Abstract For short-ranged disordered quantum models in one dimension, the many-body-localization is analyzed via the adaptation to the many-body context (Serbyn et al 2015 Phys. Rev . X 5 041047) of the Thouless point of view on the Anderson transition: the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (many-body-localized phase with an area-law entanglement). The central object is thus the level of hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the strong-multifractality spectrum f ( 0 ⩽ α ⩽ 2 ) = α 2 , well known in the context of Anderson localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. Finally, the possibility of a delocalized non-ergodic phase near criticality is discussed.
For short-ranged disordered quantum models in one dimension, the Many-Body-Localization is analyzed via the adaptation to the Many-Body context [M. Serbyn, Z. Papic and D.A. Abanin, PRX 5, 041047 (2015)] of the Thouless point of view on the Anderson transition : the question is whether a local interaction between two long chains is able to reshuffle completely the eigenstates (Delocalized phase with a volume-law entanglement) or whether the hybridization between tensor states remains limited (Many-Body-Localized Phase with an area-law entanglement). The central object is thus the level of Hybridization induced by the matrix elements of local operators, as compared with the difference of diagonal energies. The multifractal analysis of these matrix elements of local operators is used to analyze the corresponding statistics of resonances. Our main conclusion is that the critical point is characterized by the Strong-Multifractality Spectrum
f(0≤α≤2)=2α, well known in the context of Anderson Localization in spaces of effective infinite dimensionality, where the size of the Hilbert space grows exponentially with the volume. As a consequence, the level statistics at criticality is the closest possible to the Poisson statistics of the Localized phase, and the Delocalized phase near the critical point is not ergodic.
As disorder strength increases in quantum many-body systems a new phase of matter, the so-called many-body localization, emerges across the whole spectrum. This transition is energy dependent, a phenomenon known as mobility edge, such that the mid-spectrum eigenstates tend to localize at larger values of disorder in comparison to eigenstates near the edges of the spectrum. Many-body localization becomes more sophisticated in long-range interacting systems. Here, by focusing on several quantities, we draw the phase diagram as a function of disorder strength and energy spectrum, for a various range of interactions. Regardless of the underlying transition type, either second-order or Kosterlitz-Thouless, our analysis consistently determines the mobility edge, i.e., the phase boundary across the spectrum. We show that, long-range interaction enhances the localization effect and shifts the phase boundary towards smaller values of disorder. In addition, we establish a hierarchy among the studied quantities concerning their corresponding transition boundary and critical exponents. Interestingly, we show that deliberately discarding some information of the system can mitigate finite-size effects and provide results in line with the analytical predictions at the thermodynamic limit.
Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If
g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ can detect a QCP provided that the correlation length exponent satisfies $\ensuremath{\nu}<2$. We then show that ${\ensuremath{\chi}}_{F}$ can be used to locate a QCP even if $\ensuremath{\nu}\ensuremath{\ge}2$ if we introduce boundary conditions labeled by a twist angle $N\ensuremath{\theta}$, where
N is the system size. If the QCP lies at
g=0, we find that if
N is kept constant, ${\ensuremath{\chi}}_{F}$ has a scaling form given by ${\ensuremath{\chi}}_{F}\ensuremath{\sim}{\ensuremath{\theta}}^{\ensuremath{-}2/\ensuremath{\nu}}f(g/{\ensuremath{\theta}}^{1/\ensuremath{\nu}})$ if $\ensuremath{\theta}\ensuremath{\ll}2\ensuremath{\pi}/N$. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude
h and period
2q in which $\ensuremath{\nu}=q$, and in a
XY spin-1/2 chain in which $\ensuremath{\nu}=2$. Finally we show that when
q is very large, the model has two additional QCPs at $h=\ifmmode\pm\else\textpm\fi{}2$ which cannot be detected by studying the energy spectrum but are clearly detected by ${\ensuremath{\chi}}_{F}$. The peak value and width of ${\ensuremath{\chi}}_{F}$ seem to scale as nontrivial powers of
q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.
We numerically investigate the distribution of Drude weights
D of many-body states in disordered one-dimensional interacting electron systems across the transition to a many-body localized phase. Drude weights are proportional to the spectral curvatures induced by magnetic fluxes in mesoscopic rings. They offer a method to relate the transition to the many-body localized phase to transport properties. In the delocalized regime, we find that the Drude weight distribution at a fixed disorder configuration agrees well with the random-matrix-theory prediction $P(D)\ensuremath{\propto}{({\ensuremath{\gamma}}^{2}+{D}^{2})}^{\ensuremath{-}3/2}$, although the distribution width $\ensuremath{\gamma}$ strongly fluctuates between disorder realizations. A crossover is observed towards a distribution with different large-
D asymptotics deep in the many-body localized phase, which however differs from the commonly expected Cauchy distribution. We show that the average distribution width $\ensuremath{\langle}\ensuremath{\gamma}\ensuremath{\rangle}$, rescaled by $L\mathrm{\ensuremath{\Delta}},\phantom{\rule{0.28em}{0ex}}\mathrm{\ensuremath{\Delta}}$ being the average level spacing in the middle of the spectrum and
L the systems size, is an efficient probe of the many-body localization transition, as it increases (vanishes) exponentially in the delocalized (localized) phase.
We introduce characterizations for many-body phase transitions between delocalized and localized phases based on the system's sensitivity to boundary conditions. In particular, we change boundary conditions from periodic to antiperiodic and calculate a shift in the system's energy and shifts in the single-particle density-matrix eigenvalues in the corresponding energy window. We employ the typical model for studying MBL, a one-dimensional disordered system of fermions with nearest-neighbor repulsive interaction where disorder is introduced as randomness on on-site energies. By calculating numerically the shifts in the system's energy and eigenvalues of the single-particle density matrix, we observe that in the localized regime, both shifts are vanishing; while in the extended regime, both shifts are on the order of the corresponding level spacing. We also applied these characterizations of the phase transition to the case of having next-nearest-neighbor interactions in addition to the nearest-neighbor interactions and studied its effect on the transition.