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arXiv:1709.06238 · Progress of Theoretical and Experimental Physics

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

Shota Tamura, Hosho Katsura

Published Oct 16, 2017• 29 citations
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Abstract

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.

DOI
10.1093/ptep/ptx147

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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 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.
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.
By making use of conformal mapping, we construct various time-evolution operators in (1+1)-dimensional conformal field theories (CFTs), which take the form $\ensuremath{\int}dx\phantom{\rule{0.16em}{0ex}}f(x)\mathcal{H}(x)$, where H(x)\mathcal{H}(x)H(x) is the Hamiltonian density of the CFT and f(x)f(x)f(x) is an envelope function. Examples of such deformed evolution operators include the entanglement Hamiltonian and the so-called sine-square deformation of the CFT. Within our construction, the spectrum and the (finite-size) scaling of the level spacing of the deformed evolution operator are known exactly. Based on our construction, we also propose a regularized version of the sine-square deformation, which, in contrast to the original sine-square deformation, has the spectrum of the CFT defined on a spatial circle of finite circumference LLL, and for which the level spacing scales as 1/L21/{L}^{2}1/L2, once the circumference of the circle and the regularization parameter are suitably adjusted.
The nonequilibrium process where the system does not evolve to the featureless state, that without quantum properties, is one of the new central objects in the nonequilibrium phenomena. In this paper, starting from the short-range entangled state in the two-dimensional conformal field theories (2D CFTs), the boundary state with a regularization, we evolve the system with the inhomogeneous Hamiltonians called Möbius/sine-square-deformed (SSD) ones. Regardless of the details of CFTs considered in this paper, during the Möbius evolution, the entanglement entropy exhibits the periodic motion called quantum revival. During SSD time evolution, except for some subsystems, in the large time regime, entanglement entropy and mutual information are approximated by those for the vacuum state. We argue the time regime for the subsystem to cool down to vacuum one is t 1 ≫ O ( L l A ) , where t 1 , L , and l A are time, system, and subsystem sizes. This finding suggests the inhomogeneous quench induced by the SSD Hamiltonian may be used as the preparation for the approximately vacuum state. We propose the gravity dual of the systems considered in this paper, furthermore, and generalize it. In addition to them, we discuss the relation between the inhomogeneous quenches and continuous multiscale entanglement renormalization ansatz (cMERA). Published by the American Physical Society 2024
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.