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arXiv:1712.09823 · Progress of Theoretical and Experimental Physics

Conformal quantum mechanics and sine-square deformation

Tsukasa Tada

Published Apr 26, 2018• 29 citations
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Abstract

We revisit conformal quantum mechanics (CQM) from the perspective of sine-square deformation (SSD) and the entanglement Hamiltonian. The operators related to SSD and the entanglement Hamiltonian are identified. Thus, the nature of SSD and the entanglement can be discussed in a much simpler CQM setting than higher-dimensional field theories.

DOI
10.1093/ptep/pty058

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We revisit conformal quantum mechanics (CQM) from the perspective of sine-square deformation (SSD) and the entanglement Hamiltonian. The operators related to SSD and the entanglement Hamiltonian are identified. Thus, the nature of SSD and the entanglement can be discussed in a much simpler CQM setting than higher-dimensional field theories.
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.
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.
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.
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.
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.