Session T22: Localization and Disorder in Strongly Correlated Systems

Many-body localization as percolation in d$>$1

Statistical mechanics is the framework that connects thermodynamics to the microscopic world. It hinges on the assumption of equilibration. Isolated quantum systems need not equilibrate; this is the phenomenon of many-body localization (MBL). While a detailed understanding of MBL and the associated delocalization transition is beginning to emerge in one dimension, relatively little is known about higher dimensions. In this work, we present a minimal tractable model for MBL in all spatial dimensions. Specifically, we analyze a disordered Floquet circuit composed of Clifford gates. In one dimension, the system is always localized, while in higher dimensions, it exhibits both delocalized and localized phases. The localized phase consists of well-defined metallic puddles embedded in an insulating matrix. When the puddles percolate, the system delocalizes; this maps the dynamical transition to critical percolation. We also comment on the stability of the phases to generic perturbations away from the Clifford class.   

Quasi Many-Body Localization in Translation Invariant Systems

We examine localization phenomena associated with generic, high entropy, states of a translation invariant, one-dimensional spin ladder. At intermediate time scales, we find slow growth of entanglement entropy consistent with the known phenomenology of many-body localization in disordered, interacting systems. At longer times, however, anomalous diffusion sets in, leading to full spin polarization decay on a time-scale exponential in system size. We identify a single length scale which parametrically controls both the eventual spin transport times and the divergence of the susceptibility to spin glass ordering. We dub this pre-thermal dynamical behavior, quasi many-body localization.   

Signatures of many-body localization transition in entanglement and particle number fluctuation following a global quench

The presence of disorder in a non-interacting system can localize all the energy eigenstates, a well-known phenomena known as Anderson localization. In recent years understanding the effect of disorder on quantum systems in the presence of interactions has gained a lot of interest and has been termed many-body localization (MBL). Effects of interactions show up as the logarithmic growth of entanglement entropy after a global quench. We perform a systematic study of the evolution and saturation of entanglement and particle number fluctuations, and show that they can be used to detect the localization transition. The particle number fluctuations can potentially be measured in experiments, thus giving us the first experimental signature of MBL.   

Energy dependence of localization with interactions and disorder: The generalized inverse participation ratio of an ensemble of two-site Anderson-Hubbard systems

We explore the effect of interactions on novel features found in non-interacting disordered systems. Johri and Bhatt [PRL {\bf 109} 076402 (2012), PRB {\bf 86} 125140 (2012)] showed that for non-interacting particles moving in a disordered potential Lifshitz states lead to a decrease in localization at the band edges. This is reflected in an abrupt decline in the inverse participation ratio following a sharp peak. We consider an ensemble of two-site Anderson-Hubbard systems and study a generalization of the inverse participation ratio applicable to interacting systems. With on-site Coulomb repulsion $U$, two types of resonances can occur: As in the non-interacting case, the potentials at the two sites may be similar. In addition, the potential at one site may differ from its neighbor by $U$. We demonstrate that these two types of resonance and the diversity of transitions in the interacting case result in much more varied dependence of localization on energy, with multiple local minima, including a strong suppression and more structure near the Fermi level. Opportunities for experimental observation are considered.   

Quantum revivals and many-body localization

We show that the interaction-induced dephasing that distinguishes many-body localized phases from Anderson insulators has a striking consequence for quantum revivals in the time evolution of local observables. We examine the magnetization dynamics of a single ``qubit'' spin weakly coupled to an otherwise isolated disordered spin chain and first demonstrate that in the localized regime the spin chain is unable to act as a source of dissipation for the qubit, which therefore retains an imprint of its initial magnetization at infinite time. For Anderson localization, the magnetization exhibits periodic revivals, whose rate is strongly suppressed upon adding interactions after a time scale corresponding to the onset of dephasing. In contrast, the ergodic phase acts as a bath for the qubit, with no revivals visible on the time scales studied. The suppression of quantum revivals provides a quantitative, experimentally observable alternative to entanglement growth as a measure of the ``non-ergodic but dephasing'' nature of many-body localized systems.   

Probing the many-body localization transition with matrix elements of local operators

We propose the statistics of matrix elements of local operators as a new probe of the many-body localized (MBL) phase. Matrix elements of a given local operator $V$ encode many physical properties, such as the response of the system to a local perturbation induced by the action of $V$, spectral functions, and dynamics of the system. The distribution of matrix elements of a local operator between system's eigenstates exhibits qualitatively different behavior in the many-body localized and ergodic phases, allowing for an accurate determination of the two phases. To characterize this distribution, for a given system size $L$, we introduce a parameter $g(L)=\langle\log\frac{V_{i,i+1}}{\Delta}\rangle$, which is a disorder-averaged ratio of the matrix element of operator $V$ between adjacent eigenstates, and $\Delta$ is the level spacing. We find that $g(L)$ decreases with $L$ in the MBL phase, and grows in the ergodic phase. We propose that at the MBL-delocalization transition $g(L)$ is independent of system size, $g(L)=g_c\sim1$, and use this criterion to map out the phase diagram of a disordered 1D XXZ spin-1/2 chain. By studying the scaling of $g(L)$ as a function of energy density, we locate the many-body mobility edge. We discuss implications for delocalization phase transition.   

Strong-randomness phenomena in quantum Ashkin-Teller models

The $N$-color quantum Ashkin-Teller spin chain is a prototypical model for the study of strong-randomness phenomena at first-order and continuous quantum phase transitions. This talk discusses strong-disorder renormalization group approaches to this system in the weak-coupling as well as the strong-coupling regimes. Specifically, we introduce a novel general variable transformation that unifies the treatment of the strong-coupling regime. This allows us to determine the phase diagram for all color numbers $N$, and the critical behavior for all $N \ne 4$. In the case of two colors, $N=2$, a partially ordered product phase separates the paramagnetic and ferromagnetic phases in the strong-coupling regime. This phase is absent for all $N>2$, i.e., there is a direct phase boundary between the paramagnetic and ferromagnetic phases. In agreement with the quantum version of the Aizenman-Wehr theorem, all phase transitions are continuous, even if their clean counterparts are of first order. We also discuss the various critical and multicritical points. They are all of infinite-randomness type, but depending on the coupling strength, they belong to different universality classes.   

Efros-Shklovskii Coulomb gap in the absence of disorder

Certain models of frustrated electron systems have been shown to self-generate glassy behavior, in the absence of disorder. Possible candidate materials contain quarter-filled triangular lattices with long-range Coulomb interactions, as found in the $\theta$-family of organic BEDT-TTF crystals. In disordered insulators with localized electronic states, the so-called Coulomb glass, the single particle excitation spectrum displays the well-known Efros-Shklovskii gap. The same excitation spectrum is investigated in a class of models that display self-generated electronic glassiness, showing pseudogap formation related to the Efros-Shklovskii Coulomb gap. Our study suggests universal characteristics of all electron glasses, regardless of disorder.   

Many-body mobility edge due to symmetry-constrained dynamics and strong interactions

Many-body localization at a finite energy density inhibits thermalization and opens the possibility to study macroscopic quantum phenomena in highly excited states. The system transitions from an ergodic to a nonergodic phase at a critical energy density defined to be the many-body mobility edge. We present a mechanism for the formation of a many-body mobility edge in disordered systems with strong interactions, that satisfy conservation laws. The strong interaction spectrally differentiates eigenstates at positive temperature from those at negative temperature based on correlations, whose quantum dynamics differ dramatically due to the conservation laws. Upon introducing disorder, this difference in the dynamics can lead to an energy-dependent onset of many-body localization, thus leading to the formation of a many-body mobility edge. We exemplify this mechanism in the strongly anisotropic spin-$1/2$ XXZ model in a random field, whose dynamics is constrained by the conservation of total spin projection. We compute a set of diagnostic quantities that verify the presence of a mobility edge in this model. Furthermore, we discuss how introducing correlated disorder in the model can enhance this effect and stabilize the mobility edge itself.   

Lattice aspect ratio effects on transport in two-dimensional quantum percolation

In a previous work [Dillon and Nakanishi, E.Phys.J.B, to be published (2014)], we calculated the transmission coefficient of the two-dimensional quantum percolation problem and concluded that there are three regimes, namely, exponentially localized, power-law localized, and delocalized. However, this remains a controversial problem and works by many others fall either in a group claiming that quantum percolation in 2D is always exponentially localized (as one-parameter scaling would suggest) or in one claiming that there is a transition to a less localized (perhaps power-law localized or delocalized) state. Among the many different types of calculations, it stood out that most works based on two-dimensional strips of highly anisotropic aspect ratios fall in the first group, whereas our previous calculations and most others in the second group were based on isotropic square geometry. In order to understand the deviations between our results and those based on strip geometry, we applied our direct calculation of the transmission coefficient to strips of a wide range of aspect ratios, and report on how aspect ratio influences transmission and localization length.   

Heat diffusion in the disordered Fermi and electron liquids: the role of inelastic processes

We study thermal transport in the disordered Fermi and electron liquids at low temperatures. Gravitational potentials are used as sources for finding the heat density and its correlation function. For a comprehensive study, we extend the renormalization group (RG) analysis developed for electric transport by including the gravitational potentials into the RG scheme. The analysis reveals that for the disordered Fermi liquid the Wiedemann-Franz law remains valid even in the presence of quantum corrections caused by the interplay of diffusion modes and the electron-electron interaction. In the present scheme this fundamental relation is closely connected with a fixed point in the multi-parametric RG flow of the gravitational potentials. For the disordered electron liquid we additionally analyze inelastic processes induced by the Coulomb interaction at sub-temperature energies. While the general form of the correlation function has to be compatible with energy conservation, these inelastic processes are at the origin of logarithmic corrections violating the Wiedemann-Franz law. The interplay of various terms in the heat density-heat density correlation function therefore differs from that for densities of other conserved quantities, such as total number of particles or spin.   

Effect of impurities on strongly-correlated superconductivity with inhomogeneous cluster dynamical mean field theory

We study the problem of an out-of-plane impurity in the square-lattice Hubbard model using inhomogeneous cluster dynamical mean field theory (I-CDMFT). This problem simulates the effect of impurities in superconducting cuprates. The impurity is located at the center of a 2x2 plaquette, surrounded by 8 or 24 other plaquettes without impurities. This system constitutes the repeated unit treated with cluster dynamical mean field theory. We find that the impurity shifts the onset of superconductivity towards higher doping. We study the effect of the impurity on the pseudogap as it appears in the local density of states. We also discuss its effect on the extent of the antiferromagnetic phase.   

Disorder Driven Quantum Criticality in Three Dimensional Dirac Semi-Metals

We study the nature of the quantum phase transition between a three dimensional Dirac semi-metal and a disorder controlled diffusive metal. We analyze a lattice model using numerical and field theoretical methods to explore the phase diagram and quantum critical behavior. We determine the scaling properties of the density of states and various thermodynamic observables for sufficiently large system sizes and extract the relevant critical exponents. As a result, we show the scaling functions obey energy over temperature scaling and the quantum critical point is an interacting fixed point.   

Microscopic driving force in electronic smectic-nematic transition in La$_{1/3}$Ca$_{2/3}$MnO$_{3}$

Electronic liquid crystal (ELC) phases provide unique descriptions to characterize the electronic structures and elucidate the underlying physics in correlated materials from symmetry perspective. Although ELC phases have been proposed to play a key role in interpreting the structure-property relationship in a wide range of correlated materials, the experimental manifestations of the nature of the transition between such phases have been waiting to be explored. Using transmission electron microscopic tools with recently developed techniques, we studied the electronic smectic-nematic phase transition in La$_{1/3}$Ca$_{2/3}$MnO$_{3}$ by monitoring the evolution of charge ordering and orbital ordering superstructures as a function of temperature. We observed that the transition is driven by the formation of defects and electronic phase separation. In addition, we found that charge inhomogeneity is responsible for the electronic smectic-nematic phase transition in this material.