Relativistic fermions on a lattice are shown to correspond to the
fluctuations in the localized nonrelativistic fermions. Therefore, in contrast to nonrelativistic case, the relativistic fermions are critical with universal exponents described by the strong coupling limit of the nonrelativistic problem. The fluctuations also describe
anisotropic spin chain at the onset to {\it long range magnetic order} whose universality class is the Ising model. This generalizes the universality in spin models to include multifractal exponents. Finally, analogous to the nonrelativistic case, the relativistic fermions may exhibit ballistic character due to correlations.
We present some exact results for the effect of disorder on the critical properties of an anisotropic
XY spin chain in a transverse field. The continuum limit of the corresponding fermion model is taken and in various cases results in a Dirac equation with a random mass. Exact analytic techniques can then be used to evaluate the density of states and the localization length. In the presence of disorder the ferromagnetic-paramagnetic or Ising transition of the model is in the same universality class as the random transverse field Ising model solved by Fisher using a real-space renormalization-group decimation technique (RSRGDT). If there is only randomness in the anisotropy of the magnetic exchange then the anisotropy transition (from a ferromagnet in the x direction to a ferromagnet in the y direction) is also in this universality class. However, if there is randomness in the isotropic part of the exchange or in the transverse field then in a nonzero transverse field the anisotropy transition is destroyed by the disorder. We show that in the Griffiths' phase near the Ising transition that the ground-state energy has an essential singularity. The results obtained for the dynamical critical exponent, typical correlation length, and for the temperature dependence of the specific heat near the Ising transition agree with the results of the RSRGDT and numerical work.
We theoretically investigate the many-body localization phase transition in a one-dimensional Ising spin chain with random long-range spin-spin interactions, ${V}_{ij}\ensuremath{\propto}{\left|i\ensuremath{-}j\right|}^{\ensuremath{-}\ensuremath{\alpha}}$, where the exponent of the interaction range $\ensuremath{\alpha}$ can be tuned from zero to infinitely large. By using exact diagonalization, we calculate the half-chain entanglement entropy and the energy spectral statistics and use them to characterize the phase transition towards the many-body localization phase at infinite temperature and at sufficiently large disorder strength. We perform finite-size scaling to extract the critical disorder strength and the critical exponent of the divergent localization length. With increasing $\ensuremath{\alpha}$, the critical exponent experiences a sharp increase at about ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.2$ and then gradually decreases to a value found earlier in a disordered short-ranged interacting spin chain. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the system is mostly localized and the increase in the disorder strength may drive a transition between two many-body localized phases. In contrast, for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the transition is from a thermalized phase to the many-body localization phase. Our predictions could be experimentally tested with an ion-trap quantum emulator with programmable random long-range interactions, or with randomly distributed Rydberg atoms or polar molecules in lattices.
We consider the one-dimensional lattice model of interacting fermions with disorder studied previously by Oganesyan and Huse [Phys. Rev. B 75, 155111 (2007)]. To characterize a possible many-body localization transition as a function of the disorder strength
W, we use an exact renormalization procedure in configuration space that generalizes the Aoki real-space renormalization procedure for Anderson localization one-particle models [H. Aoki, J. Phys. C 13, 3369 (1980)]. We focus on the statistical properties of the renormalized hopping
VL between two configurations separated by a distance
L in configuration space (distance being defined as the minimal number of elementary moves to go from one configuration to the other). Our numerical results point toward the existence of a many-body localization transition at a finite disorder strength
Wc. In the localized phase
W>Wc, the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in
L as $(\text{ln}\text{ }{V}_{L}^{typ})\ensuremath{\simeq}\ensuremath{-}\frac{L}{{\ensuremath{\xi}}_{loc}}$ and the localization length diverges as ${\ensuremath{\xi}}_{loc}(W)\ensuremath{\sim}{(W\ensuremath{-}{W}_{c})}^{\ensuremath{-}{\ensuremath{\nu}}_{loc}}$ with a critical exponent of order ${\ensuremath{\nu}}_{loc}\ensuremath{\simeq}0.45$. In the delocalized phase
W<Wc, the renormalized hopping remains a finite random variable as $L\ensuremath{\rightarrow}\ensuremath{\infty}$ and the typical asymptotic value ${V}_{\ensuremath{\infty}}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{\ensuremath{\infty}}}}$ presents an essential singularity $(\text{ln}\text{ }{V}_{\ensuremath{\infty}}^{typ})\ensuremath{\sim}\ensuremath{-}{({W}_{c}\ensuremath{-}W)}^{\ensuremath{-}\ensuremath{\kappa}}$ with an exponent of order $\ensuremath{\kappa}\ensuremath{\sim}1.4$. Finally, we show that this analysis in configuration space is compatible with the localization properties of the simplest two-point correlation function in real space.
Understanding the nature of the transition from the delocalized to the many-body localized (MBL) phase is an important unresolved issue. To probe the nature of the MBL transition, we investigate the universal properties of single-particle excitations produced in highly excited many-body eigenstates of a disordered interacting quantum many-body system. In a class of one-dimensional spinless fermionic models with random disorder, we study the finite-size scaling of the ratio of typical to average values of the single-particle local density of states and the scattering rates across the MBL transition. Our results indicate that the MBL transition in this class of one-dimensional models of spinless fermions is continuous in nature. For various ranges of interactions in the system, the critical exponent $\ensuremath{\nu}$ with which the correlation length $\ensuremath{\xi}$ diverges at the transition point ${W}_{c}, \ensuremath{\xi}\ensuremath{\sim}|W\ensuremath{-}{W}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, satisfies the Chayes-Chayes-Fisher-Spencer (CCFS) bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ where
d is the physical dimension of the system. We also discuss why the critical exponent obtained from finite-size scaling of the conventional diagnostic of many-body localization, the level spacing ratio, strongly violates the CCFS bound while the single-particle density of states and scattering rates are consistent with the CCFS criterion.
Thermal and many-body localized phases are separated by a dynamical phase transition of a new kind. We analyze the distribution of off-diagonal matrix elements of local operators across this transition in two different models of disordered spin chains. We show that the behavior of matrix elements can be used to characterize the breakdown of thermalization and to extract the many-body Thouless energy. We find that upon increasing the disorder strength the system enters a critical region around the many-body localization transition. The properties of the system in this region are: (i) the Thouless energy becomes smaller than the level spacing, (ii) the matrix elements show critical dependence on the energy difference, and (iii) the matrix elements, viewed as amplitudes of a fictitious wave function, exhibit strong multifractality. This critical region decreases with the system size, which we interpret as evidence for a diverging correlation length at the many-body localization transition. Our findings show that the correlation length becomes larger than the accessible system sizes in a broad range of disorder strength values and shed light on the critical behavior near the many-body localization transition.