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Emergence of conformal symmetry in critical quantum spin chains: Applying the Koo-Saleur formula to nonintegrable models

Ashley Milsted, Guifré Vidal

Published Jun 5, 2017• 0 citations
OpenAlex

Abstract

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 $H = \sum_j h_j$ as an input, we construct operators $H_n$ (Fourier modes of the local Hamiltonian terms $h_j$) that transform the low-energy eigenstates of $H$ 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 $H_n$ 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 $H$ corresponding to Virasoro primary operators, as must be done in order to identify the CFT.

Related papers

Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula

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We study the emergence of two-dimensional 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. As a means to this end, we propose and demonstrate automated procedures which, using only the lattice Hamiltonian $H={\ensuremath{\sum}}_{j}{h}_{j}$ as an input, systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators, and assign the remaining low-energy eigenstates to conformal towers. The energies and momenta of the primary operator states are needed to determine the primary operator scaling dimensions and conformal spins, an essential part of the conformal data that specifies the CFT. Our techniques use the action, on the low-energy eigenstates of $H$, of the Fourier modes ${H}_{n}$ of the Hamiltonian density ${h}_{j}$. The ${H}_{n}$ were introduced as lattice representations of the Virasoro generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper, we demonstrate that these operators can be used to extract conformal data in a nonintegrable quantum spin chain.

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Emergence of conformal symmetry in quantum spin chains: Antiperiodic boundary conditions and supersymmetry

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Tensor networks, quantum spin chains, and quantum field theory

Yijian Zou

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Emergence of Kac-Moody symmetry in critical quantum spin chains

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Given a critical quantum spin chain with a microscopic Lie group symmetry, corresponding, e.g., to $U(1)$ or $SU(2)$ 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.

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Emergent universality in critical quantum spin chains: entanglement Virasoro algebra

Hu, Qi, Adrián Franco-Rubio, Guifré Vidal

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.

arXiv:2009.11383•Sep 23, 2020•5 citations
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.
We study the emergence of two-dimensional 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. As a means to this end, we propose and demonstrate automated procedures which, using only the lattice Hamiltonian $H={\ensuremath{\sum}}_{j}{h}_{j}$ as an input, systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators, and assign the remaining low-energy eigenstates to conformal towers. The energies and momenta of the primary operator states are needed to determine the primary operator scaling dimensions and conformal spins, an essential part of the conformal data that specifies the CFT. Our techniques use the action, on the low-energy eigenstates of HHH, of the Fourier modes Hn{H}_{n}Hn​ of the Hamiltonian density hj{h}_{j}hj​. The Hn{H}_{n}Hn​ were introduced as lattice representations of the Virasoro generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper, we demonstrate that these operators can be used to extract conformal data in a nonintegrable quantum spin chain.
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.
At a quantum critical point, the low-energy physics of a quantum spin chain is described by conformal field theory (CFT). Given the Hamiltonian of a critical spin chain, we propose a variational method to build an approximate lattice representation ϕ_{α} of the corresponding primary CFT operators ϕ_{α}^{CFT}. We then show how to numerically compute the operator product expansion coefficients C_{αβγ}^{CFT} governing the fusion of two primary fields. In this way, we complete the implementation of Cardy's program, outlined in the 1980s, which aims to extract the universality class of a phase transition, as encoded in the so-called conformal data of the underlying CFT, starting from a microscopic description. Our approach, demonstrated here for the critical quantum Ising model, only requires a generic (i.e., in general, nonintegrable) critical lattice Hamiltonian as its input.
Understanding the universality class of continuous phase transition is of central importance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the conformal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. \n \nOur method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are extracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a systematic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condition. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. \n \nWe test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the underlying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains.
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.
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.