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arXiv:1009.4731 · Journal of Physics Condensed Matter

Quantum criticality in inter-band superconductors

Aline Ramires, M. A. Contínentino

Published Nov 16, 2010• 7 citations
arXivHTMLPDF

Abstract

In fermionic systems with different types of quasi-particles, attractive interactions can give rise to exotic superconducting states, such as pair density wave (PDW) superconductivity and breached pairing. In recent years the search for these new types of ground states in cold atoms and in metallic systems has been intense. In the case of metals the different quasi-particles may be the up and down spin bands in an external magnetic field or bands arising from distinct atomic orbitals that coexist at a common Fermi surface. These systems present a complex phase diagram as a function of the difference between the Fermi wavevectors of the different bands. This can be controlled by external means, varying the density in the two-component cold atom system or, in a metal, by applying an external magnetic field or pressure. Here we study the zero temperature instability of the normal system as the Fermi wavevector mismatch of the quasi-particles (bands) is reduced and find a second order quantum phase transition to a PDW superconducting state. From the nature of the quantum critical fluctuations close to the superconducting quantum critical point (SQCP), we obtain its dynamic critical exponent. It turns out to be z = 2 and this allows us to fully characterize the SQCP for dimensions d ≥ 2.

DOI
10.1088/0953-8984/22/48/485701

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In fermionic systems with different types of quasi-particles, attractive interactions can give rise to exotic superconducting states, such as pair density wave (PDW) superconductivity and breached pairing. In recent years the search for these new types of ground states in cold atoms and in metallic systems has been intense. In the case of metals the different quasi-particles may be the up and down spin bands in an external magnetic field or bands arising from distinct atomic orbitals that coexist at a common Fermi surface. These systems present a complex phase diagram as a function of the difference between the Fermi wavevectors of the different bands. This can be controlled by external means, varying the density in the two-component cold atom system or, in a metal, by applying an external magnetic field or pressure. Here we study the zero temperature instability of the normal system as the Fermi wavevector mismatch of the quasi-particles (bands) is reduced and find a second order quantum phase transition to a PDW superconducting state. From the nature of the quantum critical fluctuations close to the superconducting quantum critical point (SQCP), we obtain its dynamic critical exponent. It turns out to be z = 2 and this allows us to fully characterize the SQCP for dimensions d ≥ 2.
Different types of superfluid ground states have been investigated in systems of two species of fermions with Fermi surfaces that do not match. This study is relevant for cold atomic systems, condensed matter physics, and quark matter. In this paper we consider this problem when the fermionic quasi-particles can transmute into one another and only their total number is conserved. We use a Bardeen–Cooper–Schrieffer (BCS) approximation to study superconductivity in two-band metallic systems with inter-and intra-band interactions. Tuning the hybridization between the bands varies the mismatch of the Fermi surfaces and produces different instabilities. For inter-band attractive interactions, we find a first-order normal–superconductor transition and a homogeneous metastable phase with gapless excitations. In the case of intra-band interactions, the transition from the superconductor to the normal state is continuous as hybridization increases and associated with a quantum critical point. The case when both interactions are present is also considered.
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.
The study of superconductivity in correlated systems is an exciting area of condensed matter physics. In this paper we consider superconducting ground states in systems described by two-band models with different effective masses. These two bands are coupled through an effective hybridization that can be directly tuned by pressure. We consider the cases of s-wave superconductivity associated with the electrons in a narrow band and also with inter-band pairing. To study the system in the strong coupling regime we introduce the s-wave scattering length a(s), and obtain the superconducting order parameters and the chemical potential as functions of the interaction strength 1/k(F)a(s) along the BCS-BEC crossover at T = 0. Finally, we discuss the phase diagram of this model as a function of external pressure and how our results can be applied for two-band systems as Fe pnictides or heavy fermions. The main result of this study is the occurrence of a superconducting quantum critical point (SQCP) in this two-band model.
Motivated by exploring superconductivity in multi-orbital systems, we study two orbital models of spinful fermions representing (px,pyp_x,p_ypx​,py​) or (dxz,dyz)d_{xz}, d_{yz})dxz​,dyz​) orbitals on the square lattice. For minimal interorbital ttt-JJJ or ttt-VVV on-site interactions, a random phase approximation uncovers regimes of instability towards incommensurate dxyd_{xy}dxy​ pair density wave (ddd-PDW) superconductivity with driven by interband pairing. We study the competition of PDW order with uniform nodal dxyd_{xy}dxy​ pairing states and magnetic and charge density wave (CDW) instabilities. At strong coupling, we derive an effective hard-core Cooper pair Hamiltonian which we study using a bosonic Gutzwiller ansatz to reveal a period-222 PDW over a wide range of fillings as well as a checkerboard CDW at quarter-filling. Our results apply to correlated multi-orbital materials with quasi-1D bands, Hubbard models on the square-octagon lattice, and atomic fermions in ppp-orbitals. Our work highlights the role of the orbital content and multiband Fermi surfaces in stabilizing interband PDW states.
We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.
The study of multi-band superconductivity is relevant for a variety of systems, from ultra cold atoms with population imbalance to particle physics, and condensed matter. As a consequence, this problem has been widely investigated bringing to light many new and interesting phenomena. In this work we point out and explore a correspondence between a two-band metal with a kkk-dependent hybridization and a uniformly polarized fermionic system in the presence of spin-orbit coupling (SOC). We study the ground state phase diagram of the metal in the presence of an attractive interaction. We find a remarkable behavior of its superconducting properties whenever hybridization mixes orbitals of different parities in neighboring sites. We show this enhances superconductivity and drives the crossover from weak to strong coupling in analogy with SOC in cold atoms. We obtain the quantum phase transitions between the normal and superfluid states, as the intensity of different parameters characterizing the metal are varied, including Lifshitz transitions, with no symmetry breaking, associated with the appearance of soft modes in the Fermi surface.