Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2203.14992 · Physical review. B./Physical review. B

Operator fusion from wave-function overlap: Universal finite-size corrections and application to the Haagerup model

Yuhan Liu, Yijian Zou, Shinsei Ryu

Published Apr 13, 2023• 16 citations
arXivHTMLPDF

Abstract

Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c\ensuremath{\approx}2.1$ and the lowest spin-1 operator in the twisted sector has scaling dimension $1<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.4$.

DOI
10.1103/physrevb.107.155124

Related papers

Numerical Evidence for a Haagerup Conformal Field Theory

Tzu-Chen Huang, Ying-Hsuan Lin, Kantaro Ohmori +2 more

We numerically study an anyon chain based on the Haagerup fusion category and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ∼2. Fusion categories generalize the concept of finite group symmetries to noninvertible symmetry operations, and the Haagerup fusion category is the simplest one which comes from neither finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries. Basically the same result was independently obtained in the preceding Letter [Phys. Rev. Lett. 128, 231602 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.231602].

arXiv:2110.03008•Physical Review Letters•Jun 10, 2022•59 citations

Conformal Fields and Operator Product Expansion in Critical Quantum Spin Chains

Yijian Zou, Ashley Milsted, Guifré Vidal

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.

arXiv:1901.06439•Physical Review Letters•Jan 28, 2020•45 citations

Universal information of critical quantum spin chains from wavefunction overlap

Yijian Zou

Given a critical quantum spin chain, we show how universal information about its quantum critical point can be extracted from wavefunction overlaps. More specifically, we consider overlap between low-energy eigenstates of the spin chain Hamiltonian with different boundary conditions, namely periodic boundary conditions and open boundary conditions. We show that such overlaps decay polynomially with the system size, where the exponent only depends on the central charge. Furthermore, the bulk-to-boundary operator product expansion (OPE) coefficients can be extracted from the overlaps involving excited states. We illustrate the proposal with the Ising model and the three-state Potts model.

arXiv:2104.00103•Physical review. B./Physical review. B•Apr 18, 2022•13 citations

Tensor networks, quantum spin chains, and quantum field theory

Yijian Zou

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.

•Jul 7, 2020•0 citations

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

Ashley Milsted, Guifré Vidal

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.

•Jun 5, 2017•0 citations

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

Ashley Milsted, Guifré Vidal

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.

arXiv:1706.01436•Physical review. B./Physical review. B•Dec 4, 2017•48 citations
Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou and Vidal, Phys. Rev. B 105, 125125 (2022)] that the OPE coefficients can be computed from overlaps of low-energy wave functions of the spin chain. In this paper, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wave-function overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge c\ensuremath≈2.1c\ensuremath{\approx}2.1c\ensuremath≈2.1 and the lowest spin-1 operator in the twisted sector has scaling dimension 1<\ensuremathΔJ\ensuremath≤1.41<{\mathrm{\ensuremath{\Delta}}}_{J}\ensuremath{\le}1.41<\ensuremathΔJ​\ensuremath≤1.4.
We numerically study an anyon chain based on the Haagerup fusion category and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is ∼2. Fusion categories generalize the concept of finite group symmetries to noninvertible symmetry operations, and the Haagerup fusion category is the simplest one which comes from neither finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries. Basically the same result was independently obtained in the preceding Letter [Phys. Rev. Lett. 128, 231602 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.231602].
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.
Given a critical quantum spin chain, we show how universal information about its quantum critical point can be extracted from wavefunction overlaps. More specifically, we consider overlap between low-energy eigenstates of the spin chain Hamiltonian with different boundary conditions, namely periodic boundary conditions and open boundary conditions. We show that such overlaps decay polynomially with the system size, where the exponent only depends on the central charge. Furthermore, the bulk-to-boundary operator product expansion (OPE) coefficients can be extracted from the overlaps involving excited states. We illustrate the proposal with the Ising model and the three-state Potts model.
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.
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.