Sarma Phase Research Articles

Connection between the semiconductor-superconductor transition and the spin-polarized superconducting phase in the honeycomb lattice

The band structure of noninteracting fermions in the honeycomb lattice exhibits the Dirac cones at the corners of the Brillouin zone. As a consequence, fermions in this lattice manifest a semiconducting behavior below some critical value of the onsite attraction, $U_{c}$. However, above $U_{c}$, the superconducting phase can occur. We discuss an interplay between the semiconductor--superconductor transition and the possibility of realization of the spin-polarized superconductivity (the so-called Sarma phase). We show that the critical interaction can be tuned by the next-nearest-neighbor (NNN) hopping in the absence of the magnetic field. Moreover, a critical value of the NNN hopping exists, defining a range of parameters for which the semiconducting phase can emerge. In the weak coupling limit case, this quantum phase transition occurs for the absolute value of the NNN hopping equal to one third of the hopping between the nearest neighbors. Similarly, in the presence of the magnetic field, the Sarma phase can appear, but only in a range of parameters for which initially the semiconducting state is observed. Both of these aspects are attributed to the Lifshitz transition, which is induced by the NNN hopping as well as the external magnetic field.

Three-dimensional QCD phase diagram with a pion condensate in the NJL model

With the isovector coupling constants adjusted to reproduce the physical pion mass and lattice QCD results in baryon-free quark matter, we have carried out rigourous calculations for the pion condensate in the 3-flavor Nambu-Jona-Lasinio model, and studied the 3-dimensional QCD phase diagram. With the increasing isospin chemical potential $\mu_I$, we have observed two nonzero solutions of the pion condensate at finite baryon chemical potentials $\mu_B$, representing respectively the pion superfluid phase and the Sarma phase, and their appearance and disappearance correspond to a second-order (first-order) phase transition at higher (lower) temperatures $T$ and lower (higher) $\mu_B$. Calculations by assuming equal constituent mass of $u$ and $d$ quarks would lead to large errors of the QCD phase diagram within $\mu_B \in (500, 900)$ MeV, and affect the position of the critical end point.

Enhanced Fulde-Ferrell-Larkin-Ovchinnikov and Sarma superfluid states near an orbital Feshbach resonance

We investigate the inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and homogeneous Sarma superfluid states in alkaline-earth-like $^{173}$Yb atomic gases near an orbital Feshbach resonance at zero temperature with population imbalances in both the open and closed channels (or bands). We find that in homogeneous space by adjusting the open-channel Zeeman energy $h_o$, both the Sarma and Fulde-Ferrell superfluid states are greatly enhanced by the spin-exchange interaction while the closed-channel Zeeman energy $h_c$ remains small. In the presence of an external harmonic trap, the trapped gas features a shell structure of separated phases, where the Sarma phase leaves detectable valley structure in the columnar-integrated momentum distribution, and the Fulde-Ferrell state acquires enhanced spatial anisotropy. As both signatures can be easily detected in time-of-flight images, our findings are helpful to realize and detect the long-sought FFLO and Sarma superfluid states at the same time in experiments.

Q-deformed fermion in many-particle systems and its application to BCS theory

In recent decades, there have been increasing interests in quantum statistics beyond the standard Fermi–Dirac and Bose–Einstein statistics, such as the fractional statistics, quon statistics, anyon statistics and quantum groups, since they can provide new insights into the cosmology, nuclear physics and condensed matter. In this paper, we study the many-particle system formed by the q-deformed fermions (q-fermion), which is realized by deforming the quantum algebra of the anticommutation relations. From the standard perspective of the finite temperature field theory, we first construct the finite temperature Green’s function formalism for the free many-q-fermion system, then generalize it to the well known interacting fermionic system, the superconductor, and finally obtain a consistent q-deformed BCS (qBCS) theory. Interesting features can be implied from this model. For example, it predicts a Sarma-like ordered phase, which we call the q-deformed Sarma phase, at low temperature. It also presents a symmetric phase diagram in the parameter space and modified thermodynamic relations.

Spin-imbalanced Fermi superfluidity in a Hubbard model on a Lieb lattice

We obtain a phase diagram of the spin imbalanced Hubbard model on the Lieb lattice, which is known to feature a flat band in its single-particle spectrum. Using the BCS mean-field theory for multiband systems, we find a variety of superfluid phases with imbalance. In particular, we find four different types FFLO phases, i.e. superfluid phases with periodic spatial modulation. They differ by the magnitude and direction of the centre-of-mass momentum of Cooper pairs. We also see a large region of stable Sarma phase, where the density imbalance is associated with zero Cooper pair momentum. In the mechanism responsible for the formation of those phases, the crucial role is played by the flat band, wherein particles can readjust their density at zero energy cost. The multiorbital structure of the unit cell is found to stabilize the Sarma phase by allowing for a modulation of the order parameter within a unit cell. We also study the effect of finite temperature and a lattice with staggered hopping parameters on the behaviour of these phases.

Sarma phase in relativistic and non-relativistic systems

We investigate the stability of the Sarma phase in two-component fermion systems in three spatial dimensions. For this purpose we compare strongly-correlated systems with either relativistic or non-relativistic dispersion relation: relativistic quarks and mesons at finite isospin density and spin-imbalanced ultracold Fermi gases. Using a Functional Renormalization Group approach, we resolve fluctuation effects onto the corresponding phase diagrams beyond the mean-field approximation. We find that fluctuations induce a second-order phase transition at zero temperature, and thus a Sarma phase, in the relativistic setup for large isospin chemical potential. This motivates the investigation of the cold atoms setup with comparable mean-field phase structure, where the Sarma phase could then be realized in experiment. However, for the non-relativistic system we find the stability region of the Sarma phase to be smaller than the one predicted from mean-field theory. It is limited to the BEC side of the phase diagram, and the unitary Fermi gas does not support a Sarma phase at zero temperature. Finally, we propose an ultracold quantum gas with four fermion species that has a good chance to realize a zero-temperature Sarma phase.

Phase structure of spin-imbalanced unitary Fermi gases

We investigate the phase structure of spin-imbalanced unitary Fermi gases beyond mean-field theory by means of the Functional Renormalization Group. In this approach, quantum and thermal fluctuations are resolved in a systematic manner. The discretization of the effective potential on a grid allows us to accurately account for both first- and second-order phase transitions that are present on the mean-field level. We compute the full phase diagram in the plane of temperature and spin-imbalance and discuss the existence of other conjectured phases such as the Sarma phase and a precondensation region. In addition, we explain on a qualitative level how we expect that in-situ density images are affected by our findings and which experimental signatures may potentially be used to probe the phase structure.

Phase Diagram of a $$^6Li$$ 6 L i – $$^{40}K$$ 40 K Mixture in a Square Lattice

We calculate the mean-field phase diagram for an attractive interacting Fermi mixture of Lithium-6 ( $$^6Li$$ ) and Potassium-40 ( $$^{40}K$$ ) atoms in a two-dimensional optical lattice at finite temperatures. The polarization versus temperature diagrams show that there are three phases: the Sarma phase (in which the condensed pairs have zero net momentum), the Fulde–Ferrell (FF) phase (in which the pairs have non-zero net momentum), and the normal phase (in which the Helmholtz free energy is minimized for gapless phase). The zero polarization line is the conventional Bardeen–Cooper–Schrieffer state. The phase diagram contains a Lifshitz point. When the interaction strength is increased, the Lifshitz point moves toward the higher temperatures and larger polarizations. Moreover, contrary to the phase diagram of population-imbalanced $$^6Li$$ Fermi gas, where the phase separation appears for low polarizations, we found the existence of a polarization window for the FF phase. This means that as soon as the system is polarized it goes into the FF phase if the temperature is low enough. This polarization window is larger for a majority of $$^{40}K$$ atoms compared to the majority of $$^6Li$$ atoms. We also find that the largest polarization that the system can support before it becomes a normal fluid is larger for majority of $$^6Li$$ atoms.

Condensate fraction of asymmetric three-component Fermi gas

In this paper, we investigate the condensate fraction (CF) of fermionic pairs in the BCS—BEC crossover for three-component Fermi gas with both asymmetric interactions and unequal chemical potentials in two-dimensional free space. By using the functional-path-integral method, we have analytically derived the number densities and bound-state energy, from which the off-diagonal long-range order is analyzed in terms of the asymptotic behavior of the two-body density matrix. The explicit formula of CF is obtained as a function of the bound-state energy and population imbalance. It is demonstrated that the CF spectrum with respect to the bound-state energy can be used to characterize the quantum phase transition between the two kinds of Sarma phases as well as the transition from three-component to two-component superfluid. Moreover we obtain the same analytic formula of CF in the BCS superfluid phase as that of homogeneous Fermi gas with equal chemical potentials.

Collective Excitations of an Imbalanced Fermion Gas in a 1D Optical Lattice

The collective excitations that minimize the Helmholtz free energy of a population-imbalanced mixture of a $^{6}$Li gas loaded in a quasi one-dimensional optical lattice are obtained. These excitations reveal a rotonic branch after solving the Bethe-Salpeter equation under a generalized random phase approximation based on a single-band Hubbard Hamiltonian. The phase diagram describing stability regions of Fulde-Farrell-Larkin-Ovchinnikov and Sarma phases is also analyzed.

Superfluidity of a spin-imbalanced Fermi gas in a three-dimensional optical lattice

We study fermion pairing in a population-imbalanced mixture of $^{6}$Li atomic gas loaded in a three-dimensional lattice at very low temperatures. Using the number equation for each population, the gap equation and the equation for the Helmholtz free energy, we determine the gap, chemical potentials and pair-momentum as functions of polarization. These parameters define the stability regions for: a Fulde-Ferrell-Larkin-Ovchinnikov phase; a phase separation region where BCS and normal phases coexist; a Sarma phase when the pair-momentum vanishes, and the transition to the normal phase when the gap disappears. The collective-mode energies are then calculated using a Bethe-Salpeter approach in the generalized random phase approximation assuming that the system is well described by the single-band Hubbard model. A novel result is that this fermionic gas has a superfluid phase revealed by rotonlike minima in the asymmetric collective-mode energy spectrum.

Pairing imbalance in the BCS-BEC crossover of an inhomogeneous three-component Fermi gas in two dimensions

We in this paper investigate the phase diagram associated with the BCS-BEC crossover of a three-component ultracold superfluid-Fermi-gas of different chemical-potentials and equal masses in two dimensions. The gap order parameter and number densities are found analytically by using the functional path-integral method. The balance of paring will be broken in the free space due to the unequal chemical-potentials. We obtain the same particle number-density and condensed fraction in the BCS superfluid phase as that in a recent paper (Phys. Rev. A 83, 033630), while the Sarma phase of coexistence of normal and superfluid Fermi gases is the characteristics of inhomogeneous system. The minimum ratio of BCS superfluid phase becomes 1/3 in the BCS limit corresponding to the zero-ratio in the two-component system in which the critical point of phase separation is {\epsilon}B/{\epsilon}F = 2 but becomes 3 in the three-component case.

Bicritical point in multi-bands inhomogeneous superconductors

Multi-band systems as inter-metallic and heavy fermion compounds have quasi-particles arising from different orbitals at their Fermi surface. Since these quasi-particles have different masses or densities, there is a natural mismatch of the Fermi wave-vectors associated with different orbitals. This makes these materials potential candidates to observe exotic superconducting phases as Sarma or FFLO phases, even in the absence of an external magnetic field. The distinct orbitals coexisting at the Fermi surface are generally hybridized and their degree of mixing can be controlled by external pressure. In this work we investigate the existence of an FFLO type of phase in a two-band BCS superconductor controlled by hybridization analyzing the phase diagram at finite temperature, the system has similarities to conventional FFLO phase induced by field but has an entirely different nature, like a presence of a bicritical point.

Stability of Sarma phases in density imbalanced electron-hole bilayer systems

We study excitonic condensation in an electron-hole bilayer system with unequal layer densities at zero temperature. Using mean-field theory we solve the BCS gap equations numerically and investigate the effects of intra-layer interactions. We analyze the stability of the Sarma phase with $\bk,-\bk$ pairing by calculating the superfluid mass density and also by checking the compressibility matrix. We find that with bare Coulomb interactions the superfluid density is always positive in the Sarma phase, due to a peculiar momentum structure of the gap function originating from the singular behavior of the Coulomb potential at zero momentum and the presence of a sharp Fermi surface. Introducing a simple model for screening, we find that the superfluid density becomes negative in some regions of the phase diagram, corresponding to an instability towards a Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) type superfluid phase. Thus, intra-layer interaction and screening together can lead to a rich phase diagram in the BCS-BEC crossover regime in electron-hole bilayer systems.

Asymmetric two-component Fermi gas in two dimensions

We analytically investigate superfluid phase in two dimension at zero temperature for asymmetric Fermi gas. Analytic expressions for both gap and chemical potentials at mean-field level are obtained as a function of binding energy and population imbalance when mass of different types of fermions is equal. We identify regions corresponding to normal, Sarma phase for different polarization, or mass ratio and discuss topological phase diagram in the BCS-BEC crossover and find that energy gap and chemical potential behave differently in these topological quantum phase. We also obtain the low-frequency and long-wavelength collective excitation spectrum and find sound velocity has a maximum when two types of mass are equal.

Pion superfluidity beyond mean field approximation in the Nambu–Jona-Lasinio model

We investigate pion superfluidity in the frame of a two flavor Nambu--Jona-Lasinio model beyond mean field approximation. We calculate the thermodynamics to the next to leading order in an expansion in the inverse number of colors, including both quark and meson contributions at finite temperature and baryon and isospin density. Because of the meson fluctuations, the Sarma phase which exists at the mean field level is washed away, and the Bose-Einstein condensation region at low isospin density is highly suppressed.

Pressure induced FFLO instability in multi-band superconductors

Multi-band systems such as inter-metallic and heavy fermion compounds havequasi-particles arising from different orbitals at their Fermi surface. Since thesequasi-particles have different masses or densities, there is a natural mismatch of the Fermiwavevectors associated with different orbitals. This makes these materials potentialcandidates to observe exotic superconducting phases as Sarma or FFLO phases, even in theabsence of an external magnetic field. The distinct orbitals coexisting at the Fermi surfaceare generally hybridized and their degree of mixing can be controlled by external pressure.In this work we investigate the existence of an FFLO type of phase in a two-band BCSsuperconductor controlled by hybridization. At zero temperature, as hybridization(pressure) increases we find that the BCS state becomes unstable with respect toan inhomogeneous superconducting state characterized by a single wavevectorq.

Polarized Superfluidity in the Attractive Hubbard Model with Population Imbalance

We study a two-component Fermi system with attractive interactions and different populations of the two species in a cubic lattice. For an intermediate coupling, we find a uniformly polarized superfluid which is stable down to very low temperatures. The momentum distribution of this phase closely resembles that of the Sarma phase, characterized by two Fermi surfaces. This phase is shown to be stabilized by a potential energy gain, as in a BCS superfluid, in contrast with the unpolarized Bose-Einstein condensate which is stabilized by kinetic energy. We present general arguments suggesting that preformed pairs in the unpolarized superfluid favor the stabilization of a polarized superfluid phase.

Theory of superfluids with population imbalance: Finite-temperature and BCS-BEC crossover effects

In this paper we present a very general theoretical framework for addressing fermionic superfluids over the entire range of BCS to Bose Einstein condensation (BEC) crossover in the presence of population imbalance or spin polarization. Our emphasis is on providing a theory which reduces to the standard zero temperature mean field theories in the literature, but necessarily includes pairing fluctuation effects at non-zero temperature within a consistent framework. Physically, these effects are associated with the presence of pre-formed pairs (or a fermionic pseudogap) in the normal phase, and pair excitations of the condensate, in the superfluid phase. We show how this finite $T$ theory of fermionic pair condensates bears many similarities to the condensation of point bosons. In the process we examine three different types of condensate: the usual breached pair or Sarma phase and both the one and two plane wave Larkin- Ovchinnikov, Fulde-Ferrell (LOFF) states. The last of these has been discussed in the literature albeit only within a Landau-Ginzburg formalism, generally valid near $T_c$. Here we show how to arrive at the two plane wave LOFF state in the ground state as well as at general temperature $T$.

Analytical results for cold asymmetrical fermion superfluids at the mean-field level

We present the analytical results at the mean-field level for the asymmetrical fermion system with attractive contact interaction at zero temperature. The results can be expressed in terms of linear combinations of the elliptic integrals of the first and second kinds. In the limit of small gap parameter, we discuss how the asymmetry in fermion species affects the phases of the ground state of the system. In the limit of large gap parameter, we show that two candidate phases are competing for the system's ground state. The Sarma phase containing a pure Fermi fluid and a mixed condensate is favored at a large degree of asymmetry. The separated phase consisting of a pure Fermi fluid and a boson condensate supports the system at a small degree of asymmetry. The two phases are degenerate in the limit of infinite pairing gap.