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arXiv:1802.08393 · Physical review. B./Physical review. B

Boltzmann transport theory for many-body localization

Jae-Ho Han, Ki‐Seok Kim

Published Jun 25, 2018• 9 citations
arXivHTMLPDF

Abstract

We investigate a many-body localization transition based on Boltzmann transport theory. Introducing weak-localization corrections into a Boltzmann equation, Hershfield and Ambegaokar rederived the Wolfle-Vollhardt self-consistent equation for the diffusion coefficient [Phys. Rev. B 34, 2147 (1986)]. We generalize this Boltzmann equation framework, introducing electron-electron interactions into the Hershfield-Ambegaokar Boltzmann transport theory based on the study of Zala-Narozhny-Aleiner [Phys. Rev. B 64, 214204 (2001)]. Here, not only Altshuler-Aronov corrections but also dephasing effects are taken into account. As a result, we obtain a self-consistent equation for the diffusion coefficient in terms of the disorder strength and temperature, which extends the Wolfle-Vollhardt self-consistent equation in the presence of electron correlations. Solving our self-consistent equation numerically, we find a many-body localization insulator-metal transition, where a metallic phase appears from dephasing effects dominantly instead of renormalization effects at high temperatures. Although this mechanism is consistent with that of recent seminal papers [Ann. Phys. (NY) 321, 1126 (2006); Phys. Rev. Lett. 95, 206603 (2005)], we find that our three-dimensional metal-insulator transition belongs to the first-order transition, which differs from the Anderson metal-insulator transition described by the Wolfle-Vollhardt self-consistent theory. We speculate that a bimodal distribution function for the diffusion coefficient is responsible for this first-order phase transition.

DOI
10.1103/physrevb.97.214206

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We investigate a many-body localization transition based on Boltzmann transport theory. Introducing weak-localization corrections into a Boltzmann equation, Hershfield and Ambegaokar rederived the Wolfle-Vollhardt self-consistent equation for the diffusion coefficient [Phys. Rev. B 34, 2147 (1986)]. We generalize this Boltzmann equation framework, introducing electron-electron interactions into the Hershfield-Ambegaokar Boltzmann transport theory based on the study of Zala-Narozhny-Aleiner [Phys. Rev. B 64, 214204 (2001)]. Here, not only Altshuler-Aronov corrections but also dephasing effects are taken into account. As a result, we obtain a self-consistent equation for the diffusion coefficient in terms of the disorder strength and temperature, which extends the Wolfle-Vollhardt self-consistent equation in the presence of electron correlations. Solving our self-consistent equation numerically, we find a many-body localization insulator-metal transition, where a metallic phase appears from dephasing effects dominantly instead of renormalization effects at high temperatures. Although this mechanism is consistent with that of recent seminal papers [Ann. Phys. (NY) 321, 1126 (2006); Phys. Rev. Lett. 95, 206603 (2005)], we find that our three-dimensional metal-insulator transition belongs to the first-order transition, which differs from the Anderson metal-insulator transition described by the Wolfle-Vollhardt self-consistent theory. We speculate that a bimodal distribution function for the diffusion coefficient is responsible for this first-order phase transition.
Transport process of non-interacting electrons is reformulated, in view of the exact treatment of interacting fermions reported by the present authors in this conference. It makes correspondence to the Boltzmann theory, covering the weakly localized regime, and also eliminates the hidden ultraviolet divergence by incorporating the particle-particle and hole-hole pairs. A mass renormalization of purely two-particle nature appears in the transport equation, giving rise to the conductivity of the Drude form. Taking the maximally crossed diagrams, the 2D conductivity is shown to vanish in the elastic scattering limit, while the distribition function preserves its sharp peak and the transport mass reduces to the free-electron value. The description differs from the self-consistent theory by Vollhardt and Wölfle, in which the Bethe-Salpeter structure of the scattering vertex is abandoned in favour of the time-reversal symmetry so the correspondence to the Boltzmann theory is lost.
A description of quantum transport for electrons in the presence of a random impurity potential is provided in terms of a kinetic equation, which is obtained from the diagrammatic structure for the linear response functions. When the electron-electron interaction is neglected, the kinetic equation is of the Boltzmann type, with an effective scattering kernel that signals localization effects. With the further inclusion of the interaction, a suitable distribution function for the energy of the quasiparticles can be introduced even in the presence of strong disorder. In this case the kinetic equation has the diffusive form, with diffusion coefficient and Landau interaction parameters that get renormalized by the interplay of disorder and interaction. The quasiparticles kinetic equation is also shown to contain all known results for the hydrodynamic correlation functions.
The DOS, the dynamic (optical) conductivity, and the phase diagram of a strongly correlated and strongly disordered paramagnetic Anderson-Hubbard model are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are taken into account by the DMFT, and disorder is taken into account via an appropriate generalization of the self-consistent theory of localization. The DMFT effective single-impurity problem is solved by a numerical renormalization group (NRG); we consider the three-dimensional system with a semielliptic DOS. The correlated metal, Mott insulator, and correlated Anderson insulator phases are identified via the evolution of the DOS and dynamic conductivity, demonstrating both the Mott-Hubbard and Anderson metal-insulator transition and allowing the construction of the complete zero-temperature phase diagram of the Anderson-Hubbard model. Rather unusual is the possibility of a disorder-induced Mott insulator-to-metal transition.
The interplay between Mott and Anderson routes to localization in disordered interacting systems gives rise to different transitions and transport regimes. Here, we investigate the phase diagram at finite temperatures using dynamical mean-field theory combined with typical medium theory, which is an effective theory of the Mott-Anderson metal-insulator transition. We mainly focus on the properties of the coexistence region associated with the Mott phase transition. For weak disorder, the coexistence region is found to be similar to that in the clean case. However, as we increase disorder, Anderson localization effects are responsible for shrinking the coexistence region, and at sufficiently strong disorder (approximately equal to twice the bare bandwidth) it drastically narrows, the critical temperature Tc{T}_{c}Tc​ abruptly goes to zero, and we observe a phase transition in the absence of a coexistence of the metallic and insulating phases. In this regime, the effects of interaction and disorder are found to be of comparable importance for charge localization.
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 WWW, 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{V}_{L}VL​ between two configurations separated by a distance LLL 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{W}_{c}Wc​. In the localized phase W>WcW>{W}_{c}W>Wc​, the typical renormalized hopping ${V}_{L}^{typ}\ensuremath{\equiv}{e}^{\overline{\text{ln}\text{ }{V}_{L}}}$ decays exponentially in LLL 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<WcW<{W}_{c}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.
Effects of electron correlations on Anderson insulators have been one of the central themes for recent two decades, suggesting that the Anderson insulating phase turns into a novel insulating state referred to as many body localization (MBL). However, the role of spin degrees of freedom in this dynamical phase transition still remains unclarified as a function of the interaction strength. In this study, we perform real-space spin-resolved Hartree-Fock-Anderson simulations to investigate metal-insulator transitions above a critical disorder strength in three spatial dimensions, where all single-particle states are Anderson-localized without interactions. Here, relatively weak correlations below the Mott regime are taken into account in the mean-field fashion but disorder effects are introduced essentially exactly. We find two types of single-particle mobility edges, where the multifractal spectrum of the interaction-driven low-energy mobility edge deviates from that of the high-energy one smoothly connected with the multifractal spectrum of the metal-insulator transition without interactions. We show that the weakly interacting insulating phase remains to be a paramagnetic Anderson insulating state up to the temperature of the order of the band width. On the other hand, we uncover that the relatively strongly interacting insulating phase still below the Mott regime is ferromagnetic, which turns into a ferromagnetic metallic state at a critical temperature much lower than the order of the bandwidth. Based on all these results, we propose a quantum phase transition from a paramagnetic Anderson insulating state to a ferromagnetic MBL insulating phase via an intermediate ferromagnetic metallic state, which intervenes between these two insulators at the Fermi energy.