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Physical Review A

Emergent infrared conformal dynamics: Applications to strongly interacting quantum states

Jeff Maki, Fei Zhou

Published May 16, 2024• 3 citations
Publisher

Abstract

The authors exploit a connection between the theoretical renormalization group equation flow and the expansion dynamics of strongly interacting quantum gases. This analysis shows that there shall be an emergent conformal symmetry that restricts the dynamics and limits entropy production in one dimension, while it is absent in three-dimensions.

DOI
10.1103/physreva.109.l051303

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The authors exploit a connection between the theoretical renormalization group equation flow and the expansion dynamics of strongly interacting quantum gases. This analysis shows that there shall be an emergent conformal symmetry that restricts the dynamics and limits entropy production in one dimension, while it is absent in three-dimensions.
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.
In this article we study the quench dynamics of Galilean and scale-invariant many-body systems which can be prepared using interacting atomic gases. The far-from-equilibrium dynamics is investigated by employing mmm-body density matrices, which are most conveniently defined in terms of a special basis, the conformal tower states. We explicitly illustrate that, although during the initial stage of the dynamics all symmetries can be broken and absent in the unitary evolution because of the initialization of the state, there is always an emergent conformal symmetry in the long-time limit. The emergence of this dynamic conformal symmetry is robust and always occurs, even when scale and other symmetries (such as rotational symmetry) are still fully broken in the many-body states; it uniquely defines the characteristics of the asymptotic dynamics near a scale-invariant strong-coupling fixed point. As an immediate application of the asymptotic dynamics of the microscopic density matrices, we focus on the effects of this emergent conformal symmetry on two observables: the moment of inertia tensor Iij(t),i,j=x,y,z{I}_{ij}(t), i,j=x,y,zIij​(t),i,j=x,y,z, and the entropy density field S(r,t)S(\mathbf{r},t)S(r,t) in the hydrodynamic flow of strongly interacting particles. We show that the long-time behavior of these observables is completely set by conformal symmetry, while the leading long-time corrections depend on interference effects between different conformal tower states. The emergent conformal symmetry naturally leads to entropy conservation and conformal cooling, an energy-conserving cooling of a strongly interacting gas during free expansion. When the interaction Hamiltonian breaks the scale symmetry, we further demonstrate that there is a direct cause-effect relation between conformal symmetry breaking in the long-time limit and a nonvanishing entropy production. This suggests that the entropy production rate is a natural parameter for categorizing the breaking of conformal symmetry.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.
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.