Theory of the Many-Body Localization Transition in One-Dimensional Systems

Abstract

We formulate a theory of the many-body localization transition based on a novel real-space renormalization group (RG) approach. The results of this theory are corroborated and intuitively explained with a phenomenological effective description of the critical point and of the "badly conducting" state found near the critical point on the delocalized side. The theory leads to the following sharp predictions: (i) The delocalized state established near the transition is a Griffiths phase, which exhibits subdiffusive transport of conserved quantities and sub-ballistic spreading of entanglement. The anomalous diffusion exponent α < 1=2 vanishes continuously at the critical point. The system does thermalize in this Griffiths phase. (ii) The many-body localization transition is controlled by a new kind of infinite-randomness RG fixed point, where the broadly distributed scaling variable is closely related to the eigenstate entanglement entropy. Dynamically, the entanglement grows as ∼ log t at the critical point, as it does in the localized phase. (iii) In the vicinity of the critical point, the ratio of the entanglement entropy to the thermal entropy and its variance (and, in fact, all moments) are scaling functions of L=ξ, where L is the length of the system and ξ is the correlation length, which has a power-law divergence at the critical point.

Rare‐region effects and dynamics near the many‐body localization transition

“…The definition of Griffiths regions in a system close to the MBL critical point requires more care. In this regime, even the thermal bulk only appears thermal when probed on lengths scales larger than the correlation length ξ , expected to diverge at the critical point, and on timescales that are long compared with its correlation time τ (ξ ) [which is thought to go as exp(ξ/ζ ) [58,67]]. Hence the boundary between the Griffiths region and the thermal bulk is smeared over the scale ξ and for isolated Griffiths regions to be well defined they must be of size R 0 ξ .…”

“…Thus, the construction of the MBL critical point is similar to an iterated version of the Griffiths analyses we have gone through here. This is the basic idea underlying the strong-randomness renormalization-group treatments of the MBL transition [58,67,100].…”

“…In the thermal phase of a system with quenched randomness there may be rare regions that are more disordered and thus locally MBL. Ultimately, these rare regions do thermalize due to being surrounded by thermal regions, which act as a "bath," but their slow dynamics can dominate the long-time or low-frequency dynamics of such a system [58][59][60]. These effects are most dramatic in one dimension, where such inclusions of the MBL phase can bottleneck transport, making the diffusion constant vanish even within the thermal phase, as has been seen in some numerical studies [61,62].…”

“…Therefore it may suffice to (i) count inclusions, and (ii) understand the influence of a single, isolated wrong-phase inclusion as a function of its size and properties. For MBL inclusions in the thermal phase, both steps are relatively straightforward: assuming spatially uncorrelated disorder and a one-dimensional geometry, the density of localized inclusions of linear dimension L is exponentially small in L. Further, the transit time t for information or particles to traverse a localized inclusion of length L can be seen to be t(L, ζ ) ∼ e L/ζ , with a finite decay length ζ [58,59]. Thus, inclusions with transit times ∼ t have length L ∝ ζ log t, and the density of such inclusions falls off as a power of t with an exponent that varies throughout the thermal phase; as we discuss in Sec.…”

Low-frequency conductivity in many-body localized systems

“…Instead, an isolated system in the MBL phase is a "quantum memory", retaining some local memory of its local initial conditions at arbitrarily late times [9][10][11][12][13][14][15][16][17][18]. The existence of the MBL phase can be proved with minimal assumptions [20]; many of its properties are phenomenologically understood [10,11,16], and some cases can be explored using strong-randomness renormalization group methods [9,[21][22][23]. While the eigenstate properties of MBL systems are in some respects similar to those of noninteracting Anderson insulators, there are important differences in the dynamics, such as the logarithmic spreading of entanglement in the MBL phase [6,9,11,18,19,24,25].…”