FFLO State Research Articles

New orientations of the Cooper pair momentum in FFLO state for asymmetric D-wave fermion superfluids

The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state in asymmetric D-wave fermion superfluids is investigated generally. The angle dependence of the pairing gap (angle-dependent gap) for D-wave is also considered. Except the two special orientations of the Cooper pair momentum which are parallel/orthogonal to the symmetry-axis of ADG (called as FFLO-ADG-O and FFLO-ADG-P states), new orientations of the Cooper pair momentum between the two special ones are also found. These new orientations of the Cooper pair momentum correspond to new possible locally stable state (defined as FFLO-ADG-B state). Furthermore, the FFLO-ADG-O state is favored for small asymmetry and the FFLO-ADG-P state is located at large asymmetry, whereas the FFLO-ADG-B state is stable for moderate asymmetry. Moreover, the range of the asymmetry where the superfluids exists is enhanced strongly by taking the ADG to the FFLO state.

Reentrant Fulde-Ferrell-Larkin-Ovchinnikov state in small-sized superconductors

We study the effect of a parallel magnetic field in a thin and small superconductor. The field suppresses superconductivity through Zeeman coupling while stabilizes the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state at high fields before superconductivity is destroyed. When the spatial period of FFLO state is comparable to the size of the superconductor, there is a strong commensuration effect, which modifies the superconducting phase diagram. We investigate the FFLO state and the phase diagram in the presence of strong commensuration effect both for the $s$-wave and $d$-wave superconductors using the Bogoliubov de Gennes equation, Green function approach, and Ginzburg-Landau theory. We found that the superconducting phase diagram is strongly modulated. Interestingly, there is re-entrance of superconductivity upon increasing the magnetic field. The commensuration effect of the FFLO state can be used to detect the FFLO state in experiments.

Fulde\u2013Ferrell\u2013Larkin\u2013Ovchinnikov and vortex phases in a layered organic superconductor

Superconductivity is one of the most intriguing topics in solid state physics. Generally, the superconducting Cooper pairs are broken by the Zeeman effect, which gives the so-called Pauli paramagnetic limit HPauli. However, when the superconductivity is in the clean limit and the orbital effect is strongly quenched, the Cooper pairs can survive even above HPauli, which is the so-called Fulde and Ferrell, and Larkin and Ovchinnikov (FFLO) phase. Extensive efforts have been devoted to the discovery of the FFLO phase. However, vortex phase transitions have given rise to considerable ambiguity in the interpretation of the experimental data. Here, we report comprehensive magnetocaloric and torque studies of the FFLO phase transition in a highly two-dimensional organic superconductor. We observe the FFLO phase transition clearly distinct from vortex melting transitions. The phase diagram provides crucial information on the stability of the FFLO phase in magnetic fields.

Tuning the in-plane Fulde-Ferrell-Larkin-Ovchinnikov state in a superconductor/ferromagnet/normal-metal hybrid structure by current or magnetic field

Temperature-induced transition of thin superconductor/ferromagnet/normal-metal (S/F/N) hybrid structure to an in-plane Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is accompanied by vanishing of effective inverse magnetic field penetration depth ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ [S. V. Mironov et al., Phys. Rev. Lett. 121, 077002 (2018)]. Here we show that ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ goes to zero only in the limit of zero magnetic field $H\ensuremath{\rightarrow}0$ and at any finite parallel $H$ or in-plane current $I$ it is finite and positive in the FFLO state, which implies a diamagnetic response. We demonstrate that ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ has a nonmonotonic dependence on $H$ and $I$ not only in the parameter range corresponding to the FFLO phase domain but also in its vicinity. We find that for S/F/N/F/S structures with certain thicknesses of F layers there is a temperature-, current-, and magnetic-field-driven transition to and out of the FFLO phase with a simultaneous jump of ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$.

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.

Fulde-Ferrell-Larkin-Ovchinnikov pairing states of a polarized dipolar Fermi gas trapped in a one-dimensional optical lattice

We study the interplay between the long- and short-range interaction of a one-dimensional optical lattice system of two-component dipolar fermions by using the density matrix renormalization group method. The atomic density profile, pairing-pairing correlation function, and the compressibility are calculated in the ground state, from which we identify the parameter region of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing state, half-metal (HM) state, FFLO-HM state, and the normal polarized state, and thus the phase diagram in the coordinates of the long- and short-range interaction strength. The effect of the long-range dipolar interaction on the FFLO state is discussed in details. We find that the long-range part of the dipole-dipole interaction does not sweep away the FFLO superconducting region that is driven by the short-range interaction in the Hubbard model, and thus the FFLO state survives in the wide parameter space of the long-range interaction, polarization and the filling.

Stabilization of the Fulde–Ferrell–Larkin–Ovchinnikov State by Tuning In-plane Magnetic-Field Direction: Application to a Quasi-One-Dimensional Organic Superconductor

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in quasi-one-dimensional systems with warped Fermi surfaces is examined in strong parallel magnetic fields. It is shown that the state is extremely stable for field directions around nontrivial optimum directions, at which the upper critical field exhibits cusps, and that the stabilization is due to a Fermi-surface effect analogous to the nesting effect for the spin density wave and charge density wave. Interestingly, the behavior with cusps is analogous to that in a square lattice system in which the hole density is controlled. For the organic superconductor (TMTSF)_2ClO_4, when the hopping parameters obtained by previous authors based on X-ray crystallography results are assumed, the optimum directions are in quadrants consistent with the previous experimental observations. Furthermore, near this set of parameters, we also find sets of hopping parameters that more precisely reproduce the observed optimum in-plane field directions. These results are consistent with the hypothesis that the FFLO state is realized in the organic superconductor.

Temperature Controlled Fulde-Ferrell-Larkin-Ovchinnikov Instability in Superconductor-Ferromagnet Hybrids.

We show that a wide class of layered superconductor-ferromagnet (S/F) hybrids demonstrates the emergence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase well below the superconducting transition temperature. By decreasing the temperature, one can switch the system from uniform to the FFLO state which is accompanied by the damping of the diamagnetic Meissner response down to zero and also by the sign change in the curvature of the current-velocity dependence. Our estimates show that an additional layer of the normal metal (N) covering the ferromagnet substantially softens the conditions required for the predicted FFLO instability, and for existing S/F/N systems, the temperature of the transition into the FFLO phase can reach several kelvins.

Metric-Space Approach for Distinguishing Quantum Phase Transitions in Spin-Imbalanced Systems

Metric spaces are characterized by distances between pairs of elements. Systems that are physically similar are expected to present smaller distances (between their densities, wave functions and potentials) than systems that present different physical behaviors. For this reason metric spaces are good candidates for probing quantum phase transitions, since they could identify regimes of distinct phases. Here we apply metric space analysis to explore the transitions between the several phases in spin imbalanced systems. In particular we investigate the so-called FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) phase, which is an intriguing phenomenon in which superconductivity and magnetism coexist in the same material. This is expected to appear for example in attractive fermionic systems with spin-imbalanced populations, due to the internal polarization produced by the imbalance. The transition between FFLO phase (superconducting phase) and the normal phase (non-superconducting) and their boundaries have been subject of discussion in recent years. We consider the Hubbard model in the attractive regime for which Density Matrix Renormalization Group calculations allow us to obtain the exact density function of the system. We then analyze the exact density distances as a function of the polarization. We find that our distances display signatures of the distinct quantum phases in spin-imbalanced fermionic systems: with respect to a central reference polarization, systems without FFLO present a very symmetric behavior, while systems with phase transitions are asymmetric.

Resolving different pairing states in Weyl superconductors through the single-particle spectrum

We study theoretically single-particle spectra of Weyl superconductors. Three different superconducting pairing states are addressed, which are the BCS-type states with the $s$-wave pairing symmetry and the $p+ip$-wave pairing symmetry, and the FFLO pairing state. We elaborate that these three states can be resolved well based on the bulk and surface spectral functions as well as the local density of states. The single impurity effect is also explored, which may help us to differentiate the BCS-type pairing states and the FFLO state further.

Interplay between orbital-quantization effects and the Fulde-Ferrell-Larkin-Ovchinnikov instability in multiple-band layered superconductors

We explore superconducting instability for a clean two-band layered superconductor with deep and shallow bands in the magnetic field applied perpendicular to the layers. In the shallow band, the quasiclassical approximation is not applicable, and Landau quantization has to be accounted for exactly. The electronic spectrum of this band in the magnetic field is composed of the one-dimensional Landau-level minibands. With increasing magnetic field the system experiences series of Lifshitz transitions when the chemical potential enters and exits the minibands. These transitions profoundly influence the shape of the upper critical field at low temperatures. In addition, the Zeeman spin splitting may cause the nonuniform state with interlayer modulation of the superconducting order parameter (Fulde-Ferrell-Larkin-Ovchinnikov state). Typically, the quantization effects in the shallow band strongly promote the formation of this state. The uniform state remains favorable only in the exceptional resonance cases when the spin-splitting energy exactly matches the Landau-level spacing. Furthermore, for specific relations between electronic spectrum parameters, the alternating FFLO state may realize, in which the order parameter changes sign between the neighboring layers. For all above cases, the reentrant high-field superconducting states may emerge at low temperatures if the shallow band has significant contribution to the Cooper pairing.

Spin-imbalanced pairing and Fermi surface deformation in flat bands

We study the attractive Hubbard model with spin imbalance on two lattices featuring a flat band: the Lieb and kagome lattices. We present mean-field phase diagrams featuring exotic superfluid phases, similar to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, whose stability is confirmed by dynamical mean-field theory. The nature of the pairing is found to be richer than just the Fermi surface shift responsible for the usual FFLO state. The presence of a flat band allows for changes in the particle momentum distributions at null energy cost. This facilitates formation of nontrivial superfluid phases via multiband Cooper pair formation: the momentum distribution of the spin component in the flat band deforms to mimic the Fermi surface of the other spin component residing in a dispersive band. The Fermi surface of the unpaired particles that are typical for gapless superfluids becomes deformed as well. The results highlight the profound effect of flat dispersions on Fermi surface instabilities, and provide a potential route for observing spin-imbalanced superfluidity and superconductivity.

Reentrant Fulde-Ferrell-Larkin-Ovchinnikov superfluidity in the honeycomb lattice

We study superconducting properties of population-imbalanced ultracold Fermi mixtures in the honeycomb lattice that can be effectively described by the spin-imbalanced attractive Hubbard model in the presence of a Zeeman magnetic field. We use the mean-field theory approach to obtain ground state phase diagrams including some unconventional superconducting phases such as the Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) phase. We show that this phase is characterized by atypical behaviour of the Cooper pairs total momentum in the external magnetic field. We show that the momentum changes its value as well as direction with change of the system parameters. We discuss the influence of van Hove singularities on the possibility of the reentrant FFLO phase occurrence, without a BCS precursor.

Excitonic Phase Diagram of the Three-Chain Hubbard Model for Semiconducting and Semimetallic Ta2NiSe5

Transition metal chalcogenide Ta$_2$NiSe$_5$, a promising material for the excitonic insulator, is investigated on the basis of the quasi-one-dimensional three-chain Hubbard model with two conduction ($c$) bands and one valence ($f$) band. In the semimetallic case where only one of two $c$ bands and the $f$ band cross the Fermi level, the transition from the $c$-$f$ compensated semimetal to the uniform excitonic insulator takes place at low temperature as the same as in the semiconducting case. On the other hand, when another $c$ band also crosses the Fermi level, the system shows three types of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) excitonic orders characterized by the condensation of excitons with finite center-of-mass momentum $q$ corresponding to the three types of nesting vectors between the imbalanced two $c$ and one $f$ Fermi surfaces. The obtained FFLO states are metallic in contrast to the excitonic insulator and are expected to be observed in semimetallic Ta$_2$NiSe$_5$ under high pressure.

Spatially Nonuniform Superconductivity in Quasi-Two-Dimensional Organic Charge-Transfer Salts

In the following, a brief overview on the recently found robust experimental evidence for the existence of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state in layered organic superconductors is given. These electronically quasi-two-dimensional (2D) clean-limit superconductors are ideally suited for observing FFLO states. Applying a magnetic field parallel to the layers suppresses orbital effects and superconductivity is observed beyond the Pauli paramagnetic limit. Both, thermodynamic as well as microscopic experimental data show the existence of an additional high-field low-temperature superconducting state having a one-dimensionally modulated order parameter.

Instability of Fulde-Ferrell-Larkin-Ovchinnikov states in atomic Fermi gases in three and two dimensions

The exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states have been actively searched for experimentally since the mean-field based FFLO theories were put forward half a century ago. Here we investigate the stability of FFLO states against unavoidable pairing fluctuations, and conclude that FFLO superfluids cannot exist due to their intrinsic instability in three and two dimensions. This explains their absence in experimental observations in both condensed matter systems and the most recent, more promising ultracold atomic Fermi gases with a population imbalance.

Orbital effect for the Fulde-Ferrell-Larkin-Ovchinnikov phase in a quasi-two-dimensional superconductor in a parallel magnetic field

We theoretically study the orbital destructive effect against superconductivity in a parallel magnetic field in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO or LOFF) phase at zero temperature in a quasi-two-dimensional (Q2D) conductor. We demonstrate that at zero temperature a special parameter, $\lambda = l_{\perp}(H)/d$, is responsible for strength of the orbital effect, where $l_{\perp}(H)$ is a typical "size" of the quasi-classical electron orbit in a magnetic field and $d$ is the inter-plane distance. We discuss applications of our results to the existing experiments on the FFLO phase in the organic Q2D conductors $\kappa$-(ET)$_2$Cu(NCS)$_2$ and $\kappa$-(ET)$_2$Cu[N(CN)$_2$]Cl.

Magnetic-field-induced 1st order transition to FFLO state at paramagnetic limit in 2D superconductors

We have recently reported the first direct calorimetric observation of a magnetic-field-induced first-order phase transition into a high-field FFLO superconducting state at the Clogston-Chandrasekar ‘Pauli’ paramagnetic limitHp in a 2D superconductor κ − (BEDT-TTF)2Cu(NCS)2. The high-field state is both higher entropy and strongly paramagnetic, as thermodynamically required for the FFLO state. Here we compare our results with theoretical predictions for the field dependence of the high-field FFLO state in the 2D limit, revealing tentative evidence for transitions between FFLO states of differing order parameter. We also present calorimetric evidence for a 1st order phase transition into the FFLO state for a second 2D organic superconductor: β″ − (BEDT-TTF)2SF5(CH)2(CF)2(SO)3.

Axial anomaly and nesting in the inhomogeneous chiral phase

Axial anomaly and nesting is elucidated in the context of the inhomogeneous chiral phase. Using the Gross-Neveu models in 1+1 dimensions, we shall discuss axial anomaly and nesting from two different points of view: one is homogeneous chiral transition and the other is the Ferrel-Fulde-Larkin-Ovchinnikov (FFLO) state in superconductivity, which are closely related to each other by way of duality. It is shown that axial anomaly leads to a particular kind of the FFLO state within the two dimensional Nambu-Jona Lasinio model, where axial anomaly is manifested in a different mode. Nesting is a driving mechanism for both phenomena, but its realization has different features. We reconsider the effect of nesting in the context of duality.

Thermodynamic Evidence for the Fulde-Ferrell-Larkin-Ovchinnikov State in the KFe_{2}As_{2} Superconductor.

We investigate the magnetic phase diagram near the upper critical field of KFe_{2}As_{2} by magnetic torque and specific heat experiments using a high-resolution piezorotary positioner to precisely control the parallel alignment of the magnetic field with respect to the FeAs layers. We observe a clear double transition when the field is strictly aligned in the plane and a characteristic upturn of the upper critical field line, which goes far beyond the Pauli limit at 4.8T. This provides firm evidence that a Fulde-Ferrell-Larkin-Ovchinnikov state exists in this iron-based KFe_{2}As_{2} superconductor.