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arXiv:cond-mat/0606335 · Physical Review B

Mixed state of a lattice d -wave superconductor

Ashot Melikyan, Zlatko Tešanović

Published Oct 4, 2006• 13 citations
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Abstract

We study the mixed state in an extreme type-II lattice ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductor in the experimentally most relevant regime of intermediate magnetic fields ${H}_{c1}⪡H⪡{H}_{c2}$. We analyze the low energy spectrum of the problem, dominated by nodal Dirac-like quasiparticles with momenta near ${\mathbf{k}}_{F}=(\ifmmode\pm\else\textpm\fi{}{k}_{D},\ifmmode\pm\else\textpm\fi{}{k}_{D})$, and find that the spectrum exhibits characteristic oscillatory behavior with respect to the product of ${k}_{D}$ and magnetic length $l$. The Simon-Lee scaling, predicted in this regime, is satisfied only on average, with the magnitude of the oscillatory part of the spectrum displaying the same ${l}^{\ensuremath{-}1}$ dependence as its monotonous ``envelope'' part. In general, the spectrum obeys a scaling law ${E}_{n\mathbf{k}}=(\ensuremath{\hbar}{v}_{F}∕l){\mathcal{E}}_{n}(\mathbf{k}l,t∕\ensuremath{\Delta},{k}_{D}l)$, where $\mathcal{E}$ is a dimensionless universal $2\ensuremath{\pi}$-periodic function of ${k}_{D}l$. The oscillatory behavior of the spectrum is due to the internodal interference enhanced by the singular nature of the low-energy eigenfunctions near vortices. Our results constitute an example of a finite size scaling of the Dirac-type quantum criticality. We also study a separate problem of a single vortex piercing an isolated superconducting grain of size $L\ifmmode\times\else\texttimes\fi{}L$. Here we find that the periodicity of the quasiparticle energy oscillations with respect to ${k}_{D}L$ is doubled relative to the case where the field is zero and the vortex is absent, both such oscillatory behaviors being present at the leading order in ${L}^{\ensuremath{-}1}$. Finally, we review the overall features of the tunneling conductance experiments in $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ (YBCO) and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (BSCCO), and suggest an interpretation of the peaks at $5--20\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ observed in the tunneling local density of states in these materials. We find that in the case of a pure $d$-wave superconducting order parameter with featureless vortex cores, the zero bias conductance peak (ZBCP) appears only on the sites that are the immediate nearest neighbors of vortex locations, while all the other sites in the close vicinity of vortices exhibit no such ZBCP and instead display pronounced peaks at subgap energies, typically at about a half or less of the coherence peak energy. Furthermore, we find that the on-site ZBCP can be strongly suppressed by enhanced local pairing near a vortex.

DOI
10.1103/physrevb.74.144501

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For the $t$-$J$ model, variational wave functions can generally be constructed based on an accurate description of antiferromagnetism (AFM) at half-filling and an exact phase-string sign structure under doping. The single-hole-doped and two-hole-doped states, as determined by variational Monte Carlo (VMC) simulations, display sharply contrasting behaviors. The single-hole state constitutes a ``cat state'' that resonates strongly between a quasiparticle component and a local loop-current component, with approximately equal weights. In the ground state, the quasiparticle spectral weight $Z_{\mathbf{k}}$ peaks at momenta $\mathbf{k}_0 \equiv (\pm\fracπ{2},\pm\fracπ{2})$. The total-energy dispersion versus $\mathbf{k}$ agrees remarkably well with the Green function Monte Carlo results. However, Landau's one-to-one correspondence hypothesis for quasiparticles breaks down here with the incoherent component exhibiting intrinsic magnetization originating from a minimal $2\times2$ loop current that forms a $4\times4$ pattern on the square lattice--a finding in excellent agreement with density matrix renormalization group (DMRG) calculations. In the two-hole ground state, a new pairing mechanism is revealed: the two holes are automatically fused into a tightly bound object consisting of an incoherent $d_{xy}$ pairing along the diagonal direction by compensating the local loop currents. This hole pair is again a ``cat state'' that resonates strongly between the incoherent $d_{xy}$ and a coherent $d_{x^2-y^2}$ Cooper channel to gain substantial hopping energy. Its size extends over an area of about $4\times 4$ lattice spacings, much smaller than the divergent AFM correlation length, implying that it should survive as a minimal superconducting building block even in the dilute doping regime. Experimental implications and the generalization to the finite-doping case are briefly addressed.

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The problem of dx2-y2-wave quasiparticles in a weakly disordered Abrikosov vortex lattice is studied. Starting with a periodic lattice, the topological structure of the magnetic crystal momenta of gapless fermions is found for the particle-hole symmetric case. If in addition the site centered inversion symmetry is present, both the location and the number of the gapless fermions can be determined using an index theorem. In the case of spatially aperiodic vortex array, Simon and Lee scaling is found to be violated due to a quantum anomaly. The electronic density of states is found to scale with the root-mean-square vortex displacement as sqrt[H]f(u2rms/xi2), while thermal conductivity is H independent, but different from the H=0 case.
We use a number of large-N limits to explore the competition between ground states of square lattice doped antiferromagnets which break electromagnetic U(1),U(1),U(1), time-reversal, or square lattice space-group symmetries. Among the states we find are d-, ${(s}^{*}+id)\ensuremath{-},$ and ${(d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}})$-wave superconductors, Wigner crystals, Wigner crystals of hole pairs, orbital antiferromagnets (or staggered-flux states), and states with spin-Peierls and bond-centered charge stripe order. In the vicinity of second-order quantum phase transitions between the states, we go beyond the large-N limit by identifying the universal quantum field theories for the critical points, and computing the finite temperature, quantum critical damping of fermion spectral functions. We identify candidate critical points for the recently observed quantum critical behavior in photoemission experiments on ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{Ca}\mathrm{}\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ by Valla et al. [Science 285, 2110 (1999)]. These involve onset of a charge-density wave, or of broken time-reversal symmetry with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}}$ or s∗+id{s}^{*}+ids∗+id pairing, in a d-wave superconductor. It is not required (although it is allowed) that the stable state in the doped cuprates be anything other than the d-wave superconductor---the other states need only be stable nearby in parameter space. At finite temperatures, fluctuations associated with these nearby states lead to the observed fermion damping in the vicinity of the nodal points in the Brillouin zone. The cases with broken time-reversal symmetry are appealing because the order parameter is not required to satisfy any special commensurability conditions. The observed absence of inelastic damping of quasiparticles with momenta $(\ensuremath{\pi},k),$ $(k,\ensuremath{\pi})$ (with $0<~k<~\ensuremath{\pi})$ also appears very naturally for the case of fluctuations to ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}{+id}_{\mathrm{xy}}$ order.
Focusing on a quantum-limit behavior, we study a single vortex in a clean sss-wave type-II superconductor by self-consistently solving the Bogoliubov--de Gennes equation. The discrete energy levels of the vortex bound states in the quantum limit are discussed. The vortex core radius shrinks monotonically up to an atomic-scale length on lowering the temperature TTT, and the shrinkage stops to saturate at a lower TTT. The pair potential, supercurrent, and local density of states around the vortex exhibit Friedel-like oscillations. The local density of states has particle-hole asymmetry induced by the vortex. These are potentially observed directly by scanning tunneling microscopy.
What are zero energy excitations in high-TcT_{c}Tc​ superconductors?
The long disputed issue of the limiting value of quasiparticle density of states $\r(0) = \r (\e \to 0)$ in a \emph{d-}wave superconductor with impurities (\emph{vs} its linear vanishing, $\r_0(\e) \propto |\e|/\D$, near the nodal point $\e = 0$ in a pure system with the gap parameter $\D$) is discussed. Using the technique of group expansions of Green functions in complexes of interacting impurities, it is shown that finite (˚0)\r(0)(˚​0) value is possible if the (finite) impurity perturbation VVV is spin-dependent (magnetic). The found value has a power law dependence on the impurity concentration ccc: $\r(0) \propto \r_N c^{n}$, where $\r_N$ is the normal metal density of states and n≥2n \geq 2n≥2 is the least number of impurities in a complex that can localize nodal quasiparticle. This result essentially differs from the known predictions of self-consistent approximation: $\r(0) \propto \r_N \sqrt {c/\r_N\D}$ (for the unitary limit V→∞V \to \inftyV→∞) or $\r(0) \propto (\D/cV^2)\exp(-\D/cV^2\r_N)$ (for the Born limit $|V|\r_N \ll 1$). We predict also existence of a narrow interval (mobility gap) around the Fermi energy, where all the states are localized on proper impurity clusters, leading to exponential suppression of low-temperature kinetics.
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.
For the ttt-JJJ model, variational wave functions can generally be constructed based on an accurate description of antiferromagnetism (AFM) at half-filling and an exact phase-string sign structure under doping. The single-hole-doped and two-hole-doped states, as determined by variational Monte Carlo (VMC) simulations, display sharply contrasting behaviors. The single-hole state constitutes a ``cat state'' that resonates strongly between a quasiparticle component and a local loop-current component, with approximately equal weights. In the ground state, the quasiparticle spectral weight ZkZ_{\mathbf{k}}Zk​ peaks at momenta k0≡(±π2,±π2)\mathbf{k}_0 \equiv (\pm\fracπ{2},\pm\fracπ{2})k0​≡(±2π​,±2π​). The total-energy dispersion versus k\mathbf{k}k agrees remarkably well with the Green function Monte Carlo results. However, Landau's one-to-one correspondence hypothesis for quasiparticles breaks down here with the incoherent component exhibiting intrinsic magnetization originating from a minimal 2×22\times22×2 loop current that forms a 4×44\times44×4 pattern on the square lattice--a finding in excellent agreement with density matrix renormalization group (DMRG) calculations. In the two-hole ground state, a new pairing mechanism is revealed: the two holes are automatically fused into a tightly bound object consisting of an incoherent dxyd_{xy}dxy​ pairing along the diagonal direction by compensating the local loop currents. This hole pair is again a ``cat state'' that resonates strongly between the incoherent dxyd_{xy}dxy​ and a coherent dx2−y2d_{x^2-y^2}dx2−y2​ Cooper channel to gain substantial hopping energy. Its size extends over an area of about 4×44\times 44×4 lattice spacings, much smaller than the divergent AFM correlation length, implying that it should survive as a minimal superconducting building block even in the dilute doping regime. Experimental implications and the generalization to the finite-doping case are briefly addressed.