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.
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.
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.
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase is an unconventional superconducting state found under the influence of a strong Zeeman field. This phase is identified by finite center-of-mass momenta in the Cooper pairs, causing the pairing amplitude to oscillate in real space. Repulsive correlations, on the other hand, smear out spatial inhomogeneities in d-wave superconductors. We investigate the FFLO state in a strongly correlated d-wave superconductor within a consolidated framework of Hartree-Fock-Bogoliubov theory and the Gutzwiller approximation. We find that the crucial effects of strong correlations lie in shifting the BCS-FFLO phase boundary towards a lower Zeeman field and thereby enlarging the window of the FFLO phase. In the FFLO state, our calculation features a sharp midgap peak in the density of states, indicating the formation of strongly localized Andreev bound states. We also find that the signatures of the FFLO phase survive even in the presence of an additional translational-symmetry-breaking competing order in the ground state. This is demonstrated by considering a broken-symmetry ground state with a simultaneous presence of the d-wave superconducting order and a spin-density wave order, often found in unconventional superconductors.
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.
Introduction The quantum phase transitions considered so far in this book are mostly continuous, second order. They are associated with the divergence of a correlation length and of a characteristic time that renders the quantum critical point scale invariant. However, there are many important phase transitions at which the order parameter goes abruptly to zero and, more important, at which the characteristics of length and critical time do not diverge. In general there is also a latent heat , or work required, to transform one phase into another. In this and the next chapter we study these discontinuous quantum phase transitions. We proceed by considering initially some important cases of what is known as fluctuation induced quantum phase transitions to next treat a more general case of a first-order quantum phase transition. We will show through the study of these problems that the concepts of critical exponents and scaling remain useful, even though these systems do not exhibit scale invariance at their critical point. In Chapter 15 we present the scaling theory of first-order quantum phase transitions and use its results to treat some general problems. One of the most interesting cases of fluctuation-induced transitions occurs in superconductors due to the coupling of the charged carriers to the electromagnetic field. We consider initially a wider class of systems with macroscopic wave functions, the superfluids, of which superconductors are the charged version. We adopt a macroscopic approach to this problem and describe the superfluid by a complex order parameter φ , such that | φ | 2 is proportional to the density of condensed particles. The zero temperature phase diagram is expressed in terms of a control parameter, m 2 , such that, at m 2 = 0, the system presents a quantum phase transition from a superfluid state, for m 2 < 0, φ ≠ 0, to a normal state for m 2 > 0 where φ = 0. In the case of the superconductor with particles of charge q = 2 e , we will include the coupling of these particles to the electromagnetic field and study its effect in the phase diagram of these systems. We show that in the neutral superfluid, the zero temperature transition to the normal state is continuous and occurs at a quantum critical point.
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.
Quantum many body phases, in competition or coexistence has always been a fascinating area of research in condensed matter physics. One of the most widely investigated scenario of phase competition is that between superconductivity and magnetism, which in spite of several decades of research continues to bring up new surprises till date. While superconductivity and magnetism are naively considered to be antagonists, there are several circumstances where these many body phases can be found to be cooperating, coexisting and even promoting each other. The results of such cooperative behavior are found to show up as exotic quantum phases. In this chapter we discuss about such an exotic superconducting phase, which comes to life when a particular class of superconducting material viz. "Pauli limited superconductors" are subjected to an imbalance between the population of the fermionic species undergoing pairing. The consequence of such an imbalance is the emergence of a non-trivial spatially modulated superconducting phase, popularly known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Characterized by a finite momentum pairing between the fermionic species the FFLO phase hosts a non zero magnetization, leading to an unconventional coexistent phase of superconductivity and magnetism.