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arXiv:2206.01656 · Physical review. B./Physical review. B

Emergence of Kac-Moody symmetry in critical quantum spin chains

Ruoshui Wang, Yijian Zou, Guifré Vidal

Published Sep 8, 2022• 10 citations
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Abstract

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.

DOI
10.1103/physrevb.106.115111

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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.
In conformal field theories, when the conformal symmetry is enhanced by a global Lie group symmetry, the original Virasoro algebra can be extended to Kac-Moody algebra. In this paper, we extend the lattice construction of the Kac-Moody generators introduced in Wang et al., [Phys. Rev. B 106, 115111 (2022)] to continuous systems and apply it to one-dimensional continuous boson systems. We justify this microscopic construction of Kac-Moody generators in two ways. First, through phenomenological bosonization, we express the microscopic construction in terms of the boson operators in the bosonization context, which can be related to the Kac-Moody generators in conformal field theories. Second, we study the behavior of the Kac-Moody generators in the integrable Lieb-Liniger model and reveal its underlying particle-hole excitation picture through Bethe ansatz solutions. Finally, we test the computation of the Kac-Moody generator in the continuous matrix product state simulations, paving the way for more challenging nonintegrable systems.
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.
New or enlarged symmetries can emerge at the low-energy spectrum of a Hamiltonian that does not possess the symmetries, if the symmetry breaking terms in the Hamiltonian are irrelevant under the renormalization group flow. We propose a tensor network based algorithm to numerically extract lattice operator approximation of the emergent conserved currents from the ground state of any quantum spin chains, without the necessity to have prior knowledge about its low-energy effective field theory. Our results for the spin-1/2 J-Q Heisenberg chain and a one-dimensional version of the deconfined quantum critical points demonstrate the power of our method to obtain the emergent lattice Kac-Moody generators. It can also be viewed as a way to find the local integrals of motion of an integrable model and the local parent Hamiltonian of a critical gapless ground state.
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.
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.