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arXiv:1711.10514 · Physical review. B./Physical review. B

Non-Fermi liquid at the FFLO quantum critical point

Dimitri Pimenov, Ipsita Mandal, Francesco Piazza, Matthias Punk

Published Jul 16, 2018• 22 citations
arXivHTMLPDF

Abstract

When a 2D superconductor is subjected to a strong in-plane magnetic field, Zeeman polarization of the Fermi surface can give rise to inhomogeneous FFLO order with a spatially modulated gap. Further increase of the magnetic field eventually drives the system into a normal metal state. Here, we perform a renormalization group analysis of this quantum phase transition, starting from an appropriate low-energy theory recently introduced in Phys. Rev. B 93, 085112 (2016). We compute one-loop flow equations within the controlled dimensional regularization scheme with fixed dimension of Fermi surface, expanding in $\ensuremath{\epsilon}=5/2\ensuremath{-}d$. We find a new stable non-Fermi-liquid fixed point and discuss its critical properties. One of the most interesting aspects of the FFLO non-Fermi-liquid scenario is that the quantum critical point is potentially naked, with the scaling regime observable down to arbitrary low temperatures. In order to study this possibility, we perform a general analysis of competing instabilities, which suggests that only charge density wave order is enhanced in the vicinity of the quantum critical point.

DOI
10.1103/physrevb.98.024510

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When a 2D superconductor is subjected to a strong in-plane magnetic field, Zeeman polarization of the Fermi surface can give rise to inhomogeneous FFLO order with a spatially modulated gap. Further increase of the magnetic field eventually drives the system into a normal metal state. Here, we perform a renormalization group analysis of this quantum phase transition, starting from an appropriate low-energy theory recently introduced in Phys. Rev. B 93, 085112 (2016). We compute one-loop flow equations within the controlled dimensional regularization scheme with fixed dimension of Fermi surface, expanding in \ensuremathϵ=5/2\ensuremath−d\ensuremath{\epsilon}=5/2\ensuremath{-}d\ensuremathϵ=5/2\ensuremath−d. We find a new stable non-Fermi-liquid fixed point and discuss its critical properties. One of the most interesting aspects of the FFLO non-Fermi-liquid scenario is that the quantum critical point is potentially naked, with the scaling regime observable down to arbitrary low temperatures. In order to study this possibility, we perform a general analysis of competing instabilities, which suggests that only charge density wave order is enhanced in the vicinity of the quantum critical point.
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.
Resorting to a recently developed theoretical device called dimensional regularization for quantum criticality with a Fermi surface, we examine a metal-insulator quantum phase transition from a Landau's Fermi-liquid state to a U(1) spin-liquid phase with a spinon Fermi surface in two dimensions. Unfortunately, we fail to approach the spin-liquid Mott quantum critical point from the U(1) spin-liquid state within the dimensional regularization technique. Self-interactions between charge fluctuations called holons are not screened, which shows a run-away renormalization group flow, interpreted as holons remain gapped. This leads us to consider another fixed point, where the spinon Fermi surface can be destabilized across the Mott transition. Based on this conjecture, we reveal the nature of the spin-liquid Mott quantum critical point: Dimensional reduction to one dimension occurs for spin dynamics described by spinons. As a result, Landau damping for both spin and charge dynamics disappear in the vicinity of the Mott quantum critical point. When the flavor number of holons is over its critical value, an interacting fixed point appears to be identified with an inverted XYXYXY universality class, controlled within the dimensional regularization technique. On the other hand, a fluctuation-driven first order metal-insulator transition results when it is below the critical number. We propose that the destabilization of a spinon Fermi surface and the emergence of one-dimensional spin dynamics near the spin-liquid Mott quantum critical point can be checked out by spin susceptibility with a 2kF2{k}_{F}2kF​ transfer momentum, where kF{k}_{F}kF​ is a Fermi momentum in the U(1) spin-liquid state: The absence of Landau damping in U(1) gauge fluctuations gives rise to a divergent behavior at zero temperature while it vanishes in the presence of a spinon Fermi surface.
We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions (m)(m)(m) and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in ddd spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension d=dc(m)d={d}_{c}(m)d=dc​(m). The fixed points are identified from the RG flow equations, as functions of ddd and mmm. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of (d=3,m=2)(d=3,\phantom{\rule{0.16em}{0ex}}m=2)(d=3,m=2) and (d=2,m=1)(d=2,\phantom{\rule{0.16em}{0ex}}m=1)(d=2,m=1). In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.
We study the problem of disorder-free metals near a continuous quantum critical point. We depart from the standard paradigm [Hertz, Phys. Rev. B 14, 1165 (1976) and Millis, Phys. Rev. B 48, 7183 (1993)] and treat both fermions and bosons (i.e., order parameter fields) on equal footing. We construct a Wilsonian effective field theory that integrates out only high-energy boson and fermion modes. Below the upper critical dimension of the theory (d=3d=3d=3 spatial dimensions), we find new fixed points in which the bosons are described by the Wilson-Fisher fixed point and are coupled to a non-Fermi-liquid metal. We describe subtleties with the renormalization group flow of four-Fermi interactions, which can be surmounted in a controlled large-NNN limit. In this limit, we find that the theory has no superconducting instability.
We consider the emergence of a non-Fermi liquid fixed point in two-dimensional metals at the onset of a quantum phase transition from a Fermi liquid state to an incommensurate charge density wave (CDW) ordered phase. The momentum of the CDW boson is centred at the wavevector Q, which connects a single pair of antipodal points on the Fermi surface with antiparallel tangent vectors. We employ the dimensional regularization technique in which the co-dimension of the Fermi surface is extended to a generic value, while keeping the dimension of Fermi surface itself fixed at one. Although the system is strongly coupled at dimension d=2, the interactions become marginal at the upper critical dimension d=dc, whose value is found to be 5/2. Using a controlled perturbative expansion in the parameter ϵ=dc−d, we compute the critical exponents of the stable infrared fixed point characterizing the quantum critical point. The scalings of the original theory are determined by setting ϵ=1/2, where the fermion self-energy is seen to scale with the frequency with a fractional power law of 2/3, which is the telltale signature of a typical non-Fermi liquid phase.
Introduction In this chapter we examine a special type of quantum critical point related to superconducting zero-temperature instabilities in many-body systems. We consider the case where the superconductor order parameter is inhomogeneous and characterised by a wave vector q C . Inhomogeneous ground states also appear in other systems like magnetic materials, in the form of spin-density waves or helicoidal ground states, in charge density wave systems and in excitonic insulators. They share the existence of a characteristic wave vector that determines the spatial modulation of the order parameter. For superconductors, we consider two cases. First, the problem, treated independently by Fulde and Ferrell (1964)and Larkin and Ovchinnikov (1965), of an s -wave, singlet superconductor in a homogeneous external magnetic field in the absence of vortices or orbital effects. The predicted ground state is generally known as the FFLO superconductor and is described by an order parameter that oscillates in space like Δ e i q C · r . It arises from the competition between the pairing energy of electrons with opposite spins and the Zeemann energy due to the external magnetic field that forces the alignment of the spins. For sufficiently large magnetic fields the Zeemann energy always wins and the ground state of the system is a spin-polarised, normal fermionic gas. However, as the magnetic field decreases, there is a quantum phase transition to a modulated superconducting state (Samokhin and Marénko, 2006; Caldas and Continentino, 2012) whose nature we identify. We also determine the universality class of this superconducting quantum phase transition. The characteristic wave vector q C of the resulting modulated superconducting phase is related to the difference between the Fermi wave vectors of the antiparallel spin bands that in turn is proportional to the magnetic field. A related question we also investigate is the fate of the uniform superconductor when the magnetic field is increased in the absence of orbital effects. We show that these two extreme approaches, increasing the field in the homogeneous superconductor and decreasing the field starting from the polarised metal do not merge smoothly into one another.