-n})HnCFT​=21​(Ln​+L−n​) in the boundary CFT, where LnL_nLn​'s are (one copy of) the Virasoro generators. We numerically confirm our results using the critical Ising quantum spin chain and other (free-fermion equivalent) models.
We study the emergence of 2D conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. Using only the lattice Hamiltonian H=∑jhjH = \sum_j h_jH=∑j​hj​ as an input, we construct operators HnH_nHn​ (Fourier modes of the local Hamiltonian terms hjh_jhj​) that transform the low-energy eigenstates of HHH in the same way as certain combinations of the Virasoro generators do in the CFT. These operators were introduced for certain integrable lattice models by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we propose and demonstrate the use of operators HnH_nHn​ to extract conformal data in nonintegrable quantum spin chains. In particular, we explain how to estimate the central charge from a simple ground-state expectation value and to systematically identify the low-energy states of HHH corresponding to Virasoro primary operators, as must be done in order to identify the CFT.
Universal properties of a critical quantum spin chain are encoded in the underlying conformal field theory (CFT). This underlying CFT is fully characterized by its conformal data. We propose a method to extract the conformal data from a critical quantum spin chain with both periodic and antiperiodic boundary conditions (PBC and APBC) based on low-energy eigenstates, generalizing previous work on spin chains with only PBC. First, scaling dimensions and conformal spins are extracted from the energies and momenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasoro generators is generalized to APBC and used to identify conformal towers. Third, local and string operators on the lattice are identified with CFT operators with PBC and APBC, respectively. Finally, operator product expansion coefficients are extracted by computing matrix elements of lattice primary operators in the low-energy subspaces with PBC and APBC. To go beyond exact diagonalization, tensor network methods based on periodic uniform matrix product states are used. We illustrate our approach with critical and tricritical Ising quantum spin chains. In the latter case, we propose lattice operators that correspond to super-Virasoro generators and verify their action on low-energy eigenstates. In this way, we explore the emergence of superconformal symmetry in the quantum spin chain.
In this paper we study the ground-state properties of a ladder Hamiltonian with chiral SU(2)\text{SU}(2)SU(2)-invariant spin interactions, a possible first step toward the construction of truly two-dimensional nontrivial systems with chiral properties starting from quasi-one-dimensional ones. Our analysis uses a recent implementation by us of SU(2)\text{SU}(2)SU(2) symmetry in tensor network algorithms, specifically for infinite density matrix renormalization group. After a preliminary analysis with Kadanoff coarse graining and exact diagonalization for a small-size system, we discuss its bosonization and recap the continuum limit of the model to show that it corresponds to a conformal field theory, in agreement with our numerical findings. In particular, the scaling of the entanglement entropy as well as finite-entanglement scaling data show that the ground-state properties match those of the universality class of a c=1c=1c=1 conformal field theory (CFT) in (1+1)(1+1)(1+1) dimensions. We also study the algebraic decay of spin-spin and dimer-dimer correlation functions, as well as the algebraic convergence of the ground-state energy with the bond dimension, and the entanglement spectrum of half an infinite chain. Our results for the entanglement spectrum are remarkably similar to those of the spin-12\frac{1}{2}21​ Heisenberg chain, which we take as a strong indication that both systems are described by the same CFT at low energies, i.e., an SU(2)1\text{SU}{(2)}_{1}SU(2)1​ Wess-Zumino-Witten theory. Moreover, we explain in detail how to construct matrix product operators for SU(2)\text{SU}(2)SU(2)-invariant three-spin interactions, something that had not been addressed with sufficient depth in the literature.
Given a critical quantum spin chain with a microscopic Lie group symmetry, corresponding, e.g., to U(1)U(1)U(1) or SU(2)SU(2)SU(2) spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated with conformal invariance is augmented with a Kac-Moody algebra associated with conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically show that, when projected onto low-energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac-Moody algebra. The lattice version of the Kac-Moody generators allows us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac-Moody towers. We illustrate the proposal with the XXZ model and the Heisenberg model with a next-to-nearest-neighbor coupling.
Quantum critical chains are well-described and understood by virtue of conformal field theory. Still, the meaning of the real space entanglement spectrum—the eigenvalues of the reduced density matrix—of such systems remains elusive in general, even when there is an additional quantum number available such as the spin or particle number. In this paper, we explore in detail the properties and structure of the reduced density matrix of critical XXZ spin- chains. We investigate the quantum/thermal correspondence between the reduced density matrix of a T = 0 pure quantum state and the thermal density matrix of an effective entanglement Hamiltonian. Using large scale DMRG and QMC simulations, we investigate the conformal structure of the spectra, the entanglement Hamiltonian, and temperature. We then introduce the notion of spin-resolved entanglement entropies, which display interesting scaling features.
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