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arXiv:1108.2811 · Physical Review Letters

Theory of a Fermi-Liquid to Non-Fermi-Liquid Quantum Phase Transition in Dimensions d > 1

T. R. Kirkpatrick, D. Belitz

Published Feb 23, 2012• 10 citations
arXivHTMLPDF

Abstract

We develop a theory for a generic instability of a Fermi liquid in dimension d>1 against the formation of a Luttinger-liquid-like state. The density of states at the Fermi level is the order parameter for the ensuing quantum phase transition, which is driven by the effective interaction strength. A scaling theory in conjunction with an effective field theory for clean electrons is used to obtain the critical behavior of observables. In the Fermi-liquid phase the order-parameter susceptibility, which is measurable by tunneling, is predicted to diverge for 1<d<3.

DOI
10.1103/physrevlett.108.086404

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We develop a theory for a generic instability of a Fermi liquid in dimension d>1 against the formation of a Luttinger-liquid-like state. The density of states at the Fermi level is the order parameter for the ensuing quantum phase transition, which is driven by the effective interaction strength. A scaling theory in conjunction with an effective field theory for clean electrons is used to obtain the critical behavior of observables. In the Fermi-liquid phase the order-parameter susceptibility, which is measurable by tunneling, is predicted to diverge for 1<d<3.
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The mechanisms leading to instability of the non-Fermi-liquid state of a Luttinger liquid with two-level impurities are proposed. Since exchange scattering in 1D systems is two-channel scattering in a certain range of parameters, several types of non-Fermi-liquid excitations with different quantum numbers exist in the vicinity of the Fermi level. These excitations include, first, charge density fluctuations in the Luttinger liquid and, second, many-particle excitations due to two-channel exchange interaction, which are associated with band-type as well as impurity fermion states. It is shown that mutual scattering of many-particle excitations of various types leads to the emergence of an additional Fermi-liquid singularity in the vicinity of the Fermi level. The conditions under which the Fermi-liquid state with a new energy scale (which is much smaller than the Kondo temperature) is the ground state of the system are formulated.
We present an analytic theory unraveling the microscopic mechanism of instabilities within interacting DDD-dimensional Fermi liquid. Our model consists of a DDD-dimensional electron gas subject to an instantaneous electron-electron interaction of a finite range exceeding the average interparticle distance. Pair, charge, and spin susceptibilities are evaluated via the one-loop renormalization group theory and via the bosonization approach, giving identical results. In case of a repulsive interaction, we identify an intrinsic Fermi-liquid instability towards insulating spin- and charge-density-wave order when the interaction coupling strength reaches a universal critical value. If both electron and hole pockets of the same size are present, the ground state is an excitonic insulator at arbitrarily small repulsive interaction. If the interaction is attractive, the ground state is a singlet non-BCS superconductor with a uniform condensate. In case if both electron and hole Fermi surfaces are present, we predict an instability towards the interpocket pair-density-wave ordering at the critical coupling. This prediction lends strong theoretical support to the pair-density-wave scenario of superconductivity in cuprate materials. Due to its simple and universal nature, presented microscopic mechanism of intrinsic instabilities of interacting DDD-dimensional Fermi liquids constitutes a solid theoretical ground for understanding quantum phase transitions in a variety of quantum materials, from ultraclean semiconductor quantum wells to high-temperature superconductors.
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We derive the thermodynamic properties of a Fermi gas, deep into the quantum degenerate regime. We show that, if Luttinger's theorem holds, a first-order phase transition occurs in the normal phase as a function of the interaction strength. We also show that a volume change occurs at finite temperatures as we go through a divergence in the s-wave scattering length, in the normal phase. The first-order transition has a critical temperature, T *, above the BCS transition temperature. Also, we show that a paramagnetic system in equilibrium, close to the divergence of the scattering length, on the negative side, screens out any applied magnetic field.
With particular reference to the role of the renormalization group (RG) approach and Ward identities (WI's), we start by recalling some old features of the one-dimensional Luttinger liquid as the prototype of non-Fermi-liquid behavior. Its dimensional crossover to the Landau normal Fermi liquid implies that a non-Fermi liquid, as, e.g., the normal phase of the cuprate high temperature superconductors, can be maintained in d>1 only in the presence of a sufficiently singular effective interaction among the charge carriers. This is the case when, nearby an instability, the interaction is mediated by critical fluctuations. We are then led to introduce the specific case of superconductivity in cuprates as an example of avoided quantum criticality. We will disentangle the fluctuations which act as mediators of singular electron–electron interaction, enlightening the possible order competing with superconductivity and a mechanism for the non-Fermi-liquid behavior of the metallic phase. This paper is not meant to be a comprehensive review. Many important contributions will not be considered. We will also avoid using extensive technicalities and making full calculations for which we refer to the original papers and to the many good available reviews. We will here only follow one line of reasoning which guided our research activity in this field.