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arXiv:1904.11549 · Physical Review A

Quantum many-body conformal dynamics: Symmetries, geometry, conformal tower states, and entropy production

Jeff Maki, Fei Zhou

Published Aug 2, 2019• 20 citations
arXivHTMLPDF

Abstract

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 $m$-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 ${I}_{ij}(t), i,j=x,y,z$, and the entropy density field $S(\mathbf{r},t)$ 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.

DOI
10.1103/physreva.100.023601

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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.
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.
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We propose and analyze a method for preparing low entropy many-body states in isolated quantum optical systems of atoms, ions, and molecules. Our approach is based upon shifting entropy between different regions of a system by spatially modulating the magnitude of the effective Hamiltonian. We conduct two case studies, on a topological spin chain and the spinful fermionic Hubbard model, focusing on the key question: can a "conformal cooling quench" remove sufficient entropy within experimentally accessible timescales? Finite-temperature, time-dependent matrix product state calculations reveal that even moderately sized bath regions can remove enough energy and entropy density to expose coherent low-temperature physics. The protocol is particularly natural in systems with long-range interactions, such as lattice-trapped polar molecules and Rydberg-excited atoms, where the magnitude of the Hamiltonian scales directly with the interparticle spacing. To this end, we propose simple, near-term implementations of conformal cooling quenches in systems of atoms or molecules, where signatures of low-temperature phases may be observed.
Recently universal dynamic scaling is observed in several systems, which exhibit a spatiotemporal self-similar scaling behavior, analogous to the spatial scaling near phase transition. The latter one arises from the emergent continuous scaling symmetry. Motivated by this, we investigate the possible relation between the scaling dynamics and the continuous scaling symmetry in this paper. We derive a theorem that the scaling invariance of the quenched Hamiltonian and the initial density matrix can lead to the universal dynamic scaling. It is further demonstrated both in a two-body system analytically and in a many-body system numerically. For the latter one, we calculate the dynamics of quantum gases quenched from the zero interaction to a finite interaction via the non-equilibrium high-temperature virial expansion. A dynamic scaling of the momentum distribution appears in certain momentum-time windows at unitarity as well as in the weak interacting limit. Remarkably, this universal scaling dynamics persists approximately with smaller scaling exponents even if the scaling symmetry is fairly broken. Our findings may offer a new perspective to interpret the related experiments. We also study the Contact dynamics in the BEC–BCS crossover. Surprisingly, the half-way time displays a maximum near unitarity while some damping oscillations occur on the BEC side due to the dimer state, which can be used to detect possible two-body bound states in experiments.