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Modern Physics Letters A

Conformal properties of a periodically driven 1D quantum system

Anı́bal Iucci, Carlos Naón

Published Apr 20, 2025• 0 citations
Publisher

Abstract

In this paper, we investigate conformal properties of a one-dimensional quantum system in the presence of an oscillating, time-dependent barrier. We compute the conformal anomaly c. We consider two methods: the well-known low-temperature expansion of the free-energy, and the analysis of the scaling regime under a deformation by a scaling operator [Formula: see text], that explicitly breaks conformal symmetry. The first method allows us to show that, as a consequence of the oscillations, c becomes dependent on the corresponding amplitude. Then, by combining these two approaches, we show that the correlation length [Formula: see text] gets an additional interaction dependent factor.

DOI
10.1142/s0217732325500464

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In this paper, we investigate conformal properties of a one-dimensional quantum system in the presence of an oscillating, time-dependent barrier. We compute the conformal anomaly c. We consider two methods: the well-known low-temperature expansion of the free-energy, and the analysis of the scaling regime under a deformation by a scaling operator [Formula: see text], that explicitly breaks conformal symmetry. The first method allows us to show that, as a consequence of the oscillations, c becomes dependent on the corresponding amplitude. Then, by combining these two approaches, we show that the correlation length [Formula: see text] gets an additional interaction dependent factor.
We investigate conformal properties of a one-dimensional quantum system with a long-range, Coulomb-like potential of the form $1∕{\ensuremath{\mid}x\ensuremath{\mid}}^{\ensuremath{\sigma}}$, with $\ensuremath{\sigma}>0$. We compute the conformal anomaly ccc as function of $\ensuremath{\sigma}$. We obtain c=0c=0c=0 for $\ensuremath{\sigma}<1$ (Wigner crystal) and c=1c=1c=1 for $\ensuremath{\sigma}>1$ (Luttinger liquid). By studying the scaling regime of the system when it is deformed by the inclusion of a term $t\ensuremath{\rho}(x)$ [with $\ensuremath{\rho}(x)$ a scaling operator], we show that, for $\ensuremath{\sigma}>1$, the correlation length $\ensuremath{\xi}$ gets an additional interaction dependent factor. This leads to a necessary redefinition of $\ensuremath{\xi}$ in order to avoid an unphysical dependence of the central charge on the coupling constant.
The exact large time and distance behavior of the field-field correlators has been computed for one-dimensional impenetrable anyons at finite temperatures. The result reproduces known asymptotics for impenetrable bosons and free fermions in the appropriate limits of the statistics parameter. The obtained asymptotic behavior of the correlators is dominated by the singularity in the spectral density of the quasiparticle states at the bottom of the band, and differs from the predictions of the conformal field theory. One can argue, however, that the anyonic response to the low-energy probes is still determined by the subleading conformal terms in the asymptotic expansion.
Two-dimensional Fermi gases with universal short-range interactions are known to exhibit a quantum anomaly, where a classical scale and conformal invariance is broken by quantum effects at strong coupling. We argue that in a quasi two-dimensional geometry, a conformal window remains at weak interactions. Using degenerate perturbation theory, we verify the conformal symmetry by computing the energy spectrum of mesoscopic particle ensembles in a harmonic trap, which separates into conformal towers formed by so-called primary states and their center-of-mass and breathing-mode excitations, the latter having excitation energies at precisely twice the harmonic oscillator energy. In addition, using Metropolis importance sampling, we compute the hyperradial distribution function of the many-body wave functions, which are predicted by the conformal symmetry in closed analytical form. The weakly interacting Fermi gas constitutes a system where the nonrelativistic conformal symmetry can be revealed using elementary methods, and our results are testable in current experiments on mesoscopic Fermi gases.
We consider the problem of correlation functions in the stationary states of one-dimensional stochastic models having conformal invariance. If one considers the space dependence of the correlators, the novel aspect is that although one considers systems with periodic boundary conditions, the observables are described by boundary operators. From our experience with equilibrium problems one would have expected bulk operators. Boundary operators have correlators having critical exponents being half of those of bulk operators. If one studies the space-time dependence of the two-point function, one has to consider one boundary and one bulk operators. The Raise and Peel model has conformal invariance as can be shown in the spin 1/2 basis of the Hamiltonian which gives the time evolution of the system. This is an XXZ quantum chain with twisted boundary condition and local interactions. This Hamiltonian is integrable and the spectrum is known in the finite-size scaling limit. In the stochastic base in which the process is defined, the Hamiltonian is not local anymore. The mapping into an SOS model, helps to define new local operators. As a byproduct some new properties of the SOS model are conjectured. The predictions of conformal invariance are discussed in the new framework and compared with Monte Carlo simulations.
We show that the leading finite-size correction to lnZ for a two-dimensional system at a conformally invariant critical point on a strip of length L width \ensuremath{\beta}(\ensuremath{\beta}\ensuremath{\ll}L) is (\ensuremath{\pi}/6)c(L/\ensuremath{\beta}), where c is the conformal anomaly. Equivalently, the leading low-temperature correction to the free energy of a one-dimensional quantum system is -(\ensuremath{\pi}/6)cL(kT)2{)}^{2})2\ensuremath{\Elzxh}v, where v is the effective ``velocity of light.'' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.