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Physical review. B, Condensed matter

Critical behavior at the pseudomagnetic phase transition in disordered Fermi liquids

T. R. Kirkpatrick, D. Belitz

Published Feb 15, 1992• 15 citations
Publisher

Abstract

The phase transition between a Fermi liquid and a pseudomagnetic phase, which has recently been predicted to occur in disordered electron systems, is considered. We determine the exact asymptotic critical behavior at the transition, establish the connection between the integral-equation approach used earlier and a renormalization-group analysis, and determine the qualitative phase diagram. The critical behavior is found to be governed in part by log-log normal terms rather than by power laws. However, the critical region is extremely small, and on realistic scales transport and thermodynamic quantities obey preasymptotic, nonuniversal power laws. We present an extensive discussion of the properties of the renormalization group for this model, and of the physical and experimental implications of our results.

DOI
10.1103/physrevb.45.3187

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The phase transition between a Fermi liquid and a pseudomagnetic phase, which has recently been predicted to occur in disordered electron systems, is considered. We determine the exact asymptotic critical behavior at the transition, establish the connection between the integral-equation approach used earlier and a renormalization-group analysis, and determine the qualitative phase diagram. The critical behavior is found to be governed in part by log-log normal terms rather than by power laws. However, the critical region is extremely small, and on realistic scales transport and thermodynamic quantities obey preasymptotic, nonuniversal power laws. We present an extensive discussion of the properties of the renormalization group for this model, and of the physical and experimental implications of our results.
The metal–insulator transition (MIT) observed in a two-dimensional dilute electron liquid raises the question about the applicability of the scaling theory of disordered electrons, the approach pioneered by Phil Anderson and his collaborators,8 for the description of this transition. In this context, we review here the scaling theory of disordered electrons with electron–electron interactions. We start with the disordered Fermi liquid, and 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. This model has a direct relation with the disordered Fermi liquid, but can be more generally applicable, since it is a minimal model for disordered interacting electrons. The discussion is mostly about the general structure of the theory emphasizing the connection of the scaling parameters entering the NLSM with conservation laws. Next, we show that the MIT, as described by the NLSM with interactions, is a quantum phase transition and identify the parameters needed for the description of the kinetics and thermodynamics of the interacting liquid in the critical region of the transition. Finally, we discuss the MIT observed in Si -MOSFETs. We consider it as an example of the Anderson transition in the presence of the electron interactions. We demonstrate that the two-parameter RG equations, which treat disorder in the one-loop approximation but incorporate the full dependence on the interaction amplitudes, describe accurately the experimental data in Si -MOSFETs including the observed non-monotonic behavior of the resistance and its strong drop at low temperatures. The fact that this drop can be reproduced theoretically, together with the argument that Anderson localization should occur at strong disorder, justified the existence of the MIT within the scaling theory.
Quantum criticality, a manifestation of emergent scale invariance in electron wavefunctions arises from intricate many-body quantum entanglement. One of the natural venues for the criticality is clean undoped Dirac semimetals, known as a marginally-renormalized critical phase. The ground state is only slightly modified from the Slater-type product wavefunction because the scatterings from weak disorder and Coulomb interactions are suppressed. Here, using the renormalization group (RG) analysis, we show that a novel class of quantum criticality appears in Dirac semimetals when the disorder strength becomes sufficiently strong in the presence of Coulomb interactions so that a quantum phase transition from the marginally-renormalized critical phase to a disorder-dominated phase arises. The ground state at the critical point is a quantum critical non-Fermi liquid (NFL), characterized by the properties of strongly entangled low-energy states such as the absence of quasi-particles. Near the critical point, unusually wide temperature regions with NFL behaviors emerge.
The Ising fluid we have investigated is a system of hard spheres which carry an Ising spin. A short range ferromagnetic potential acts between particles. A renormalization group study of the model, based on the hierarchical reference theory of fluids, gives quantitative predictions on the phase diagram and the phase boundaries. The model can show two distinct phase transitions: a magnetic order-disorder transition and a liquid-vapor phase separation which meet at a tricritical point. The critical properties of the magnetic transition have been studied by Monte Carlo simulations at two densities in systems with up to 4000 particles. The locations of the critical points are in good agreement with the theoretical estimates. However, a finite-size scaling analysis shows effective critical exponents which deviate from the expected Ising values. These discrepancies resemble the analogous result of a previous numerical study of Heisenberg fluids. Here they are interpreted as due to crossover phenomena, partly related to Fisher exponent renormalization induced by the presence of noncritical density fluctuations.
We review some of the main applications of the renormalization‐group technique in condensed matter physics. The first relevant example is the description of critical phenomena. Here perturbation theory is affected by singularities, which are a consequence of the long‐ranged character of the dominant fluctuations when approaching criticality. The use of renormalization group allows to sum up these singularities into a power‐law behavior of the physical quantities, which is experimentally observed near a continuous phase transition. The second example is provided by the description of the physical properties of interacting Fermi and Bose systems. Here perturbation theory is affected by infrared divergences within stable liquid phases, due to the presence of massless excitations, in reduced dimensionality. However, the condition of stability of the system implies exact cancellations among the singular contributions, controlled by additional Ward identities, which must be considered besides the standard Ward identities related to the conservation of the total particle and spin density. The combined use of renormalization group and these new Ward identities allows for the closure of the renormalization‐group equations, leading to the description of the asymptotic behavior of the system at low energies.
The authors consider a generic model for a disordered Fermi liquid. By means of an infinite resummation of the loop expansion they show that there is a phase transition involving the spin degrees of freedom which can be identified with a new fixed point in the renormalisation group flow equations. The critical exponents obtained for 2<d<4 are apparently exact. The large-disorder phase is described as an incompletely frozen spin phase with no long-range order and subdiffusive spin transport. The results are in good agreement with experiments on Si:P.
We derive the equilibrium and transport properties of metals using renormalization group equations and finite-size scaling. Particular attention is given to the well-known cases of Fermi and Luttinger liquids. An important subtlety is that the temperature dependence of many transport coefficients is determined by “dangerous” irrelevant operators. We also characterize violations of Fermi or Luttinger liquid behavior in terms of indices, analogous to the critical indices describing phase transitions. We briefly consider the normal-state properties of the cuprates from this standpoint. 2