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arXiv:2205.04500 · Physical review. B./Physical review. B

Virasoro and Kac-Moody algebra in generic tensor network representations of two-dimensional critical lattice partition functions

Ruoshui Wang, Xiangdong Zeng, Ce Shen, Ling-Yan Hung

Published Sep 9, 2022• 2 citations
arXivHTMLPDF

Abstract

In this paper, we propose a general implementation of the Virasoro generators and Kac-Moody currents in generic tensor network representations of two-dimensional (2D) critical lattice models. Our proposal works even when a quantum Hamiltonian of the lattice model is not available, which is the case in many numerical computations involving numerical blockings. We tested our proposal on the 2D Ising model, and also the dimer model, which works to high accuracy even with a fairly small system size. Our method makes use of eigenstates of a small cylinder to generate descendant states in a larger cylinder, suggesting some intricate algebraic relations between lattices of different sizes.

DOI
10.1103/physrevb.106.115116

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In this paper, we propose a general implementation of the Virasoro generators and Kac-Moody currents in generic tensor network representations of two-dimensional (2D) critical lattice models. Our proposal works even when a quantum Hamiltonian of the lattice model is not available, which is the case in many numerical computations involving numerical blockings. We tested our proposal on the 2D Ising model, and also the dimer model, which works to high accuracy even with a fairly small system size. Our method makes use of eigenstates of a small cylinder to generate descendant states in a larger cylinder, suggesting some intricate algebraic relations between lattices of different sizes.
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.
We show that the recently developed pseudoparticle-operator algebra which generates the low-energy Hamiltonian eigenstates of multicomponent integrable systems with contact interactions also provides a natural operator representation for the Virasoro algebras associated with the conformal-invariant character of the low-energy spectrum of these models. Studying explicitly the Hubbard chain in a nonzero chemical potential and external magnetic field, we establish that the pseudoparticle-perturbation theory provides a correct starting point for the construction of a suitable critical-point Hamiltonian. We derive explicit expressions in terms of pseudoparticle operators for the generators of the Virasoro algebras and the energy-momentum tensor, describe the conformal-invariant character of the critical point from the point of view of the response to curvature of the two-dimensional space time, and discuss the relation to Kac-Moody algebras and dynamical separation.
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.
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.
Consider the partition function of a classical system in two spatial dimensions, or the Euclidean path integral of a quantum system in two space-time dimensions, both on a lattice. We show that the tensor network renormalization algorithm [G. Evenbly and G. Vidal Phys. Rev. Lett. 115, 180405 (2015)] can be used to implement local scale transformations on these objects, namely, a lattice version of conformal maps. Specifically, we explain how to implement the lattice equivalent of the logarithmic conformal map that transforms the Euclidean plane into a cylinder. As an application, and with the 2D critical Ising model as a concrete example, we use this map to build a lattice version of the scaling operators of the underlying conformal field theory, from which one can extract their scaling dimensions and operator product expansion coefficients.
A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.180405; S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.110504], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.