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| Fig. 1: Phase diagram for the Floquet Ising chain adapted from V. Khemani et al. [1] (Source: V. Calvera) |
The aim of this report is to review the phase diagram of the periodically-driven (a.k.a. Floquet) disordered Ising chain. [1] This model is important as it is one of the simplest models that display spontaneous discrete time-translation symmetry breaking (dTTSb). What is more, one of the phases is stable even in the absence of the Ising symmetry. [2]
Naturally, one can ask why to consider a periodic-drive or disorder. A periodic-drive is the simplest setting, aside from the well studied static framework, where there is a time-translation symmetry that could be spontaneously broken. Disorder is necessary because it is believed that translation invariant interacting systems will heat to infinite temperature under a periodic drive, i.e. all correlation functions show no dependence on the initial state in the long time regime. [3,4] The idea from a linear response point of view is that in the presence of translation invariance, extended states can easily exchange energy between each other. The role of disorder in the Ising chain is to localize the modes even in the presence of small interactions, and thus avoid the heating. [5,6]
In the following, I will review some background notions of periodically-driven (a.k.a. Floquet) systems, eigenstate order, and the (static) disordered Ising chain. After this, I will review the phase diagram of the disordered Floquet Ising chain in the presence of Ising symmetry.
Floquet systems have a periodic Hamiltonian, H(t)=H(t+T) for some T>0. [7,8] The Floquet unitary is defined as
which corresponds to time evolution for a period T. Instead of diagonalizing the Hamiltonian at every time, it is convenient to diagonalize the Floquet unitary. The Floquet eigenvectors |φα> with quasi-energy ε α are defined as
with quasi-energies defined modulo 2π/T. Studying the properties of these eigenvectors is enough to understand the dynamics of the system under discrete time evolution with time-step T as they form a basis for the state space.
As the Hamiltonian has period T, these systems have a discrete time-translation symmetry of shifting time by integer multiples of T. Similar to the standard spontaneous breaking of a symmetry, a Floquet system has spontaneous dTTSB when the correlation function in the infinite system size limit of local operators whose distance is taken to infinity have a period larger than T for late times. [9]
The traditional study of static phases of matter centered in the properties of observables restricted to the ground- state manifold or thermal states. Nevertheless, while studying many-body localized (MBL) systems it became clear that we should also consider other properties of the energies and eigenstates of the Hamiltonian, thus referred to as eigenstate order. [5,10-12] Among these properties are the entanglement structure of the eigenstates and spectral properties of the energies (e.g. distribution of the difference between consecutive energies).
As pointed out in Huse et al., the disordered Ising chain has (among other phases) an MBL paramagnet (PM) phase and a MBL spin-glass (SG) phase. [10] The eigenstates deep in the MBL PM phase correspond to eigenstates tensor products of spins polarized in the transverse direction. The MBL SG phase corresponds to the Ising dual of the PM, i.e. the domain walls (change in orientation of spins along the interaction direction) are the ones that are frozen. The main physical difference between these two phases is that all eigenstates in the SG phase have long-range magnetic order while the PM do not.
This notion is extended to the Floquet setting by studying the properties of the Floquet eigenstates evolved over a period. [1]
The model consists of a chain of L spin-1/2 degrees of freedom driven by the Hamiltonian such that for a half the period equals a Hamiltonian in the SG phase (Hz) and the other half of period equals a Hamiltonian in the PM phase (Hx), i.e.
with hi(Ji) taken from uniform distributions with mean h(J) and width δh(δ J). Kx/z are small coupling constants corresponding to interactions.
The phase diagram ( Fig. 1) was first obtained in Khemani et al.. [1] The PM and 0-SG are essentially the same as the MBL PM and SG phases of the undriven disordered Ising chain. The new phases are the 0π-Paramagnet (0π-PM) and the π-Spin glass (π-SG). Both phases display spontaneous dTTSB, while only the eigenstates of later also show long-range order. The 0 or π in front of PM and SG denote quasi-degeneracies in the spectrum of quasi-energies, i.e. eigenstates come in pairs that are related by the filling of a mode localized on the ends of the chain with quasi-energy equal (up to exponential accuracy) to 0 or π/T.
Berdanier et al. argued that the phase transitions could be understood in terms of the phase transition of emergent Ising chains: as we approach the phase boundary between two phases we expect that on average the system can be described by islands of spins of said phases. [13] The islands will have the boundary modes of the previous paragraph at quasi-energy 0 or π/T depending on the phase. As the islands have finite size, there will be non-zero couplings between same quasi-energy boundary modes that will form a chain. Each emergent chain has two phases that differ by the presence or absence of quasi-degeneracy. Then the phase diagram can be understood as the phases of two emergent chains at quasi-energies 0 or π/T.
© Vladimir Calvera. 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.
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