Many-body localization edge in the random-field Heisenberg chain

Abstract

We present a large scale exact diagonalization study of the one dimensional spin 1/2 Heisenberg model in a random magnetic field. In order to access properties at varying energy densities across the entire spectrum for system sizes up to L = 22 spins, we use a spectral transformation which can be applied in a massively parallel fashion. Our results allow for an energy-resolved interpretation of the many body localization transition including the existence of an extensive many-body mobility edge. The ergodic phase is well characterized by Gaussian orthogonal ensemble statistics, volumelaw entanglement, and a full delocalization in the Hilbert space. Conversely, the localized regime displays Poisson statistics, area-law entanglement and non ergodicity in the Hilbert space where a true localization never occurs. We perform finite size scaling to extract the critical edge and exponent of the localization length divergence. The interplay of disorder and interactions in quantum systems can lead to several intriguing phenomena, amongst which the so-called many-body localization has attracted a huge interest in recent years. Following precursors works [1][2][3][4], perturbative calculations [5,6] have established that the celebrated Anderson localization [7] can survive interactions, and that for large enough disorder, many-body eigenstates can also "localize" (in a sense to be precised later) and form a new phase of matter commonly referred to as the many-body localized (MBL) phase.The enormous boost of interest for this topic over the last years can probably be ascribed to the fact that the MBL phase challenges the very foundations of quantum statistical physics, leading to striking theoretical and experimental consequences [8,9]. Several key features of the MBL phase can be highlighted as follows. It is nonergodic, and breaks the eigenstate thermalization hypothesis (ETH) [10][11][12]: a closed system in the MBL phase does not thermalize solely following its own dynamics. The possible presence of a many-body mobility edge (at a finite energy density in the spectrum) indicates that conductivity should vanish in a finite temperature range in a MBL system [5,6]. Coupling to an external bath will eventually destroy the properties of the MBL phase, but recent arguments show that it can survive and be detected using spectral signatures for weak bath-coupling [13]. This leads to the suggestion that the MBL phase can be characterized experimentally, using e.g. controlled echo experiments on reasonably well-isolated systems with dipolar interactions [14][15][16][17]. Another appealing aspect (with experimental consequences for self-correcting memories) is that MBL systems can sustain long-range, possibly topological, order in situations where equilibrated systems would not [18][19][20][21][22]. Finally, a striking phenomenological approach [23] pinpoints that the MBL phase shares properties with integrable systems, with extensive local integrals of mo- ∈ {14, 15, 16, 17, 18, 19, 20, 22}. Red squares correspond to a visual e...

Learning phase transitions from dynamics

“…The boundary we obtain for the MBL transition is at a slightly lower disorder strength than that of the exact diagonalization results in Ref. [61], but agrees well with that obtained using the machine learned entanglement spectra of Ref. [11].…”

“…This Hamiltonian exhibits a transition between a delocalized and a many-body localized state at a critical disorder strength that depends on the energy density of the state under consideration [60][61][62][63]. The dynamics of initial product states of spin polarization differs substantially between the two phases: While spins in the many-body localized phase retain a long-term correlation with their initial configuration, in the delocalized phase this correlation is lost over time as expected from an ergodic system [64][65][66][67].…”

“…Similarly to Ref. [61], we measure the energy density by a parameter ε interpolating between the minimal and maximal eigenenergies E 0 , E max of each disorder realization. For each disorder realization we calculate E 0 , E max , and pick the product state in the S z basis (|↑, ↑, ↓, ↑, .…”

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

“…On the one hand, a mobility edge is clearly seen in all extant numerical studies [106], but this could well be "only" a crossover. On the other hand, the theoretical construction of Refs.…”

Multipoint entanglement in disordered systems

“…On the contrary, MBL phases display a -possibly partially-localized spectrum and a correspondent mobility edge [31][32][33][34][35] . The need for a full knowledge of the entire spectrum makes the study of the MBL phases numerically extremely challenging 36,37 . While these are predominant "paradigms" of transport and entanglement growth more ornate possibilities have been discussed, e.g.…”