Many-Body Localization Phase Transition

Fig. 1: An illustration of the phase diagram on disorder (h) - normalized energy density (ε) space of the disordered Heisenberg chain. The boundary is computed from the finite scalings of various physical quantities with system size L = 14 to 22. (Source: A. Yamamura, after Luitz et al. [2])

The notion of localization was first proposed by P.W. Anderson, which deprives many-body systems of ergodicity and prevents them from thermalizing. This property emerges in systems with quenched disorder. While Anderson initially analyzed a non-interacting single particle system, many-body localization (MBL) is its generalization by allowing interactions in many-body systems. Here we introduce some numerical studies on many-body localization phase transitions, i.e. dynamical phase transitions between the MBL phase and the thermal phase. [1-3]

The Model

We focus on a commonly studied model exhibiting an MBL phase transition, a spin-1/2 Heisenberg chain with a random field in the z-direction.

The fields h_i are i.i.d random variables following a uniform distribution on [-h, h]. This model shows an MBL phase transition at h = h_c = 2 ~ 4 (Fig. 1), i.e., below h_c, the dynamics of the system is governed by the neighborhood coupling showing ergodicity and converging to thermal equilibrium states. In contrast, above h_c, the system shows localization and fails to thermalize due to the strong disorder. The numerical studies of this transition rely on the method of exact diagonalization, with which we can access to system size up to L~22.

Signals of the Phase Transition

Fig. 2: The standard deviation of entanglement entropy of half-chain σS divided by the random pure state value ST. The dashed, solid, and dotted lines represent contributions of cut-to-cut, eigenstate-to-eigenstate, and sample-to-sample variations, respectively. One can see that the sample-to-sample variance has sharper and larger peak at the critical region. Since σS ≤ ST/2 by its definition, the steady growth of its peak without saturation indicates that the system sizes are still in the pre-asymptotic region. (Source: A. Yamamura, after Khemani et al. [3])

One of the principal diagnoses of this unconventional phase transition is in the relationship between adjacent eigenstates and energies. In the MBL phase, the difference of local magnetization between adjacent eigenstates remains finite even with large system sizes, and its energy level statistics follow the Poisson distribution. On the contrary, eigenstates in the thermal phase have local magnetization equivalent to the one of corresponding Gibbs ensembles, and hence its difference between adjacent eigenstates is exponentially small. Also, its level statistics follow those of the Gaussian Orthogonal Ensemble.

Another significant diagnosis can be found in the statistics of the spatial spin correlation function and entanglement entropy of the half-chain. While in the thermal phase the system has the long-range spin correlation and entanglement, in the MBL phase, they decay exponentially. As a consequence, the entanglement entropy shows the volume-law in the former and the area-law in the latter. Moreover, interestingly, the normalized standard deviation of either quantity has its peak at the critical region. Khemani et al. further discuss the variance of entanglement entropy and show that the variance is mostly dominated by the variance across realizations of the disorder, but not by the one across different eigenstates or across different cuts of the chain (Fig. 2). [3]

Since the range of entanglement entropy is between 0 and its value of the random pure state S^T, the maximum possible value of its standard deviation σ_S is S^T/2. Therefore the steady growth of the peak of the dotted line in Fig. 2 (the standard deviation across realizations of the disorder) without saturation indicates that the accessible system sizes are still in the pre-asymptotic regime, far from the fixed point governed by the randomness of the disorder. This idea agrees with the observation that the numerically computed critical exponent in Luitz et al. do not satisfy Harris criterion. [2] Because we rely on exact diagonalization, which has exponential complexity in terms of system size L, it might be challenging to see the asymptotic behavior numerically.

© Atsushi Yamamura. 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.

References

[1] A. Pal and D. A. Huse, "Many-Body Localization Phase Transition," Phys. Rev. B 82, 174411 (2010).

[2] D. J. Luitz, N. Laflorencie, and F. Alet, "Many-Body Localization Edge in the Random-Field Heisenberg Chain," Phys. Rev. B 91, 081103 (2015).

[3] V. Khemani et al., "Critical Properties of the Many-Body Localization Transition," Phys. Rev. X 7, 021013 (2017).