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:1110.2459 · Journal of Physics A Mathematical and Theoretical

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

Hosho Katsura

Published Feb 28, 2012• 82 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/45/11/115003

Related papers

Sine-square deformation of free fermion systems in one and higher dimensions

Isao Maruyama, Hosho Katsura, Toshiya Hikihara

We study free fermion systems with the sine-square deformation (SSD) in which the energy scale of local Hamiltonians is modified according to the scaling function $f(x)={\mathrm{sin}}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]$, where $x$ is the position of the local Hamiltonian and $L$ is the length of the system in the $x$ direction. It has been revealed that when applied to one-dimensional (1D) critical systems, the SSD realizes the translationally-invariant ground state which is the same as that of the uniform periodic system. In this paper, we propose a simple theory to explain how the SSD maintains the translational invariance in the ground-state wave function. In particular, for a certain 1D system with SSD, it is shown that the ground state is exactly identical to the Fermi sea of the uniform periodic chain. We also apply the SSD to 2D systems and show that the SSD is able to suppress the boundary modulations from the open edges extremely well, demonstrating that the SSD works in any dimension and in any direction.

arXiv:1108.2973•Physical Review B•Oct 28, 2011•55 citations

Zero-energy states in conformal field theory with sine-square deformation

Shota Tamura, Hosho Katsura

We study the properties of two-dimensional conformal field theories (CFTs) with sine-square deformation (SSD). We show that there are no eigenstates of the finite norm for the Hamiltonian of a unitary CFT with SSD, except for the zero-energy vacuum state |$\left| {0} \right\rangle$|⁠. We then introduce a regularized version of the SSD Hamiltonian which is related to the undeformed Hamiltonian via a unitary transformation corresponding to the Möbius quantization. The unitary equivalence of the two Hamiltonians allows us to obtain zero-energy states of the deformed Hamiltonian in a systematic way. The regularization also provides a way to compute the expectation values of observables in zero-energy states that are not necessarily normalizable.

arXiv:1709.06238•Progress of Theoretical and Experimental Physics•Oct 16, 2017•29 citations

Exact ground state of the sine-square deformed XY spin chain

Hosho Katsura

We study the sine-square deformed quantum XY chain with open boundary conditions in which the interaction strength at the position x in the chain of length L is proportional to the function . The model can be mapped onto a free spinless fermion model with site-dependent hopping amplitudes and on-site potentials via the Jordan–Wigner transformation. Although the single-particle eigenstates of this system cannot be obtained in closed form, it is shown that the many-body ground state is identical to that of the uniform XY chain with periodic boundary conditions. This proves a conjecture of Hikihara and Nishino (2011 Phys. Rev. B 83 060414) based on numerical evidence.

arXiv:1104.1721•Journal of Physics A Mathematical and Theoretical•May 19, 2011•48 citations

Sine-square deformation and supersymmetric quantum mechanics

Kouichi Okunishi, Hosho Katsura

We investigate the sine-square deformation (SSD) of free fermions in one-dimensional continuous space. On the basis of supersymmetric quantum mechanics, we prove the correspondence between the many-body ground state of the system with SSD and that of the uniform system with periodic boundary conditions. We also discuss the connection between the SSD in the continuous space and its lattice version, where the geometric correction due to the real-space deformation plays an important role in relating the eigenstates of the lattice SSD with those of the continuous SSD.

arXiv:1505.07904•Journal of Physics A Mathematical and Theoretical•Oct 15, 2015•16 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

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
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.
We study free fermion systems with the sine-square deformation (SSD) in which the energy scale of local Hamiltonians is modified according to the scaling function $f(x)={\mathrm{sin}}^{2}[\frac{\ensuremath{\pi}}{L}(x\ensuremath{-}\frac{1}{2})]$, where xxx is the position of the local Hamiltonian and LLL is the length of the system in the xxx direction. It has been revealed that when applied to one-dimensional (1D) critical systems, the SSD realizes the translationally-invariant ground state which is the same as that of the uniform periodic system. In this paper, we propose a simple theory to explain how the SSD maintains the translational invariance in the ground-state wave function. In particular, for a certain 1D system with SSD, it is shown that the ground state is exactly identical to the Fermi sea of the uniform periodic chain. We also apply the SSD to 2D systems and show that the SSD is able to suppress the boundary modulations from the open edges extremely well, demonstrating that the SSD works in any dimension and in any direction.
We study the properties of two-dimensional conformal field theories (CFTs) with sine-square deformation (SSD). We show that there are no eigenstates of the finite norm for the Hamiltonian of a unitary CFT with SSD, except for the zero-energy vacuum state |∣0⟩\left| {0} \right\rangle∣0⟩|⁠. We then introduce a regularized version of the SSD Hamiltonian which is related to the undeformed Hamiltonian via a unitary transformation corresponding to the Möbius quantization. The unitary equivalence of the two Hamiltonians allows us to obtain zero-energy states of the deformed Hamiltonian in a systematic way. The regularization also provides a way to compute the expectation values of observables in zero-energy states that are not necessarily normalizable.
We study the sine-square deformed quantum XY chain with open boundary conditions in which the interaction strength at the position x in the chain of length L is proportional to the function . The model can be mapped onto a free spinless fermion model with site-dependent hopping amplitudes and on-site potentials via the Jordan–Wigner transformation. Although the single-particle eigenstates of this system cannot be obtained in closed form, it is shown that the many-body ground state is identical to that of the uniform XY chain with periodic boundary conditions. This proves a conjecture of Hikihara and Nishino (2011 Phys. Rev. B 83 060414) based on numerical evidence.
We investigate the sine-square deformation (SSD) of free fermions in one-dimensional continuous space. On the basis of supersymmetric quantum mechanics, we prove the correspondence between the many-body ground state of the system with SSD and that of the uniform system with periodic boundary conditions. We also discuss the connection between the SSD in the continuous space and its lattice version, where the geometric correction due to the real-space deformation plays an important role in relating the eigenstates of the lattice SSD with those of the continuous SSD.
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.
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.