Fixed Points of Wegner-Wilson Flows and Many-Body Localization

Abstract

Many-body localization (MBL) is a phase of matter that is characterized by the absence of thermalization. Dynamical generation of a large number of local quantum numbers has been identified as one key characteristic of this phase, quite possibly the microscopic mechanism of breakdown of thermalization and the phase transition itself. We formulate a robust algorithm, based on Wegner-Wilson flow (WWF) renormalization, for computing these conserved quantities and their interactions. We present evidence for the existence of distinct fixed point distributions of the latter: a Gaussian white-noise-like distribution in the ergodic phase, a 1=f law inside the MBL phase, and scale-free distributions in the transition regime. DOI: 10.1103/PhysRevLett.119.075701 Recent progress on the theory of many-body localization (MBL) demonstrates clearly that the conventional quantum statistical description of interacting many-body problems is incomplete. Concrete analytic [1], numerical [2][3][4][5], and mathematical [6,7] results establish the existence and robustness of many-body localized phases in sufficiently strongly disordered and/or low-dimensional interacting models at finite extensive entropy. While the understanding of the transition between thermal and MBL phases is only beginning to emerge [8][9][10][11][12], several distinct new directions of inquiry related to MBL and the fundamental issue of ergodicity in quantum many-body systems have taken shape. These include the interplay of MBL with spontaneous symmetry breaking and topological order [13][14][15][16], selflocalization (glassiness) in translationally invariant quantum systems [17][18][19][20], and MBL in driven systems [21][22][23]. MBL has also stimulated considerable progress in developing tools for describing excited eigenstates of many-body systems [12,[24][25][26][27][28]. MBL has been realized in recent experiments [29,30] and may also have important implications for quantum engineering problems, e.g., quantum computing [31][32][33][34][35].One natural route to the breakdown of thermalization is via proliferation of a large number of conserved quasilocal quantities. The extreme version of such a proposal has gained considerable traction as a model phenomenology [36] of the so-called fully MBL regime, where the entire many-body spectrum is localized. Consider a generic system, e.g., the n-site spin-1=2 random-field Heisenberg chain [see Eq. (7)], which is diagonalized by a (nonunique) unitary matrix U.

Efficient Representation of Fully Many-Body Localized Systems Using Tensor Networks

“…The algorithm to construct such a spectral tensor network proposed by Pekker and Clark does not scale efficiently with system size [46]. Constructing the unitary using a Wegner-Wilson flow approach also appears to be limited to small system sizes [47]. A proposal with an efficient scaling was given by Pollmann et al using stacked layers of unitaries, i.e., a quantum circuit, by minimizing the fluctuations in the total energy [48].…”

Finite-size scaling with respect to interaction and disorder strength at the many-body localization transition

“…We remark that renormalization group arguments have been applied to study the asymptotic behavior of entanglement entropy [49]. We also remark that the extra exponent µ is estimated less than 1 in the gapless regime.…”

One-particle density matrix occupation spectrum of many-body localized states after a global quench

“…The eigenstates are product states of these quasiparticles and therefore satisfy an area law of entanglement [12,19,20], necessarily violating the eigenstate thermalization hypothesis [21][22][23]. The construction of the conserved quantities (commonly referred to as l-bits) is extensively studied [15][16][17][18][24][25][26][27][28][29][30][31]. As in the Fermi liquid, the MBL Hamiltonian is diagonal in the quasiparticle basis, but contains quasiparticle density-density interaction terms, which are absent in the Anderson insulator [10,13].…”