Uniform Superconducting State Research Articles

Superconducting and charge-ordered states in the anisotropic t–J–U model

Motivated by the effect of symmetry breaking in cuprates superconductors YBa2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}Cu3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}O7-δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{7-\\delta }$$\\end{document}, we employ the renormalized mean-field theory to study the presence of uniform superconducting and charge-ordered states in two anisotropic t–J–U models, either with hopping strength anisotropy or antiferromagnetic interaction anisotropy. In the case of uniform superconducting state, compared with the isotropic t–J–U model with only dx2-y2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_{x^2-y^2}$$\\end{document}-wave superconducting state, there is an additional s-wave superconducting state in the model with hopping strength anisotropy. Meanwhile, the hopping anisotropy may enhance the critical Coulomb interaction Uc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U_c$$\\end{document} at the Mott insulator to the Gossamer superconductor transition point, and strong hopping anisotropy may weaken the superconducting state. In the case of a charge-ordered state, hopping anisotropy may suppress the amplitude of the charge density waves and pair density waves, which originate from local Coulomb interactions. These results indicate that the effects of hopping anisotropy and local Coulomb interactions are competitive. Moreover, the antiferromagnetic interaction anisotropy only weakly suppresses the superconducting gap and density wave amplitude. Our results show that the t–J–U model with hopping anisotropy is qualitatively consistent with experimental superconducting pair symmetry and charge density waves in the YBa2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document}Cu3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_3$$\\end{document}O7-δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_{7-\\delta }$$\\end{document} system.

Charge, bond, and pair density wave orders in a strongly correlated system

The coexistence of multiple quasi-degenerate orders is the hallmark of the strongly correlated materials. Experiments often reveal several spatially modulated orders in the underdoped cuprates. This has come to the forefront with the possible detection of the pair density wave states. However, microscopic calculations often struggle to stabilize such spatially modulating orders as the ground state in the strong correlation limit. This work uses the $t-t^\prime-J$-model with an additional nearest-neighbor repulsion to stabilize spatially oscillating charge, bond, and pairing orders in the underdoped regime. We employ the standard Gutzwiller approach while treating the inhomogeneity for the spatial orders using the self-consistent Hartree-Fock-Bogoliubov methodology. Our calculations reveal that unidirectional bond density states coexisting with charge and pairing modulations can have lower energy than the uniform superconducting state over an extensive doping range. These modulating states vanish monotonically as the modulation wavevector becomes shorter with increased dopings. The finite momentum orders melt upon increasing doping to a vestigial nematic state which breaks the rotational symmetry of the system. The spatial features of the ground state at each doping reveal multiple wavevectors, which potentially drives the incommensuration of charge orders. Interestingly, the spatially modulating states are absent when the strong correlations criteria are relaxed, suggesting that the removal of double occupancy aids the stabilization of density wave orders.

Thermodynamic evidence for the formation of a Fulde-Ferrell-Larkin-Ovchinnikov phase in the organic superconductor λ−(BETS)2GaCl4

In this work, the thermodynamic properties of the organic superconductor $\lambda$-(BETS)$_2$GaCl$_4$ are investigated to study a high-field superconducting state known as the putative Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We observed a small thermodynamic anomaly in the field $H_{\rm FFLO}$ $\sim$ 10~T, which corresponds to the Pauli limiting field $H_{\rm P}$. This anomaly probably originates from a transition from a uniform superconducting state to the FFLO state. $H_{\rm FFLO}$ does not show a strong field-angular dependence due to a quasi-isotropic paramagnetic effect in $\lambda$-(BETS)$_2$GaCl$_4$. The thermodynamic anomaly at $H_{\rm FFLO}$ is smeared out and low-temperature upper critical field $H_{\rm c2}$ changes significantly if fields are not parallel to the conducting plane even for a deviation of $\sim$0.5$^{\circ}$. This behavior indicates that the high-field state is very unstable, as it is influenced by the strongly anisotropic orbital effect. Our results are consistent with the theoretical predictions on the FFLO state, and show that the high-field superconductivity is probably an FFLO state in $\lambda$-(BETS)$_2$GaCl$_4$ from a thermodynamic point of view.

Nonuniform superconductivity and Josephson effect in a conical ferromagnet

Using the Gorkov equations, we provide an exact solution for a one-dimensional model of superconductivity in the presence of a conical helicoidal exchange field. Due to the special type of symmetry of the system, the superconducting transition always occurs into a nonuniform superconducting phase (in contrast with the Fulde-Ferrell-Larkin-Ovchinnikov state, which appears only at low temperatures). We directly demonstrate that the uniform superconducting state in our model carries a current and thus does not correspond to the ground state. We study in the framework of the Bogoliubov-de Gennes approach the properties of the Josephson junction with a conical ferromagnet as a weak link. In our numerical calculations, we do not use any approximations (such as, e.g., a quasiclassical approach), and we show a realization of an anomalous $\phi_{0}$ junction (with a spontaneous phase difference $\phi_{0}$ in the ground state). The spontaneous phase difference $\phi_{0}$ strongly increases at high values of the exchange field near the borderline with a half-metal, and it exists also in the half-metal regime.

Heat transport in nonuniform superconductors

We calculate electronic energy transport in inhomogeneous superconductors using a fully self-consistent nonequilibrium quasiclassical Keldysh approach. We develop a general theory and apply it to a superconductor with an order parameter that forms domain walls of the type encountered in the Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a domain wall is inherently anisotropic and nonlocal. The bound states in the nonuniform region play a crucial role and control heat transport in several ways: (i) they modify the spectrum of quasiparticle states and result in Andreev reflection processes and (ii) they hybridize with the impurity band and produce a local transport environment with properties very different from those in a uniform superconductor. As a result of this interplay, heat transport becomes highly sensitive to temperature, magnetic field, and disorder. For strongly scattering impurities, we find that the transport across domain walls at low temperatures is considerably more efficient than in the uniform superconducting state.

Quasi-one-dimensional pair density wave superconducting state

We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model consists of an array of strongly-interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions and tunneling of the electrons and Cooper pairs between them. Within the interchain mean-field theory, we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain mean field theory. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles.

Finite isospin chiral perturbation theory in a magnetic field

The phase diagram of finite isospin, zero temperature QCD with the pions coupled to photons in a uniform external magnetic field is explored in the low field, small isospin density regime for which chiral perturbation theory is a valid description. For realistic pion masses, the system behaves as a type-II superconductor: a uniform superconducting state is formed at low-enough magnetic fields, a vortex state for intermediate magnetic fields and finally a normal state for large magnetic fields. In each these phases (including the vortex phase), the $\pi^{0}$ remains uncondensed just as in the zero-external field problem. The critical magnetic field where the phase transition from the uniform superconducting state to a vortex state occurs was found numerically.

Field-induced reentrant superconductivity in thin films of nodal superconductors

Previous works on nodal d-wave superconductors have shown that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) like modulated superconducting (SC) state can be realized with no magnetic field when quasiparticles acquire an additional linear term in the wavevector in their dispersion. In the present work, stability of such a novel modulated SC state in an artificial film against an applied magnetic field is studied. As a reflection of the presence of the two different FFLO states, one close to zero field and the other at the high field end, in a single field v.s. temperature phase diagram of thin films, the conventional uniform SC state generally tends to appear as a reentrant ordered phase bounded by the normal phase in {\it lower} fields.

Double-stage continuous-discontinuous superconducting phase transition in the Pauli paramagnetic limit of a 3D superconductor: the URu2Si2 case

The sharp suppression of the de-Haas van-Alphen oscillations observed in the mixed superconducting (SC) state of the heavy fermion compound URu$_{2}$Si$% _{2}$ is shown to confirm a theoretical prediction of a narrow double-stage SC phase transition, smeared by fluctuations, in a 3D paramagnetically-limitted superconductor. The predicted scenario of a second order transition to a nonuniform (FFLO) state followed by a first order transition to a uniform SC state, obtained by using a non-perturbative approach, is also found to be consistent with recent thermal conductivity measurements performed on this material.

Theory of Unusual Superconducting Phase Transitions in Heavy Fermion Metals at High Magnetic Fields

We present a non-perturbative theory of paramagnetically-driven superconducting (SC) phase transitions in heavy-fermion metals, which reliably determines their stable SC phases, treats properly the corresponding finite jumps of the order parameter, and can account for unusual features reported recently for this type of materials. It is found that for quasi-2D heavy-fermion metals, such as CeCoIn5, at high magnetic fields oriented perpendicular to the highly conducting planes, the effect of the Fulde-Ferrel (FF) modulation is too weak to prevent a direct first-order phase transition from the normal to the uniform SC state. For 3D heavy-fermion metals, such as URu2Si2, the FF modulation stabilizes, under a decreasing magnetic field, a non-uniform SC state via a second-order phase transition from the normal state. However, at a slightly lower field the modulated phase becomes unstable, transforming to a uniform SC state via a first-order transition.

Scaling of the superfluid density in strongly underdopedYBa2Cu3O6+y: Evidence for a Josephson phase

Recent measurements on extremely underdoped ${\text{YBa}}_{2}{\text{Cu}}_{3}{\text{O}}_{6+y}$ [Phys. Rev. Lett. 99, 237003 (2007)] have allowed the critical temperature ${T}_{c}$, superfluid density ${\ensuremath{\rho}}_{s0}\ensuremath{\equiv}{\ensuremath{\rho}}_{s}(T⪡{T}_{c})$, and dc conductivity ${\ensuremath{\sigma}}_{\text{dc}}(T\ensuremath{\gtrsim}{T}_{c})$ to be determined for a series of electronic dopings for ${T}_{c}\ensuremath{\simeq}3--17\text{ }\text{K}$. The general scaling relation ${\ensuremath{\rho}}_{s0}/8\ensuremath{\simeq}4.4{\ensuremath{\sigma}}_{\text{dc}}{T}_{c}$ is observed, extending the validity of both the $ab$-plane and $c$-axis scaling an order of magnitude and creating a region of overlap. This suggests that strongly underdoped materials may constitute a Josephson phase; as the electronic doping is increased a more uniform superconducting state emerges.

Discontinuous superconducting transitions in the paramagnetic limit: a non-perturbative approach

The unusual superconducting (SC) phase transitions occuring under competing orbital and spin pair breaking, which characterize clean strongly type-II superconductors at low temperatures, are investigated within a non-perturbative approach, which avoides the difficulties encountered in various perturbative approaches and enables comparison with recent experimental data. It is shown that in a 3D system with strong spin-splitting, a spatial (FFLO) modulation of the order parameter along the magnetic-field direction preserves the continuous nature of the SC transition. However, at a magnetic field slightly below Hc2 the FFLO state becomes unstable, transforming discontinuously into a uniform SC state via a first-order phase transition. Our calculation shows that the entropy jump at the first-order phase transition is siginificantly larger than its total variation in the continuous region between the two transitions, in agreement with recent thermal-conductivity measurements performed on the heavy-fermion compound URu2Si2.

Fermi-liquid effects in the Fulde-Ferrell-Larkin-Ovchinnikov state of two-dimensionald-wave superconductors

We study the effects of Fermi-liquid interactions on quasi-two-dimensional $d$-wave superconductors in a magnetic field. The phase diagram of the superconducting state, including the periodic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in high magnetic fields, is discussed for different strengths of quasiparticle many-body interactions within Landau's theory of Fermi liquids. Decreasing the Fermi-liquid parameter ${F}_{0}^{a}$ causes the magnetic spin susceptibility of itinerant electrons to increase, which in turn leads to a reduction of the FFLO phase. It is shown that a negative ${F}_{0}^{a}$ results in a first-order phase transition from the normal to the uniform superconducting state in a finite temperature interval. Finally, we discuss the thermodynamic implications of a first-order phase transition for $\mathrm{Ce}\mathrm{Co}{\mathrm{In}}_{5}$.

Interplay of paramagnetic, orbital, and impurity effects on the phase transition of a normal metal to the superconducting state

We derive the generalized Ginzburg-Landau free energy functional for conventional and unconventional singlet superconductors in the presence of paramagnetic, orbital and impurity effects. Within the mean field theory, we determine the criterion for appearence of the non uniform (Fulde-Ferrell-Larkin-Ovchinnikov) superconducting state, with vortex lattice structure and additional modulation along the magnetic field. We also discuss the possible change of the order of transition from normal to superconducting state. We find that the superconducting phase diagram is very sensitive to geometrical effects such as the nature of the order parameter and the shape of the Fermi surface. In particular, we obtain the qualitative phase diagrams for three-dimensional isotropic s-wave superconductors and in quasi two-dimensional d-wave superconductors under magnetic field perpendicular to the conducting layers. In addition, we determine the criterion for instability toward non uniform superconducting state in s-wave superconductors in the dirty limit.

Physics of vortex core in chiral p-wave superconductor

We describe the essence of quantum effects inside s-wave and chiral p-wave vortices, without explicit use of quasiclassical Green function formalism. Physical quantities such as the impurity scattering rate and nuclear spin relaxation rate contain the coherence factor of the Andreev bound states in the matrix element of the transitions. The coherence factor of the Andreev bound state is different from that of a quasiparticle in bulk superconductors. Consequently, the physics within vortex core is different from that in normal state or that in spatially uniform superconducting state.

Induced magnetization and inhomogeneous superconductivity in presence of external magnetic field

Thermodynamic characteristics of the one-dimensional attractive Hubbard model are calculated by means of the generalized self-consistent field (GSCF) approach for a wide range of interaction strength U⩽0, in the presence of external magnetic field h⩾0, temperature T⩾0 and electron concentration 0⩽ n⩽1. The GSCF theory shows a sharp transition driven by h>0 and T from the uniform superconducting state into a spatially inhomogeneous state in which the order parameter Δ q (−) has wave vector ( 0<| q |⩽π ). The concentration of double occupied sites and the bound electron pairs with opposite spins n pair are calculated and we found a simple relationship between the order parameters Δ (−) q and average spin s, valid for the arbitrary U, all n and h.

Phase Diagram of the Fulde-Ferrell-Larkin-Ovchinnikov State in a Three-Dimensional Superconductor

The phase diagram of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in a 3-dimensional superconductor is discussed. We use the quasi-classical Green's function to calculate the free energy. It is shown that the phase transition from the normal state is of first order at high temperatures but is of second order near T =0. To describe the phase transition in the Ginzburg-Landau theory, one has to take into account up to eighth order term with respect to the order parameter. The phase transition from the FFLO state to the uniform superconducting state is of second order in all the temperature range T <0.561 T c in accordance with Burkhardt and Rainer's result in a 2-dimensional system.

Fulde‐Ferrell‐Larkin‐Ovchinnikov state in layered superconductors

AbstractThe phase diagram of a layered superconductor in a large parallel magnetic field is calculated. This includes a calculation of the lower critical field beyond which the superconductor is in the FFLO‐state and possesses a spatially modulated order parameter and spin polarization. The order parameter, spin polarization, free‐energy density, and phase boundaries of this unconventional superconducting phase are evaluated numerically in the complete B–T plane. The analysis suggests that the transition at the lower critical field is of second order, and not of first order, as previously assumed. The order parameter of the FFLO‐state merges continuously into the uniform superconducting state.

Dynamics of holes and spins in the Hubbard t-J model and high-temperature superconductivity.

We study the t-J model as a microscopic model of high-temperature superconductors. The local constraints in the slave-fermion representation are solved by the projection operator onto the spin states and the ${\mathrm{CP}}^{1}$ spin variables. This resolution reveals that the antiferromagnetic (AF) J coupling of spins is accompanied with a four-Fermi interaction of charged holes. It describes attraction of fermionic holes on nearest-neighbor (NN) sites and induces formation of pairs of NN holes. Each pair carries two units of charge. Condensation of such hole pairs breaks the global U(1) charge symmetry and generates the superconductivity. The possible phase-separation state may become unstable against the uniform superconducting state due to the kinetic energy of holes. We obtain an effective lattice model in the short-range AF state by integrating out the odd-site spin fluctuations in the short-range AF order. Spin fluctuations induce attractions between next-NN holes. The model includes the effective model of Shraiman and Siggia for mobile holes in AF magnets as its normal-state part.Global magnetic properties, such as the N\'eel state and the spiral state, are described by this model. To study the superconducting phase, we develop a hole-pair theory by introducing auxiliary link variables as a hole-pair field. Their effective action gives the ``gap'' equation. The Ginzburg-Landau theory of it describes the superconducting-normal phase transition. We identify its general structure, which takes a form very similar to U(1) lattice gauge theory coupled with Higgs field. The pure gauge part reflects the magnetic J coupling, while the Higgs part is supplied by the elctron-hopping t term. The hole-pair condensation is expected at sufficiently large hole density because the phases of pair field are to be stabilized there coherently through a Higgs mechanism. A superconducting ``flux'' state of pairs is favored energetically. In this background condensation, low-energy excitations of holes are described by normal hopping of Fermi-liquid-type plus relativistic massive Dirac particles in a parity doublet. They are coupled with the gauge fields of ${\mathrm{CP}}^{1}$ spins in a U(1) gauge-invariant manner. This composes the hole part of our effective field theory in the superconducting phase. The spin part is the familiar nonlinear ${\mathrm{CP}}^{1}$ \ensuremath{\sigma} model. Our microscopic derivation of the effective field theory at low-temperature relates its parameters to those of the original t-J model. Throughout the study, we recognize a global and strong correlation between the magnetism and the superconductivity, or spins and holes, reflecting their common origin, i.e., the J coupling. The study of a simple solvable model presents a plausible critical line of superconductivity in the temperature-concentration plane.

Interplay of Impurity Pinning and Superconducting States in Quasi-One-Dimensional Conductors

Effect of impurity pinning on quasi-one-dimensional superconducting systems is studied by extending a single impurity problem treated in the previous work [J. Phys. Jpn. 53 (1984) 1206] to the case of many impurities. The ground state energy and related properties are calculated numerically based on the self-consistent harmonic approximation. It is shown that a domain of the pinned state followed by the soliton-formation of the Josephson phase can exist in the spatially uniform superconducting state.