Beta‐functions of non‐linear σ‐models for disordered and interacting electron systems

Dell’Anna, Luca

doi:10.1002/andp.201600317

Cited by 6 publications

(5 citation statements)

References 32 publications

(89 reference statements)

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“…In other words, the considered here FNLSM is extension of NLSM for the symmetry classes A, AI, and AII to the case of interacting systems. In general, noninteracting NLSM for the other 7 symmetry classes [98,99] can be extended to include the terms describing electron-electron interactions [100][101][102].…”

Section: Discussion and Outlookmentioning

confidence: 99%

Finkel’stein Nonlinear Sigma Model: Interplay of Disorder and Interaction in 2D Electron Systems

Burmistrov

2019

J. Exp. Theor. Phys.

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In this paper 1 I briefly review recent theoretical results derived within the Finkel'stein nonlinear sigma model approach for description of two-dimensional interacting disordered electron systems. The examples include an electron system with two valleys, electrons in a double quantum well, electrons on the surface of a topological insulator, an electron system with superconducting correlations, and the integer quantum Hall effect.

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Section: Discussion and Outlookmentioning

confidence: 99%

Finkel’stein Nonlinear Sigma Model: Interplay of Disorder and Interaction in 2D Electron Systems

Burmistrov

2019

J. Exp. Theor. Phys.

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“…Generalizations of the e-e interaction description in the NLSM for different symmetry classes were considered both in the Matsubara and the Keldysh techniques in Refs. [122] and [123], respectively. N. 15 Compared to the standard definition of the quantum resistance, this unit contains an additional factor π.…”

Section: Personal Notementioning

confidence: 99%

Scale-dependent theory of the disordered electron liquid

Finkel'stein,

Schwiete

2023

Preprint

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We review the scaling theory of disordered itinerant electrons with e-e interactions. We first show how to adjust the microscopic Fermi-liquid theory to the presence of disorder. Then we describe the non-linear sigma model (NLSM) with interactions (Finkel'stein's model). This model is closely connected to the Fermi liquid, but is more generally applicable, since it can also be viewed as a minimal effective functional describing disordered interacting electrons. Our discussion emphasizes the general structure of the theory, and the connection of the scaling parameters to conservation laws. We then move on to discuss the metal-insulator transition (MIT) in the disordered electron liquid in two and three dimensions. Generally speaking, this MIT is a non-trivial example of a quantum phase transition. The NLSM approach allows to identify the dynamical exponent connecting the spatial and energy scales, which is central for the description of the kinetic and thermodynamic behavior of the system in the critical region of the MIT in three dimensions. In two dimensions, the system can be discussed in terms of a flow in the disorder-interaction phase plane, which is controlled by a fixed point. We demonstrate that the two-parameter RG-equations accurately describe electrons in Si-MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The theory can also be applied to systems with an attractive interaction in the Cooper channel, where it describes the suppression of superconductivity in disordered amorphous films. We extend our discussion to heat transport in the two-dimensional electron liquid. Similar to the electric conductivity, the thermal conductivity also acquires logarithmic corrections induced by the interplay of the electron interaction and disorder. The resulting thermal conductivity can be calculated in the NLSM formalism after introducing so-called gravitational potentials. It may be concluded that the combined effect of disorder and e-e interactions determines all aspects of the physics of itinerant electrons in the presence of disorder.

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“…Recent progress includes understanding the interplay of wave function multifractality and interactions [30,31,32] as well as the effects of disorder on interacting surface states of topological insulators [33]. Yet interacting versions of the nonstandard classes greatly expand the possibilities for understanding critical delocalization and interaction-driven quantum phase transitions, as shown by Dell'Anna [34,35] and others [36,37,38,39,40]. In addition, some nonstandard class models in low dimensions can be solved exactly in the absence of interactions [22,41,42], enabling a nonperturbative starting point (with respect to disorder) for analyzing interaction effects.…”

Section: The Ergodic-mbl Transition In 2d and Nonstandard Classesmentioning

confidence: 99%

Response theory of the ergodic many-body delocalized phase: Keldysh Finkel’stein sigma models and the 10-fold way

2017

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We derive the finite temperature Keldysh response theory for interacting fermions in the presence of quenched short-ranged disorder, as applicable to any of the 10 Altland-Zirnbauer classes in an Anderson delocalized phase with at least a U(1) continuous symmetry. In this formulation of the interacting Finkel'stein nonlinear sigma model, the statistics of one-body wave functions are encoded by the constrained matrix field, while physical correlations follow from the hydrodynamic density or spin response field, which decouples the interactions. Integrating out the matrix field first, we obtain weak (anti)localization and Altshuler-Aronov quantum conductance corrections from the hydrodynamic response function. This procedure automatically incorporates the correct infrared cutoff physics, and in particular gives the Altshuler-Aronov-Khmelnitsky (AAK) equations for dephasing of weak (anti)localization due to electron-electron collisions. We explicate the method by deriving known quantum corrections in two dimensions for the symplectic metal class AII, as well as the spin-SU(2) invariant superconductor classes C and CI. We show that quantum conductance corrections due to the special modes at zero energy in nonstandard classes are automatically cut off by temperature, as previously expected, while the Wigner-Dyson class Cooperon modes that persist to all energies are cut by dephasing. We also show that for short-ranged interactions, the standard self-consistent solution for the dephasing rate is equivalent to a particular summation of diagrams via the self-consistent Born approximation. This should be compared to the corresponding AAK solution for long-ranged Coulomb interactions, which exploits the Markovian noise correlations induced by thermal fluctuations of the electromagnetic field. We discuss prospects for exploring the many-body localization transition as a dephasing catastrophe in short-range interacting models, as encountered by approaching from the ergodic side.

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Beta‐functions of non‐linear σ‐models for disordered and interacting electron systems

Cited by 6 publications

References 32 publications

Finkel’stein Nonlinear Sigma Model: Interplay of Disorder and Interaction in 2D Electron Systems

Finkel’stein Nonlinear Sigma Model: Interplay of Disorder and Interaction in 2D Electron Systems

Scale-dependent theory of the disordered electron liquid

Response theory of the ergodic many-body delocalized phase: Keldysh Finkel’stein sigma models and the 10-fold way

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