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arXiv:2305.19061

Emergent Infrared Conformal Dynamics in Strongly Interacting Quantum Gases

Jeff Maki, Fei Zhou

Published May 30, 2023• 0 citations
arXivHTMLPDF

Abstract

Conformal dynamics can appear in quantum gases when the interactions are fine tuned to be scale symmetric. One well-known example of such a system is a three-dimensional Fermi gas at a Feshbach resonance. In this letter, we illustrate how conformal dynamics can also emerge in the infrared limit in one-dimensional harmonically trapped Fermi gases, even though the system may not have exactly scale symmetric interactions. The conformal dynamics are induced by strong renormalization effects due to the nearby infrared stable scale invariant interaction. When the system approaches the infrared limit, or when the external harmonic trapping frequency $ω_f \rightarrow 0$, the dynamics are characterized by a unique vanishingly small dissipation rate, $Γ\propto ω_f$, rather than a constant as in generic interacting systems. We also examine the work done in a two-quench protocol, $W$, and the average power $\mathcal{P}$. In one dimension, the average power, $\mathcal{P} \propto ω_f$, becomes vanishingly small in the infrared limit, a signature of emergent conformal dynamics.

DOI
10.48550/arxiv.2305.19061

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Conformal dynamics can appear in quantum gases when the interactions are fine tuned to be scale symmetric. One well-known example of such a system is a three-dimensional Fermi gas at a Feshbach resonance. In this letter, we illustrate how conformal dynamics can also emerge in the infrared limit in one-dimensional harmonically trapped Fermi gases, even though the system may not have exactly scale symmetric interactions. The conformal dynamics are induced by strong renormalization effects due to the nearby infrared stable scale invariant interaction. When the system approaches the infrared limit, or when the external harmonic trapping frequency ωf→0ω_f \rightarrow 0ωf​→0, the dynamics are characterized by a unique vanishingly small dissipation rate, Γ∝ωfΓ\propto ω_fΓ∝ωf​, rather than a constant as in generic interacting systems. We also examine the work done in a two-quench protocol, WWW, and the average power P\mathcal{P}P. In one dimension, the average power, P∝ωf\mathcal{P} \propto ω_fP∝ωf​, becomes vanishingly small in the infrared limit, a signature of emergent conformal dynamics.
Conformal symmetry heavily constrains the dynamics of nonrelativistic quantum gases tuned to a nearby quantum critical point. One important consequence of this symmetry is that entropy production can be absent in far-away-from-equilibrium dynamics of strongly interacting three-dimensional (3D) and one-dimensional (1D) quantum gases placed inside an adjustable harmonic trapping potential. This can lead to an oscillatory fully revivable many-body dynamic state, which is reflected in many physical observables. In this article we further investigate the consequences of conformal symmetry on (a) the zero-temperature autocorrelation function, (b) the Wigner distribution function, and (c) the von Neumann entanglement entropy. A direct calculation of these quantities for generic strongly interacting systems is usually extremely difficult. However, we have derived the general structures of these functions in the nonequilibrium dynamics when their dynamics are constrained by conformal symmetry. We obtain our results for (a) by utilizing an operator-state correspondence which connects the imaginary time evolution of primary operators to different initial states of harmonically trapped gases, while the dynamics of the functions in (b) and (c) are derived from conformal invariant density matrices.
In this Letter, we investigate the effects of a time-dependent, short-ranged interaction on the long-time expansion dynamics of Fermi gases. We show that the effects of the interaction on the dynamics is dictated by how it changes under a conformal transformation, and derive an explicit criterion for the relevancy of time-dependent interactions near both the strongly and noninteracting scale invariant limits. In addition, we show that it is possible to engineer interactions that give rise to nonexponential thermalization dynamics in trapped Fermi gases. To supplement the symmetry analysis, we perform hydrodynamic simulations to show that the moment of inertia of the trapped gas indeed follows a universal time dependence that is determined jointly by the conformal symmetry and time-dependent scattering length a(t). Our results should also be relevant to the dynamics of other systems that are nearly scale invariant and that are governed by a nonrelativistic conformal symmetry.
The study of the quantum dynamics of ultra-cold atomic gases has become a forefront of atomic research. Experiments studying dynamics have become routine in laboratories, and a plethora of phenomena have been studied. Theoretically, however, the situation is often intractable unless one resorts to numerical or semiclassical calculations. In this thesis we apply the symmetry associated with scale invariance to study the dynamics of atomic gases, and discuss the implications of this symmetry on the full quantum dynamics. In particular we study the time evolution of an expanding two-dimensional Bose gas with attractive contact interactions, and the three-dimensional Fermi gas at unitarity. To do this we employ a quantum variational approach and exact symmetry arguments. It is shown that the time evolution due to a scale invariant Hamiltonian produces an emergent conformal symmetry. This emergent conformal symmetry has implications on the time evolution of an expanding quantum gas. In addition, we examine the effects of broken scale symmetry on the expansion dynamics. To do this, we develop a non-perturbative formalism that classifies the possible dynamics that can occur. This formalism is then applied to two systems, an ensemble of two-body systems, and for the compressional and elliptic flow of a unitary Fermi gas, both in three spatial dimensions.
Conformal field theory (CFT) has been extremely successful in describing large-scale universal effects in one-dimensional (1D) systems at quantum critical points. Unfortunately, its applicability in condensed matter physics has been limited to situations in which the bulk is uniform because CFT describes low-energy excitations around some energy scale, taken to be constant throughout the system. However, in many experimental contexts, such as quantum gases in trapping potentials and in several out-of-equilibrium situations, systems are strongly inhomogeneous. We show here that the powerful CFT methods can be extended to deal with such 1D situations, providing a few concrete examples for non-interacting Fermi gases. The system's inhomogeneity enters the field theory action through parameters that vary with position; in particular, the metric itself varies, resulting in a CFT in curved space. This approach allows us to derive exact formulas for entanglement entropies which were not known by other means.
Deterministic preparation of an ultracold harmonically trapped one-dimensional Fermi gas consisting of a few fermions has been realized by the Heidelberg group. Using Floquet formalism, we study the time dynamics of two- and three-fermion systems in a harmonic trap under an oscillating magnetic field. The oscillating magnetic field produces a time-dependent interaction strength through a Feshbach resonance. We explore the dependence of these dynamics on the frequency of the oscillating magnetic field for noninteracting, weakly interacting, and strongly interacting systems. We identify the regimes where the system can be described by an effective two-state model and an effective three-state model. We find an unbounded coupling to all excited states at the infinitely strong interaction limit and several simple relations that characterize the dynamics. Based on our findings, we propose a technique for driving transition from the ground state to the excited states using an oscillating magnetic field.