FFLO State Research Articles

FFLO States in Layered Organic Superconductors

AbstractIn this short review, the recently found experimental evidence that Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) states are realized in quasi‐two‐dimensional (2D) organic superconductors is reported. At low temperatures and when a high magnetic field is aligned parallel to the conducting organic layers, an upturn of the upper critical field much beyond the Pauli limit is observed, as proven by thermodynamic measurements. Under certain conditions, a second thermodynamic transition emerges inside the FFLO state. Nuclear magnetic resonance (NMR) work has added strong microscopic support for the realization of the FFLO state. The NMR spectra in the FFLO phase can very well be explained by a nonuniform one‐dimensionally modulated superconducting order parameter. All these features, appearing only in a very narrow angular region close to parallel‐field orientation, give robust evidence for the realization of the FFLO state in organic superconductors.

The influence of the dimensionality of the system on the realization of unconventional Fulde–Ferrell–Larkin–Ovchinnikov pairing in ultra-cold Fermi gases

The recent development of experimental techniques in ultracold atomic Fermi gases is extremely helpful in the progress of the realization of the unconventional Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superfluid phase in quasi-one dimensional systems (Liao et al 2010 Nature 467 567). Due to a Fermi surface nesting, which is enhanced in 1D, the low-dimensional systems are particularly good candidates to find the FFLO phase stable. We investigate the influence of a dimensional crossover (from one dimension (1D) to two dimensions (2D) or three dimensions (3D)) on the stability of the FFLO state in the spin-imbalanced attractive Hubbard model.

Topological superfluid in a two-dimensional polarized Fermi gas with spin–orbit coupling and adiabatic rotation

We study the properties of superfluid in a two-dimensional (2D) polarized Fermi gas with spin–orbit coupling and adiabatic rotation which are trapped in a harmonic potential. Due to the competition between polarization, spin–orbit coupling, and adiabatic rotation, the Fermi gas exhibits many intriguing phenomena. By using the Bardeen–Cooper– Schrieffer (BCS) mean-field method with local density approximation, we investigate the dependence of order parameter solution on the spin–orbit coupling strength and the rotation velocity. The energy spectra with different rotation velocities are studied in detail. Besides, the conditions for the zero-energy Majorana fermions in topological superfluid phase to be observed are obtained. By investigating distributions of number density, we find that the rotation has opposite effect on the distribution of number density with different spins, which leads to the enhancement of the polarization of Fermi gas. Here, we focus on the region of BCS pairing and ignore the Fulde–Ferrell–Larkin–Ovchinnikov state.

In Search of Unambiguous Evidence of the Fulde–Ferrell–Larkin–Ovchinnikov State in Quasi-Low Dimensional Superconductors

In layered conductors with a sufficiently weak interlayer coupling in-plane magnetic field cause only small diamagnetic currents and the orbital depairing is strongly suppressed. Therefore, the Zeeman effect predominantly governs the spin-singlet superconductivity making the formation of the spatially modulated Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase possible in such materials. Despite decades of strenuous effort, this state still remains a profound mystery. In the last several years, however, there have been observed several hints indicating the experimental realization of the FFLO state in organic layered superconductors. The emergence of the FFLO phase has been demonstrated mainly based on thermodynamic quantities or microscopically with spin polarization distribution that exhibit anomalies within the superconducting state in the presence of the in-plane magnetic field. However, the direct observation of superconducting order parameter modulation is so far missing. Recently, there have been proposed theoretically several hallmark signatures for FFLO phase, which are a direct consequence of its main feature, the spatial modulation of the order parameter, and hence can provide incontrovertible evidence of FFLO. In this article, a review of these signatures and the underlying theoretical framework is given with the purpose to summarize the results obtained so far, omitting duplications, and to emphasize the ideas and physics behind them.

Inhomogeneous Chiral and Coulomb Crystal in Neutron Stars

AbstractWe study the possibility of two types of inhomogeneous phases in core of neutron stars: one is the Coulomb crystal, which is known as quark-hadron pasta structures, and another one is chiral crystal. In the Coulomb crystal, the inhomogeneous phase appears as the result of the balance between the surface tension and the Coulomb interaction. In chiral crystal, we study the inhomogeneous chiral condensate, which has spatial modulation. In the simple model in 1+1 dimensions, this condensate has the same feature with the FFLO state, which is well known in the condensed matter physics.

Protocol to engineer Fulde-Ferrell-Larkin-Ovchinnikov states in a cold Fermi gas

We propose a two-step experimental protocol to directly engineer Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states in a cold two-component Fermi gas loaded into a quasi-one-dimensional trap. First, one uses phase imprinting to create a train of domain walls in a superfluid with equal number of $\uparrow$- and $\downarrow$-spins. Second, one applies a radio-frequency sweep to selectively break Cooper pairs near the domain walls and transfer the $\uparrow$-spins to a third spin state which does not interact with the $\uparrow$- and $\downarrow$-spins. The resulting FFLO state has exactly one unpaired $\downarrow$-spin in each domain wall and is stable for all values of domain-wall separation and interaction strength. We show that the protocol can be implemented with high fidelity at sufficiently strong interactions for a wide range of parameters available in present-day experimental conditions.

Calorimetric Measurements of Magnetic-Field-Induced Inhomogeneous Superconductivity Above the Paramagnetic Limit.

We report the first magnetocaloric and calorimetric observations of a magnetic-field-induced phase transition within a superconducting state to the long-sought exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting state, first predicted over 50years ago. Through the combination of bulk thermodynamic calorimetric and magnetocaloric measurements in the organic superconductor κ-(BEDT-TTF)_{2}Cu(NCS)_{2} as a function of temperature, magnetic field strength, and magnetic field orientation, we establish for the first time that this field-induced first-order phase transition at the paramagnetic limit H_{p} is a transition to a higher-entropy superconducting phase, uniquely characteristic of the FFLO state. We also establish that this high-field superconducting state displays the bulk paramagnetic ordering of spin domains required of the FFLO state. These results rule out the alternate possibility of spin-density wave ordering in the high-field superconducting phase. The phase diagram determined from our measurements-including the observation of a phase transition into the FFLO phase at H_{p}-is in good agreement with recent NMR results and our own earlier tunnel-diode magnetic penetration depth experiments but is in disagreement with the only previous calorimetric report.

Fulde-Ferrell-Larkin-Ovchinnikov state to topological superfluidity transition in bilayer spin-orbit-coupled degenerate Fermi gases

Recently a scheme has been proposed for generating the 2D Rashba-type spin-orbit coupling (SOC) for ultracold atomic bosons in a bilayer geometry [S.-W. Su et al, Phys. Rev. A \textbf{93}, 053630 (2016)]. Here we investigate the superfluidity properties of a degenerate Fermi gas affected by the SOC in such a bilayer system. We demonstrate that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state appears in the regime of small to moderate atom-light coupling. In contrast to the ordinary SOC, the FFLO state emerges in the bilayer system without adding any external fields or spin polarization. As the atom-light coupling increases, the system can transit from the FFLO state to a topological superfluid state. These findings are also confirmed by the BdG simulations with a weak harmonic trap added.

Superconducting and density-wave instabilities of low-dimensional conductors with a Zeeman coupling to a magnetic field

In the framework of the weak coupling renormalization group technique we examine the possible instabilities of the extended quasi-one-dimensional electron gas model with both intrachain and interchain electron-electron interactions, including the influence of umklapp scattering and the coupling of spins to a magnetic field. In the limit of purely repulsive intrachain interactions, we confirm the passage from singlet d-wave like superconductivity to an inhomogeneous FFLO state under magnetic field. The passage is accompanied by an anomalous increase of the upper critical field that scales with the antinesting distance from the quantum critical point joining superconductivity to antiferromagnetism in the phase diagram, as well as the strength of interactions. Adding weak repulsive interchain interactions promotes the passage from singlet to triplet $f$-wave superconductivity which is expanded under field by the development of a triplet FFLO state with zero angular momentum projection for the Cooper pairs. The connection between theory and experiments on the anomalous upper critical field in the Bechgaard salts is discussed.

Critical behavior in one dimension: Unconventional pairing, phase separation, BEC-BCS crossover, and magnetic Lifshitz transition

We study the superconducting properties of population-imbalanced ultracold Fermi mixtures in one-dimensional (1D) optical lattices that can be effectively described by the spin-imbalanced attractive Hubbard model (AHM) in the presence of a Zeeman magnetic field. We use the mean-field theory approach to obtain the ground state phase diagrams including some unconventional superconducting phases such as the Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) phase, and the $\eta$ phase (an extremal case of the FFLO phase), both for the case of a fixed chemical potential and for a fixed number of particles. It allows to determine optimal regimes for the FFLO phase as well as $\eta$-pairing stability. We also investigate the evolution from the weak coupling (BCS-like limit) to the strong coupling limit of tightly bound local pairs (BEC) with increasing attraction, at $T=0$. Finally, the obtained results show that despite of the occurrence of the Lifshitz transition induced by an external magnetic field, the superconducting state can still exist in the system, at higher magnetic field values.

Influence of Finite Size Effects on the Fulde-Ferrell-Larkin-Ovchinnikov State

AbstractThe Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is the superconducting phase for which the Cooper pairs have a non-zero total momentum, depending on the splitting of the Fermi surface sheets for electrons with opposite spin. In infinite systems the momentum is a continuous function of the temperature. In this paper, we have shown how the finite size of the system, through the discretized geometry of the Fermi surface, affects the physical properties of the FFLO state by introducing discontinuities in the Cooper pair momentum. Our calculation in an isotropic system show that the superconducting state with two opposite Cooper pair momenta is more stable than state with one momentum also in nano-size systems, where finite size effects play a crucial role.

Floquet engineering of Haldane Chern insulators and chiral bosonic phase transitions

The realization of synthetic gauge fields has attracted a lot of attention recently in relation with periodically driven systems and the Floquet theory. In ultra-cold atom systems in optical lattices and photonic networks, this allows to simulate exotic phases of matter such as quantum Hall phases, anomalous quantum Hall phases and analogs of topological insulators. In this paper, we apply the Floquet theory to engineer anisotropic Haldane models on the honeycomb lattice and two-leg ladder systems. We show that these anisotropic Haldane models still possess a topologically non-trivial band structure associated with chiral edge modes (without the presence of a net unit flux in a unit cell), then referring to the quantum anomalous Hall effect. Focusing on (interacting) boson systems in s-wave bands of the lattice, we show how to engineer through the Floquet theory, a quantum phase transition between a uniform superfluid and a BEC (Bose-Einstein Condensate) analog of FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) states, where bosons condense at non-zero wave-vectors. We perform a Ginzburg-Landau analysis of the quantum phase transition on the graphene lattice, and compute observables such as chiral currents and the momentum distribution. The results are supported by exact diagonalization calculations and compared with those of the isotropic situation. The validity of high-frequency expansion in the Floquet theory is also tested using time-dependent simulations for various parameters of the model. Last, we show that the anisotropic choice for the effective vector potential allows a bosonization approach in equivalent ladder (strip) geometries.

Soliton formation in the FFLO phase

There is increasing body of experimental evidences of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase in quasi-low-dimensional organic and heavy-fermion superconductors. The emergence of the FFLO phase has been demonstrated mainly based on a thermodynamic quantity or microscopically with spin polarization distribution that exhibit anomalies within the superconducting state in the presence of the in-plane magnetic field. However, the direct observation of superconducting order parameter modulation is so far (still) missing. Within the quasiclassical approach and Ginzburg-Landau formalism we study how the orbital effect of the in-plane field influences the FFLO instability in quasi-one-dimensional superconductors with a sufficiently weak interlayer coupling locking the magnetic flux to Josephson-type vortices. By making use of the continuum limit approximation of the Frenkel-Kontorova model for competing periodicities, we find and characterize the locking behavior of the modulation wave vector, when it remains equal to the magnetic length through some range of values of the external field.

Probing unconventional superconductivity in inversion-symmetric doped Weyl semimetal

Unconventional superconductivity has been predicted to arise in the topologically non-trivial Fermi surface of doped inversion symmetric Weyl semimetals (WSM). In particular, Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) and nodal BCS states are theoretically predicted to be possible superconductor pairing states in inversion symmetric doped WSM. In an effort to resolve preferred pairing state, we theoretically study two separate four terminal quantum transport methods that each exhibit a unique electrical signature in the presence of FFLO and nodal BCS states in doped WSMs. We first introduce a Josephson junction that consists of a doped WSM and an s-wave superconductor in which we show that the application of a transverse uniform current in s-wave superconductor effectively cancels the momentum carried by FFLO states in doped WSM. From our numerical analysis, we find a peak in Josephson current amplitude at finite uniform current in s-wave superconductor that serves as an indicator of FFLO states in doped WSMs. Furthermore, we show using a four terminal measurement configuration that the nodal points may be shifted by an application of transverse uniform current in doped WSM. We analyze the topological phase transitions induced by nodal pair annihilation in non-equilibrium by constructing the phase diagram and we find a characteristic decrease in the density of states that serves as a signature of the quantum critical point in the topological phase transition, thereby identifying nodal BCS states in doped WSM.

Superconductivity in magnetic multipole states

Stimulated by recent studies of superconductivity and magnetism with local and global broken inversion symmetry, we investigate the superconductivity in magnetic multipole states in locally noncentrosymmetric metals. We consider a one-dimensional zigzag chain with sublattice-dependent antisymmetric spin-orbit coupling and suppose three magnetic multipole orders: monopole order, dipole order, and quadrupole order. It is demonstrated that the Bardeen-Cooper-Schrieffer state, the pair-density wave (PDW) state, and the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state are stabilized by these multipole orders, respectively. We show that the PDW state is a topological superconducting state specified by the nontrivial $\mathbb{Z}_2$ number and winding number. The origin of the FFLO state without macroscopic magnetic moment is attributed to the asymmetric band structure induced by the magnetic quadrupole order and spin-orbit coupling.

Momentum space imaging of the FFLO state

In a magnetic field superconductors (SC) with small orbital effect exhibit the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase above the Pauli limiting field. It is characterized by Cooper pairs with finite center of mass momentum and is stabilized by the gain in Zeeman energy of depaired electrons in the imbalanced Fermi gas. The ground state is a coherent superposition of paired and depaired states. This concept, although central to the FFLO state lacks a direct experimental confirmation. We propose that STM quasiparticle interference (QPI) can give a direct momentum space image of the depaired states in the FFLO wave function. For a proof of principle we investigate a 2D single orbital tight binding model with a SC s-wave order parameter. Using the equilibrium values of pair momentum and SC gap we calculate the spectral function of quasiparticles and associated QPI spectrum as function of magnetic field. We show that the characteristic depaired Fermi surface parts appear as a fingerprint in the QPI spectrum of the FFLO phase and we demonstrate its evolution with field strength. Its observation in STM experiments would constitute a direct proof for FFLO ground state wave function.

Phase transition to the Fulde–Ferrell–Larkin–Ovchinnikov state in a quasi-one-dimensional organic superconductor with anion order

A theoretical study is presented on the effect of anion order on the phase transition from the normal state to a Fulde–Ferrell–Larkin –Ovchinnikov superconducting state in a quasi-one-dimensional organic superconductor (TMTSF)2ClO4. The temperature dependence of the upper critical field Hc2(T) is examined. In the absence of anion order, the Hc2(T) curve exhibits a one- to two-dimensional crossover with decreasing temperature T. For a sufficiently large anion order parameter, the dimensional crossover disappears and then a kink appears in the Hc2(T) curve.

On the new Continuous Matrix Product Ansatz

The fertile new field of quantum information theory is inspiring new ways to study correlated quantum systems by providing fresh insights into the structure of their Hilbert spaces. One of the latest developments in this direction was the extension of the ubiquitous matrix-product-state constructions, epitomized by the density-matrix renormalization-group algorithm, to continuous space-time; so as to be able to describe low-dimensional field theories within a variational approach. Following the earlier success achieved for bosonic theories, we present the first implementation of a continuous matrix product state (cMPS) for spinfull non-relativistic fermions in 1D. We propose a construction of variational matrices with an efficient parametrization that respects the translational symmetry of the problem (without being overly constraining) and readily meets the regularity conditions that arise from removing the ultraviolet divergences in the kinetic energy. We tested the validity of our approach on an interacting spin-1/2 system with spin imbalance. We observe that the ansatz correctly predicts the ground-state magnetic properties for the attractive spin-1/2 Fermi gas, including a phase-oscillating pair correlation function in the partially polarized regime (the 1D correlate of the FFLO state). We shall also discuss how to generalize the cMPS ansatz to other situations.

FFLO strange metal and quantum criticality in two dimensions: Theory and application to organic superconductors

Increasing the spin imbalance in superconductors can spatially modulate the gap by forming Cooper pairs with finite momentum. For large imbalances compared to the Fermi energy, the inhomogeneous FFLO superconductor ultimately becomes a normal metal. There is mounting experimental evidence for this scenario in 2D organic superconductors in large in-plane magnetic fields; this is complemented by ongoing efforts to realize this scenario in coupled tubes of atomic Fermi gases with spin imbalance. Yet, a theory for the phase transition from a metal to an FFLO superconductor has not been developed so far and the universality class has remained unknown. Here we propose and analyze a spin imbalance driven quantum critical point between a 2D metal and an FFLO phase in anisotropic electron systems. We derive the effective action for electrons and bosonic FFLO pairs at this quantum phase transition. Using this action, we predict non-Fermi liquid behavior and the absence of quasi-particles at a discrete set of hot spots on the Fermi surfaces. This results in strange power-laws in thermodynamics and response functions, which are testable with existing experimental set-ups on 2D organic superconductors and may also serve as signatures of the elusive FFLO phase itself. The proposed universality class is distinct from previously known quantum critical metals and, because its critical fluctuations appear already in the pairing channel, a promising candidate for naked metallic quantum criticality over extended temperature ranges.

Interplay between quantum confinement and Fulde–Ferrell–Larkin–Ovchinnikov phase in superconducting nanofilms

In superconducting nanofilms the energy quantization induced by the confinement in the direction perpendicular to the film splits the band of single-electron states into series of subbands. The quantum size effect leads to the experimentally observed oscillations of the critical magnetic field with increasing nanofilm thickness. Here, we study the influence of the quantum confinement on the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase in superconducting nanofilms. We show that the range of the magnetic fields for which the FFLO phase is stable oscillates as a function of the film thickness with the phase shift equal to one half of the period corresponding to the critical magnetic field oscillations. Due to the multiband character of the system a division of the FFLO phase stability region appears leading to a phase diagram which is qualitatively different than the one corresponding to a single-band situation. The number of subregions created in such manner depends on the number of bands participating in the formation of the paired state.