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| Fig. 1: Phase diagram of the binary driven spin chain in the non-interacting limit. (Source: X. Huang, following Khemani et al. [3]) |
The idea of extending the notion of phase and phase transition to closed/isolated quantum systems away from equilibrium is of great interest in statistical quantum mechanics and condensed matter physics. Recently this idea has been thoroughly explored in periodically driven quantum systems, namely Floquet systems, whose Hamiltonian H is time-dependent and satisfies H(t+T)=H(t), where T is the period of the drive. [1,2] Khemani et al. demonstrate that, introducing disorders and interactions to Floquet systems can localize spatial modes and enable rich phase structures, some of which are novel to driven systems and have no equilibrium counterpart. [3]
Thermalization in a closed quantum system is different from a usual system with an external bath. While the entire system experiencing unitary time evolution, for a local subsystem, the rest of the system serves as an internal bath and eliminate local memory at late times, bringing it to thermal equilibrium. The celebrated notion of eigenstate thermalization hypothesis (ETH) ensures that eigenstate expected values agree with the expected thermodynamic ensemble averages. [4] In contrast, many-body localization (MBL) systems do not reach such thermal equilibrium and local properties fluctuate strongly between states at the same energy density. [5] The dynamics of Floquest systems are governed by the Floquet unitary operator UF, which is the time evolution operator over one period. The n-th eigenvalue of UF, e-iEn, defines the n-th quasi-energy En for Floquet systems. The Floquet version of ETH asserts that each Floquet eigenstate thermalize to infinite temperature. However, it has been shown that localization can prevent such indefinite heating in disordered many-body systems. In such systems the notion of eigensystem order can be used to identify phases in a way similar to that used in equilibrium systems.
Consider an undriven 1D disordered spin chain with Ising Symmetry composed of nearest neighour spin exchange terms in the x direction, magnetic fields in the z direction and nearest neighour spin exchange terms in the z direction. After mapping to the fermionic language, the first two terms give a p-wave superconducting free-fermion model, while the last term serves as a density-density interaction term. In the non-interacting clean model limit, the system holds two ground states: a paramagnetic (PM) state with spins aligned with the external field and a ferromagnetic state with the Ising symmetry spontaneously broken. Introducing disorders to the system alters the ferromagnetic state. Now each eigenstate becomes "glassy": the Ising symmetry is locally broken and is energetically degenerate with its Ising reversed partner. Distinct from the aforementioned ferromagnetic phase, this phase is denoted as the spin-glass (SG) phase. Specifically, consider log-normally distributed nearest neighbour exchange J in x direction and magnetic fields h in z direction. When the system is localized, both the PM phase for mean(log(J)) smaller than mean(log(h)) and the SG phase for mean(log(J)) larger than mean(log(h)) exist at all energies. [3] They are both stable against weak interactions.
Consider a binary drive with alternating mean of
log(J) over time, i.e. mean(log(J))=0 for half of the cycle and
mean(log(J))=1 for the other half of the cycle. In order to examine
whether the system reaches MBL within the parameter space studied, the
level-statistics ratio
r=min(δn,δn+1)/max(δ
Moving beyond generalization of phases already exist in equilibrium systems, two new Ising phases, the π-SG phase and the 0π-PM phase, can be found in another spin chain model with a binary drive altering between x direction and z direction. In the π-SG phase, the spectral function A(ω) shows a delta function peak at ω = π/T and the magnitude of spin-spin correlators in x direction and y direction cross each other twice during one period, meaning the SG order parameter rotating by an angle π about the z axis. The phase diagram of the Hamiltonian in the non-interacting limit is plotted in Fig.1
Contrary to naive expectation, numerical evidence provided above illustrates that Floquet systems host a rich phase diagram beyond the trivial ergodic phase. In addition to Floquet counterparts of the established phases in equilibrium system, there are several phases that are entirely new to nonequilibrium systems. Some of those phases are related to more novel concepts, e.g. time crystal. [6] The work opens up a zoo of new phenomena in Floquet systems awaiting for further exploration.
© Xuxin Huang. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] H. Sambe, "Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field," Phys.Rev. A 7, 2203 (1973).
[2] J. H. Shirley, "Solution of the Schrödinger Equation With a Hamiltonian Periodic in Time," Phys. Rev. 138, B979 (1965).
[3] V. Khemani et al., "Phase Structure of Driven Quantum Systems," Phys. Rev. Lett. 116, 250401 (2016).
[4] J. M. Deutsch, "Quantum Statistical Mechanics in a Closed System," Phys. Rev. A, 43, 2046 (1991).
[5] I. V. Gornyi, A. D. Merlin, and D. G. Polyakov, "Interacting Electrons in Disordered Wires: Anderson Localization and Low-T Transport," Phys. Rev. Lett. 95, 206603 (2005).
[6] C. W. von Keyserlingk, V. Khemani, and S. L. Sondhi, "Absolute Stability and Spatiotemporal Long-Range Order in Floquet Systems," Phys. Rev. B, 94, 085112 (2016).