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| Fig. 1: A qualitative depiction of the full phase diagram for the disordered transverse field Ising model. [1] The MBL phase extends beyond the spin glass phase into the paramagnetic phase before succumbing to thermalization. (Source: B. Marsh, after Kjall et al. [1]) |
Disordered energy landscapes give rise to distinctive phases of matter. The spin glass, a network of frustrated spins with random bonds, exhibits a low temperature, non-ergodic phase with all spins ''frozen'' in a complex metastable state. Many-body localized (MBL) quantum systems defy the usual expectation of interacting many-body systems to reach a state of thermal equilibrium, most often due to a disordered potential. The spin glass and MBL phases have qualitative similarities; both are characterized by a breakdown of the ergodic behavior on which the foundations of statistical mechanics rest. Are these two phases effectively equivalent in a model of a quantum spin glass? Here we explore the relationship between the phases in the explicit models of Kjall et al. and Laumann et al. [1, 2]. Both models show that the spin glass phase is accompanied by MBL, but that the MBL phase persists beyond the limit of the glass phase into paramagnetic phases.
The first system we review was studied by Kjall et al. and is composed of a 1D lattice of length L, with spin-1/2 degrees of freedom on each site [1]. The spins experience Ising interactions between nearest neighbors and next-nearest neighbors. To endow the model with quantum dynamics, a uniform transverse field is added. The Hamiltonian describing the system is
The nearest neighbor interactions have the form Ji = J + δJi, where each δJi is a random variable chosen from a uniform distribution on the interval [-δJ,δJ]. This term provides the disorder that is crucial to the MBL phase. In the current study, the dimensionless parameters were fixed to J = 1 and h/2 = J2 = 0.3. Note that there is a global Z2 symmetry corresponding to the parity operator P = ∏i σix which flips every spin.
To build intuition for the dynamics of the system, first consider the limit of δJ = J2 = 0, being the standard transverse field Ising model. There is a ferromagnetic phase (J > h) characterized by the ground state with all spins down, and a quantum paramagnet phase with all spins aligned with the transverse field. In the ferromagnetic phase, excitations of the system look like domain walls separating ferromagnetic regions. In the absence of disorder, states with the same number of domain walls are degenerate, thus allowing for delocalized superpositions of states with domain walls at any location. The effect of on-site disorder, setting δJ > 0, is to exponentially localize the location of domain walls in the chain. Finally, turning on the next-nearest neighbor interaction J2 > 0 introduces repulsive interactions between domain walls. This interaction, in contrast to on-site disorder, will drive thermalization.
The full phase diagram for the model is plotted in Fig. 1 as a function of the disorder strength δ J and energy density ε = 2(E - Emin)/(Emax - Emin). Three distinct phases emerge:
ETH paramagnet: In this phase the eigenstates are delocalized, the ETH is obeyed, and there is no spin glass order.
MBL paramagnet: In this phase, the disorder has become strong enough to drive the system through the MBLD transition, so that eigenstates are localized. However, the disorder is not sufficient to drive the spin glass transition.
MBL spin glass: For still stronger disorder, the system enters a fully disordered phase in which the eigenstates are both localized and show spin glass order.
Thus, we find that the MBLD transition and spin glass transition are indeed distinct, with the MBLD transition occuring before the spin glass transition as the disorder is increased.
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| Fig. 2: The qualitative phase diagram of the QREM. Dotted lines indicate first order phase transitions computed by Goldschmidt, while the solid black line depicts a distinct MBLD transition found by Laumann et al. [2,3] (Source: B. Marsh, after Laumann et al. [2]) |
The classical random energy model (REM) is in some sense the most disordered system that can be imagined. In words, the model consists of assigning a random energy to every configuration of the system. The energy assigned to a particular state is completely uncorrelated from the state itself and from any other energy level assigned to any other state. Due to the complete lack of correlations, the system becomes analytically simple to work with, and a direct calculation of thermodynamic quantities is very tractable. As we will derive below, the model shows a paramagnetic phase at high temperature and a "frozen" spin glass phase below a critical glass transition temperature.
The quantum random energy model (QREM) is achieved by adding a uniform transverse field the classical REM. The Hamiltonian describing the system, including the terms endowing the classical model with quantum dynamics, is
The classical part E({σiz}) is a random operator of size 2N that is diagonal in the σz basis. The diagonal elements are independent and identically distributed random variables with Gaussian distribution function
normalized so that the energy eigenvalues of the Hamiltonian scale extensively. The transverse field proportional Γ introduces quantum dynamics that mix classical states.
Laumann et al. construct the full phase diagram using spectral statistics of the Hamiltonian to mark the MBLD transition. The phase diagram is illustrated in Fig. 2 and, as was seen in the 1D transverse field Ising model, a distinct MBLD transition is seen that does not coincide with the spin glass transition. The four phases are characterized as follows.
ETH quantum paramagnet: With a strong transverse field, the spins simply align with the field. Excitations above the ground state are delocalized, and the ETH is satisfied. The zero-temperature quantum phase transition to this phase occurs at a critical field strength Γc = J log 2.
ETH classical paramagnet: With a weak transverse field, and at temperatures T>Tc, the system can enter a classical paramagnetic phase in which exponentially many states contribute to the free energy, as in a classical paramagnet. In this phase, due a combination of the transverse field mixing classical states and sufficiently high energies, the disorder is not sufficient to induce localization. That is, the eigenstates whose energies correspond to temperatures in this region obey the ETH.
MBL classical paramagnet: With a still weaker transverse field or lower temperature, the system can remain in the paramagnetic phase, but with disorder strong enough to induce localization. That is, eigenstates whose energies correspond to temperatures in this region are found to be localized.
MBL spin glass: For temperatures below the critical temperature Tc and transverse fields below the critical field Γc, the spin glass phase emerges. This phase is characterized by a small number of nearly degenerate ground states dominating the free energy. Low energy eigenstates falling in this region are indeed localized.
Two models that show both spin glass and MBL phases were reviewed. In both cases, the MBLD transition was found to be distinct from the spin glass phase transition. Thus, despite qualitative similarities, the two phases do not necessarily come hand in hand. However, a feature that was common to both models was that the spin glass phase was always accompanied by MBL, while the converse was not true. This motivates the question of whether any model of a quantum spin glass must show MBL in the glassy phase. This would be an intuitive and perhaps unsurprising result, but we leave the proof of a general statement to the interested reader.
© Brendan Marsh. 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] J. A. Kjall, J. H. Bardarson, and F. Pollmann, "Many-Body Localization in a Disordered Quantum Ising Chain." Phys. Rev. Lett. 113, 107204 (2014).
[2] R. A. Laumann, A. Pal, and A. Scardicchio, "Many-Body Mobility Edge in a Mean-Field Quantum Spin Glass," Phys. Rev. Lett. 113, 200405 (2014).
[3] Y. Y. Goldschmidt, "Solvable Model of the Quantum Spin Glass in a Transverse Field," Phys. Rev. B 41, 4858 (1990).