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arXiv:1911.08503 · Journal of Statistical Mechanics Theory and Experiment

Non-Landau quantum phase transitions and nearly-marginal non-Fermi liquid

Yichen Xu, Hao Geng, Xiao-Chuan Wu, Chao‐Ming Jian, Cenke Xu

Published Jul 1, 2020• 1 citation
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Abstract

Abstract Non-Fermi liquid and unconventional quantum critical points (QCP) with strong fractionalization are two exceptional phenomena beyond the classic condensed matter doctrines, both of which could occur in strongly interacting quantum many-body systems. This work demonstrates that using a controlled method one can construct a non-Fermi liquid within a considerable energy window based on the unique physics of unconventional QCPs. We will focus on the ‘nearly-marginal non-Fermi liquid’, defined as a state whose fermion self-energy scales as Σ f (i ω ) ∼ i sgn( ω )| ω | α with α close to 1 in a considerable energy window. The nearly-marginal non-fermi liquid is obtained by coupling an electron fermi surface to unconventional QCPs that are beyond the Landau’s paradigm. This mechanism relies on the observation that the anomalous dimension η of the order parameter of these unconventional QCPs can be close to 1, which is significantly larger than conventional Landau phase transitions, for example the Wilson–Fisher fixed points. The fact that η ∼ 1 justifies a perturbative renormalization group calculation proposed earlier. Various candidate QCPs that meet this desired condition are proposed.

DOI
10.1088/1742-5468/ab99a0

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Abstract Non-Fermi liquid and unconventional quantum critical points (QCP) with strong fractionalization are two exceptional phenomena beyond the classic condensed matter doctrines, both of which could occur in strongly interacting quantum many-body systems. This work demonstrates that using a controlled method one can construct a non-Fermi liquid within a considerable energy window based on the unique physics of unconventional QCPs. We will focus on the ‘nearly-marginal non-Fermi liquid’, defined as a state whose fermion self-energy scales as Σ f (i ω ) ∼ i sgn( ω )| ω | α with α close to 1 in a considerable energy window. The nearly-marginal non-fermi liquid is obtained by coupling an electron fermi surface to unconventional QCPs that are beyond the Landau’s paradigm. This mechanism relies on the observation that the anomalous dimension η of the order parameter of these unconventional QCPs can be close to 1, which is significantly larger than conventional Landau phase transitions, for example the Wilson–Fisher fixed points. The fact that η ∼ 1 justifies a perturbative renormalization group calculation proposed earlier. Various candidate QCPs that meet this desired condition are proposed.
In this thesis we study the scaling properties of unconventional metals that arise at quantum critical points using low-energy effective field theories. Due to high rate of scatterings between electrons and critical fluctuations of the order parameter associated with spontaneous symmetry breaking, Landau’s Fermi liquid theory breaks down at the critical points. The theories that describe these critical points generally flow into strong coupling regimes at low energy in two space dimensions. Here we develop and utilize renormalization group methods that are suitable for the interacting non-Fermi liquids. We focus on the critical points arising at excitonic, and commensurate spin and charge density wave transitions. By controlled analyses we find stable non-Fermi liquid and marginal Fermi liquid states, and extract the scaling behaviour. The field theories for the non-Fermi liquids are characterized by symmetry groups, local curvature of the Fermi surface, the dispersion of the order parameter fluctuations, and dimensions of space and Fermi surface.
We develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector Q⃗=0⃗\vec{Q}=\vec{0}Q​=0. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs ΛΛΛ and ΛfΛ_fΛf​, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector Q⃗Λ\vec{Q}_ΛQ​Λ​. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with Q⃗=0⃗\vec{Q}=\vec{0}Q​=0 and development of a first-order phase transition or a phase characterized by Q⃗≠0⃗\vec{Q}\neq\vec{0}Q​=0.
We examine the nature of phase transitions occurring in strongly correlated Fermi systems at the quantum critical point (QCP) associated with a divergent effective mass. Conventional scenarios for the QCP involving collective degrees of freedom are shown to have serious shortcomings. Working within the original Landau quasiparticle picture, we propose an alternative topological scenario for the QCP in systems that obey standard Fermi-liquid (FL) theory in advance of the QCP. Applying the technique of Poincar\'e mapping, we analyze the sequence of iterative maps generated by the Landau equation for the single-particle spectrum at zero temperature. It is demonstrated that the Fermi surface is subject to rearrangement beyond the QCP. If the sequence of maps converges, a multiconnected Fermi surface is formed. If it fails to converge, the Fermi surface swells into a volume that provides a measure of entropy associated with formation of an exceptional state of the system characterized by partial occupation of single-particle states and dispersion of their spectrum proportional to temperature. Based on this dual scenario, the thermodynamics of Fermi systems beyond the QCP exhibits striking departures from the predictions of standard FL theory. Mechanisms for the release of the entropy excess of the exceptional state are discussed.
One of the most successful theories of a non-Fermi-liquid metallic state is the composite Fermi-liquid (CFL) theory of the half-filled Landau level. In this paper, we study continuous quantum phase transitions out of the CFL state and into a Landau Fermi liquid, in the limit of no disorder and fixed particle number. This transition can be induced by tuning the bandwidth of the Landau level relative to the interaction energy, for instance through an externally applied periodic potential. We find a transition to the Landau Fermi liquid through a gapless Mott insulator with a Fermi surface of neutral fermionic excitations. In the presence of spatial symmetries, we also find a direct continuous transition between the CFL and the Landau Fermi liquid. The transitions have a number of characteristic observable signatures, including the presence of two crossover temperature scales, resistivity jumps, and vanishing compressibility. When the composite fermions are paired instead, our results imply quantum critical points between various non-Abelian topological states, including the $\ensuremath{\nu}=1/2$ Moore-Read Pfaffian [Ising $\ifmmode\times\else\texttimes\fi{}$ U(1)U(1)U(1) topological order], a version of the Kitaev B phase (Ising topological order), and paired electronic superconductors. To study such transitions, we use a projective construction of the CFL, which goes beyond the conventional framework of flux attachment to include a broader set of quantum fluctuations. These considerations suggest a possible route to fractionalized Mott insulators by starting with fractional quantum Hall states and tuning the Landau-level bandwidth.
Opportunities for topological phase transitions in strongly correlated Fermi systems near a quantum critical point are explored as an alternative to collective scenarios for experimentally observed departures from standard Fermi-liquid behavior. Attention is focused on a quantum critical point at which the effective mass is divergent due to vanishing of the quasiparticle group velocity at the Fermi surface. Working within the original Landau quasiparticle theory, it is demonstrated that the quasiparticle picture can remain meaningful beyond the quantum critical point through rearrangements of the unstable normal Fermi surface and quasiparticle spectrum. Two possibilities emerge at zero temperature, depending on whether the quasiparticle interaction is regular or singular at zero momentum transfer. In the regular case, one type of topological phase transformation leads to a state with a multiconnected Fermi surface. In the singular case, another type of topological phase transition leads to an exceptional state containing a fermion condensate – the Fermi surface swells into a volume in momentum space, within which partial occupation prevails and quasiparticle energies are pinned to the chemical potential. As the temperature increases from zero to a characteristic value Tm, a crossover can occur from the state with multiple Fermi surfaces to that containing a fermion condensate.
We consider several types of quantum critical phenomena from finite-density gauge-gravity duality which to different degrees lie outside the Landau-Ginsburg-Wilson paradigm. These include: (i) a ``bifurcating'' critical point, for which the order parameter remains gapped at the critical point, and thus is not driven by soft order parameter fluctuations. Rather it appears to be driven by ``confinement'' which arises when two fixed points annihilate and lose conformality. On the condensed side, there is an infinite tower of condensed states and the nonlinear response of the tower exhibits an infinite spiral structure; (ii) a ``hybridized'' critical point which can be described by a standard Landau-Ginsburg sector of order parameter fluctuations hybridized with a strongly coupled sector; (iii) a ``marginal'' critical point which is obtained by tuning the above two critical points to occur together and whose bosonic fluctuation spectrum coincides with that postulated to underly the ``Marginal Fermi Liquid'' description of the optimally doped cuprates.