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Ramp and periodic dynamics across critical points with Potts universality class

Roopayan Ghosh, Arnab Sen, K. Sengupta

Published Oct 20, 2017• 0 citations
OpenAlex

Abstract

We study ramp and periodic dynamics of ultracold bosons in an one-dimensional (1D) optical lattice which supports quantum critical points belonging to the Potts universality class. Such critical points separate a uniform and a $Z_3$ or $Z_4$ symmetry broken density-wave ground state. Our protocol involves both linear and periodic drives which takes the system from the uniform to the quantum critical point (for linear drive protocol) or to the ordered state and back (for periodic drive protocols) via controlled variation of a parameter of the system Hamiltonian. We provide exact numerical computation for finite-size boson chains with $L \le 24$ using exact-diagonalization (ED) of the excitation density $D$, the wavefunction overlap $F$, and the excess energy $Q$ at the end of the drive protocol. For the linear ramp protocol, we identify the range of ramp speeds for which $D$ and $Q$ shows Kibble-Zurek scaling and show that such scaling is consistent with that expected from critical exponents of the $q$-state Potts universality class with $q=3,4$. For periodic protocol, we show that the model display near-perfect dynamical freezing at specific frequencies; at these frequencies $D, Q \to 0$ and $|F| \to 1$. We provide a semi-analytic explanation of such freezing behavior and relate this phenomenon to a many-body version of Stuckelberg interference. We suggest experiments which can test our theory.

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We study ramp and periodic dynamics of ultracold bosons in an one-dimensional (1D) optical lattice which supports quantum critical points belonging to the Potts universality class. Such critical points separate a uniform and a Z3Z_3Z3​ or Z4Z_4Z4​ symmetry broken density-wave ground state. Our protocol involves both linear and periodic drives which takes the system from the uniform to the quantum critical point (for linear drive protocol) or to the ordered state and back (for periodic drive protocols) via controlled variation of a parameter of the system Hamiltonian. We provide exact numerical computation for finite-size boson chains with L≤24L \le 24L≤24 using exact-diagonalization (ED) of the excitation density DDD, the wavefunction overlap FFF, and the excess energy QQQ at the end of the drive protocol. For the linear ramp protocol, we identify the range of ramp speeds for which DDD and QQQ shows Kibble-Zurek scaling and show that such scaling is consistent with that expected from critical exponents of the qqq-state Potts universality class with q=3,4q=3,4q=3,4. For periodic protocol, we show that the model display near-perfect dynamical freezing at specific frequencies; at these frequencies D,Q→0D, Q \to 0D,Q→0 and ∣F∣→1|F| \to 1∣F∣→1. We provide a semi-analytic explanation of such freezing behavior and relate this phenomenon to a many-body version of Stuckelberg interference. We suggest experiments which can test our theory.
We study ramp and periodic dynamics of ultracold bosons in an one-dimensional (1D) optical lattice which supports quantum critical points separating a uniform and a Z3{Z}_{3}Z3​ or Z4{Z}_{4}Z4​ symmetry broken density-wave ground state. Our protocol involves both linear and periodic drives which takes the system from the uniform state to the quantum critical point (for linear drive protocol) or to the ordered state and back (for periodic drive protocols) via controlled variation of a parameter of the system Hamiltonian. We provide exact numerical computation, for finite-size boson chains with $L\ensuremath{\le}24$ using exact diagonalization (ED), of the excitation density DDD, the wave function overlap FFF, and the excess energy QQQ at the end of the drive protocol. For the linear ramp protocol, we identify the range of ramp speeds for which DDD and QQQ show Kibble-Zurek scaling. We find, based on numerical analysis with $L\ensuremath{\le}24$, that such scaling is consistent with that expected from critical exponents of the qqq-state Potts universality class with q=3,4q=3,4q=3,4. For the periodic protocol, we show that the model displays near-perfect dynamical freezing at specific frequencies; at these frequencies $D,Q\ensuremath{\rightarrow}0$ and $|F|\ensuremath{\rightarrow}1$. We provide a semi-analytic explanation of such freezing behavior and relate this phenomenon to a many-body version of Stuckelberg interference. We suggest experiments which can test our theory.
We analyze quantum systems under a broad class of protocols in which the temperature and a Hamiltonian control parameter are ramped simultaneously and, in general, in a nonlinear fashion toward a quantum critical point. Using an open-system version of a Kitaev quantum wire as an example, we show that, unlike finite-temperature protocols at fixed temperature, these protocols allow us to probe, in an out-of-equilibrium situation and at finite temperature, the universality class (characterized by the critical exponents ννν and zzz) of an equilibrium quantum phase transition at zero temperature. Key to this is the identification of ramps in which both coherent and incoherent parts of the open-system dynamics affect the excitation density in a non-negligible way. We also identify the specific ramps for which subleading corrections to the asymptotic scaling laws are suppressed, which serves as a guide to dynamically probing quantum critical exponents in experimentally realistic finite-temperature situations.
We study the dynamics of a tilted one-dimensional Bose-Hubbard model for two distinct protocols using numerical diagonalization for a finite sized system ($N\ensuremath{\le}18$). The first protocol involves periodic variation of the effective electric field EEE seen by the bosons which takes the system twice (per drive cycle) through the intermediate quantum critical point. We show that such a drive leads to nonmonotonic variation of the excitation density DDD and the wave function overlap FFF at the end of a drive cycle as a function of the drive frequency ${\ensuremath{\omega}}_{1}$, relate this effect to a generalized version of St\"uckelberg interference phenomenon, and identify special frequencies for which DDD and $1\ensuremath{-}F$ approach zero leading to near-perfect dynamic freezing phenomenon. The second protocol involves a simultaneous linear ramp of both the electric field EEE (with a rate ${\ensuremath{\omega}}_{1}$) and the boson hopping parameter JJJ (with a rate ${\ensuremath{\omega}}_{2}$) starting from the ground state for a low effective electric field up to the quantum critical point. We find that both DDD and the residual energy QQQ decrease with increasing ${\ensuremath{\omega}}_{2}$; our results thus demonstrate a method of achieving near-adiabatic protocol in an experimentally realizable quantum critical system. We suggest experiments to test our theory.
We study the role played by extensive degeneracy in shaping the nature of the quantum dynamics of a one-dimensional spin model for both ramp and periodic drive protocols. The model displays an extensive degenerate manifold of states for a specific value of one of the parameters of its Hamiltonian. We study a linear ramp which takes the spin model through this degenerate point and show that it leads to a deviation from the usual Kibble-Zurek behavior. We also study the Stückelberg oscillations in such a model for a ramp which passes twice through the degenerate point. Our study indicates that such oscillations are strongly suppressed leading to a distinct behavior compared to those arising from double passage through a quantum critical point. Finally, we study the periodic dynamics of the model and show, for a large drive amplitude, the existence of special drive frequencies at which the system exhibits an approximate emergent U(1)U(1)U(1) symmetry. We study the effect of this emergent symmetry on the correlators of the driven system and demonstrate the existence of dynamic symmetry restoration at these frequencies. We study the fate of the emergent symmetry when the drive amplitude is decreased and discuss possible experiments to test our theory.
We study the non-equilibrium dynamics of ultracold bosons in an optical lattice with a time dependent hopping amplitude J(t)=J_0 +δJ \cos(ωt) which takes the system from a superfluid phase near the Mott-superfluid transition (J= J_0+δJ) to a Mott phase (J=J_0-δJ) and back through a quantum critical point (J=J_c) and demonstrate dynamic freezing of the boson wavefunction at specific values of ω. At these values, the wavefunction overlap F (defect density P=1-F) approaches unity (zero). We provide a qualitative explanation of the freezing phenomenon, show it's robustness against quantum fluctuations and the presence of a trap, compute residual energy and superfluid order parameter for such dynamics, and suggest experiments to test our theory.
The emergence of equilibrium universality from intrinsically nonequilibrium dynamics is a fundamental open problem. Bose-Hubbard lattices realized in photonic and circuit-QED platforms provide a versatile setting to engineer nonlinear interactions, dissipation, and multiphoton processes. Here we investigate a Bose-Hubbard lattice subject to three-photon parametric driving, whose nonequilibrium steady state spontaneously breaks a Z3\mathbb Z_3Z3​ symmetry and realizes the criticality of the three-state Potts model, a three-state generalization of the Ising model. Using a variational phase-space approach with systematically controllable accuracy based on a Variational Multi-Gaussian ansatz, we perform finite-size scaling analyses in one and two spatial dimensions. We find that, in two-dimensional lattices with single-photon losses, the nonequilibrium steady-state transition belongs to the universality class of the 2D classical three-state Potts model. In contrast, in one-dimensional lattices with three-photon losses, the transition is governed by the one-dimensional quantum three-state Potts universality class. These results establish driven-dissipative bosonic lattices as a platform for emergent Potts criticality and identify multiphoton dissipation as a mechanism that promotes nonequilibrium critical behavior from classical to quantum universality classes.