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:2004.01930 · Progress of Theoretical and Experimental Physics

Analysis for Lorentzian conformal field theories through sine-square deformation

Xun Liu, Tsukasa Tada

Published May 1, 2020• 9 citations
arXivHTMLPDF

Abstract

Abstract We reexamine two-dimensional Lorentzian conformal field theory using the formalism previously developed in a study of sine-square deformation of Euclidean conformal field theory. We construct three types of Virasoro algebra. One of them reproduces the result by Lüscher and Mack, while another type exhibits divergence in the central charge term. The third leads to a continuous spectrum and contains no closed time-like curve in the system.

DOI
10.1093/ptep/ptaa077

Related papers

Infinite circumference limit of conformal field theory

Nobuyuki Ishibashi, Tsukasa Tada

We argue that an infinite circumference limit can be obtained in two-dimensional conformal field theory by adopting as a Hamiltonian instead of L 0 . The theory obtained has a circumference of infinite length and hence exhibits a continuous and heavily degenerated spectrum as well as the continuous Virasoro algebra. The choice of this Hamiltonian was inspired partly by the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The enigmatic behavior of sine-square deformed systems such as the sharing of their vacuum states with the closed boundary systems can be understood by the appearance of an infinite circumference.

arXiv:1504.00138•Journal of Physics A Mathematical and Theoretical•Jul 16, 2015•39 citations

Dipolar quantization and the infinite circumference limit of two-dimensional conformal field theories

Nobuyuki Ishibashi, Tsukasa Tada

Elaborating on our previous presentation, where the term dipolar quantization was introduced, we argue here that adopting [Formula: see text] as the Hamiltonian instead of [Formula: see text] yields an infinite circumference limit in two-dimensional conformal field theory. The new Hamiltonian leads to dipolar quantization instead of radial quantization. As a result, the new theory exhibits a continuous and strongly degenerated spectrum in addition to the Virasoro algebra with a continuous index. Its Hilbert space exhibits a different inner product than that obtained in the original theory. The idiosyncrasy of this particular Hamiltonian is its relation to the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The appearance of the infinite circumference explains why the vacuum states of sine-square deformed systems are coincident with those of the respective closed-boundary systems.

arXiv:1602.01190•International Journal of Modern Physics A•Nov 14, 2016•41 citations

Sine-square deformation and Möbius quantization of 2D conformal field theory

Kouichi Okunishi

Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1 dimensions, we discuss a Möbius quantization approach to the 2D conformal field theory, which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada [N. Ishibashi and T. Tada, J. Phys. A: Math. Theor. 48, 315402 (2015)], [N. Ishibashi and T. Tada, arXiv:1602.01190 [hep-th] [Search INSPIRE]. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Möbius quantization approach.

arXiv:1603.09543•Progress of Theoretical and Experimental Physics•Jun 1, 2016•49 citations

Time development of conformal field theories associated with L 1 and L −1 operators

Tsukasa Tada

Abstract In this study, we examined consequences of unconventional time development of two-dimensional conformal field theory induced by the L 1 and L −1 operators, employing the formalism previously developed in a study of sine-square deformation. We discovered that the retainment of the Virasoro algebra requires the presence of a cut-off near the fixed points. The introduction of a scale by the cut-off makes it possible to recapture the formula for entanglement entropy in a natural and straightforward manner.

arXiv:1904.12414•Journal of Physics A Mathematical and Theoretical•Apr 23, 2020•16 citations

Quantum dynamics in sine-square deformed conformal field theory: Quench from uniform to nonuniform conformal field theory

Xueda Wen, Jie-qiang Wu

Sine-square deformed conformal field theory has received extensive attention in both condensed matter physics and high-energy physics recently. This is due to its unusual properties, such as having a continuous Virasoro algebra. Here, the authors investigate its nonequilibrium dynamics by studying the entanglement evolution after a quantum quench. New physical effects, such as the absence of revivals in entanglement evolution, are observed in both analytical and numerical calculations. This effect could potentially be used when engineering a gapless system with cold atoms, in order to reduce finite-size effects. The work here also sheds new light on the nonequilibrium dynamics of conformal field theory in curved spacetime.

arXiv:1802.07765•Physical review. B./Physical review. B•May 31, 2018•72 citations

Sine-square deformation of solvable spin chains and conformal field theories

Hosho Katsura

We study solvable spin chains, one-dimensional massless Dirac fermions and conformal field theories (CFTs) with sine-square deformation (SSD), in which the Hamiltonian density is modulated by the function f ( x ) = sin 2 (π x /ℓ), where x is the position and ℓ is the length of the system. For the XY chain and the transverse field Ising chain at criticality, it is shown that the ground state of an open system with SSD is identical to that of a uniform chain with periodic boundary conditions. The same holds for the massless Dirac fermions with SSD, corresponding to the continuum limit of the gapless XY chain. For general CFTs, we find that the Hamiltonian of a system with SSD has an expression in terms of the generators of the Virasoro algebra. This allows us to show that the vacuum state is an exact eigenstate of the sine-square deformed Hamiltonian. Furthermore, for a restricted class of CFTs associated with affine Lie (Kac–Moody) algebras, including c = 1 Gaussian CFT, we prove that the vacuum is an exact ground state of the deformed Hamiltonian. This explains why the SSD has succeeded in suppressing boundary effects in one-dimensional critical systems, as observed in previous numerical studies.

arXiv:1110.2459•Journal of Physics A Mathematical and Theoretical•Feb 28, 2012•82 citations
Abstract We reexamine two-dimensional Lorentzian conformal field theory using the formalism previously developed in a study of sine-square deformation of Euclidean conformal field theory. We construct three types of Virasoro algebra. One of them reproduces the result by Lüscher and Mack, while another type exhibits divergence in the central charge term. The third leads to a continuous spectrum and contains no closed time-like curve in the system.
We argue that an infinite circumference limit can be obtained in two-dimensional conformal field theory by adopting as a Hamiltonian instead of L 0 . The theory obtained has a circumference of infinite length and hence exhibits a continuous and heavily degenerated spectrum as well as the continuous Virasoro algebra. The choice of this Hamiltonian was inspired partly by the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The enigmatic behavior of sine-square deformed systems such as the sharing of their vacuum states with the closed boundary systems can be understood by the appearance of an infinite circumference.
Elaborating on our previous presentation, where the term dipolar quantization was introduced, we argue here that adopting [Formula: see text] as the Hamiltonian instead of [Formula: see text] yields an infinite circumference limit in two-dimensional conformal field theory. The new Hamiltonian leads to dipolar quantization instead of radial quantization. As a result, the new theory exhibits a continuous and strongly degenerated spectrum in addition to the Virasoro algebra with a continuous index. Its Hilbert space exhibits a different inner product than that obtained in the original theory. The idiosyncrasy of this particular Hamiltonian is its relation to the so-called sine-square deformation, which is found in the study of a certain class of quantum statistical systems. The appearance of the infinite circumference explains why the vacuum states of sine-square deformed systems are coincident with those of the respective closed-boundary systems.
Motivated by sine-square deformation (SSD) for quantum critical systems in 1+1 dimensions, we discuss a Möbius quantization approach to the 2D conformal field theory, which bridges the conventional radial quantization and the dipolar quantization recently proposed by Ishibashi and Tada [N. Ishibashi and T. Tada, J. Phys. A: Math. Theor. 48, 315402 (2015)], [N. Ishibashi and T. Tada, arXiv:1602.01190 [hep-th] [Search INSPIRE]. We then find that the continuous Virasoro algebra of the dipolar quantization can be interpreted as a continuum limit of the Virasoro algebra for scaled generators in the SSD limit of the Möbius quantization approach.
Abstract In this study, we examined consequences of unconventional time development of two-dimensional conformal field theory induced by the L 1 and L −1 operators, employing the formalism previously developed in a study of sine-square deformation. We discovered that the retainment of the Virasoro algebra requires the presence of a cut-off near the fixed points. The introduction of a scale by the cut-off makes it possible to recapture the formula for entanglement entropy in a natural and straightforward manner.
Sine-square deformed conformal field theory has received extensive attention in both condensed matter physics and high-energy physics recently. This is due to its unusual properties, such as having a continuous Virasoro algebra. Here, the authors investigate its nonequilibrium dynamics by studying the entanglement evolution after a quantum quench. New physical effects, such as the absence of revivals in entanglement evolution, are observed in both analytical and numerical calculations. This effect could potentially be used when engineering a gapless system with cold atoms, in order to reduce finite-size effects. The work here also sheds new light on the nonequilibrium dynamics of conformal field theory in curved spacetime.
We study solvable spin chains, one-dimensional massless Dirac fermions and conformal field theories (CFTs) with sine-square deformation (SSD), in which the Hamiltonian density is modulated by the function f ( x ) = sin 2 (π x /ℓ), where x is the position and ℓ is the length of the system. For the XY chain and the transverse field Ising chain at criticality, it is shown that the ground state of an open system with SSD is identical to that of a uniform chain with periodic boundary conditions. The same holds for the massless Dirac fermions with SSD, corresponding to the continuum limit of the gapless XY chain. For general CFTs, we find that the Hamiltonian of a system with SSD has an expression in terms of the generators of the Virasoro algebra. This allows us to show that the vacuum state is an exact eigenstate of the sine-square deformed Hamiltonian. Furthermore, for a restricted class of CFTs associated with affine Lie (Kac–Moody) algebras, including c = 1 Gaussian CFT, we prove that the vacuum is an exact ground state of the deformed Hamiltonian. This explains why the SSD has succeeded in suppressing boundary effects in one-dimensional critical systems, as observed in previous numerical studies.