Session D9: Superconductivity: Charge Order and Inhomogeneity

What does charge order have to do with the mechanism of high temperature superconductivity?

Charge order clearly ``competes'' with superconductivity under many circumstances. It always tends to suppress the superfluid stiffness of the superconducting state by localizing electrons that might otherwise participate in the superconducting condensate. Thus, where the superconducting T$_{c}$ is determined by phase fluctuations, charge order suppresses T$_{c}$. However, there is suggestive experimental and theoretical evidence that charge ordering of just the right sort can enhance pairing, and hence ``assist'' superconductivity. Some of this evidence will be presented.   

How optimal inhomogeneity produces high temperature superconductivity

The role of Coulomb frustrated phase separation in doped Mott insulators, and especially the consequences of the resulting local electronic structures on the ``mechanism'' of high temperature superconductivity will be discussed. The resulting perspective on superconductivity in the cuprates, and on the more general theoretical issue of what sorts of systems can support high temperature superconductivity is discussed as are some of the general, qualitative aspects of the experimental lore which should constrain any theory of the mechanism. Finally, it is show how they are accounted for within the context of the present theory. Reference: S. A. Kivelson and E. Fradkin, ``How optimal inhomogeneity produces high temperature superconductivity,'' cond-mat/0507459, to appear as a chapter in ``Treatise of High Temperature Superconductivity'' by J. Robert Schrieffer and J. Brooks, to be published (Springer, 2006)   

Thermodynamic properties of inhomogeneous superconductors near their transition temperature

Recently, scanning tunneling spectroscopy (STS) experiments have revealed suggestive evidence of the existence of superconducting gap inhomogeneities at low temperatures in some families of cuprate materials. The consequences of such inhomogeneity near the superconducting transition, however, remain an important and unresolved issue. Here, we study the effect of intrinsic gap inhomogeneities on the mean-field electronic specific heat (and other thermodynamic properties) in the vicinity of the superconducting transition. We consider a spatially-varying pairing interaction in a d-wave BCS model, solve the mean-field equations self-consistently for the magnitude of the gap function, and determine the thermodynamic properties of the system. As T approaches Tc, the coherence length grows, causing the system to become effectively more homogeneous due to self-averaging; we explore the extent to which various types of inhomogeneity remain important or get washed out near Tc.   

Global phase diagram of the checkerboard Hubbard model

Local electronic structure (self-organized inhomogeneity) may play an essential role for the ``mechanism'' of high-T$_{c}$ superconductivity. Moreover, in the limit of large inhomogeneity, well-controlled theoretical solutions of strongly interacting models can be obtained. We have computed the phase diagram of the checkerboard Hubbard model in the limit of small inter-cluster electron hopping, t', for all doping (x=hole density per site) and for all interaction strengths, 0$<$U/t. For O(t')$<$U$<$U$_{c}$ =4.58t, and all 0$\le $ x $\le $1/2, the existence of an effective pair attraction results in one of two d-wave superconducting ground states - either with nodal or without nodal quasiparticles. For U$_{c }<$U$<$U$_{t }$=18.6t, the ground state is a Fermi liquid of spin 1/2 fermions with two possible orbital flavors. Interestingly, around x=1/4 the ground state is a spin-1/2 antiferromagnet which also possesses alternating orbital currents on every other plaquette that spontaneously break time reversal symmetry. For U$>$U$_{t}$, the ground state is a Fermi liquid of fermions with spin-3/2, with a spin-3/2 antiferromagnet is favored near x=1/4. By including next nearest neighbor hopping, t$_{2}$, within clusters, we can study the physics of particle-hole asymmetry. Strikingly, we find that increasing t$_{2}$ increases the range of U for which hole doping leads to a superconducting state, but suppresses the range of U for electron doping. (For t$_{2}\to $--t$_{2}$, the roles of electrons and holes are interchanged.)   

Superconductivity in zigzag CuO chains

Superconductivity was recently discovered in Pr$_2$Ba$_4$Cu$_7 $O$_{15- \delta}$ with a maximum T$_c$ of about 12K [1]. This material's structure is identical to that of the high T$_c$ superconductor YBCO$-247$. However, the cupper-oxide planes in this material (which are essential for the superconductivity in YBCO) are known to be insulating. Therefore it is believed that the superconductivity originates in the array of quasi-1d CuO chains and NMR experiments appear to corroborate this belief. In this work we study a microscopic model for a CuO double-chain (zigzag chain) using a combination of bosonization and numerics (DMRG). We derive a schematic phase diagram for this model, which exhibits a narrow doping region where superconducting correlations are dominant and a broader range where CDW correlations are dominant. Unlike the situation in the two-leg Hubbard ladder, superconductivity does not arise from the formation of a spin gap. Rather, it is related to a subtle ordering driven by magnetic interactions. The implications for experiment are discussed. \newline [1] See, for example, Y. Yamada and A. Matsushita, Physica C 426-431, 213(2005).   

Stripes near a Quantum Critical Point

Competing tendency in strongly correlated materials can cause spontaneous nanoscale structure, pattern formation, and even long-range spatial order. We explore the magnetic excitation spectrum in the stripe phase of high-Tc cuprates. Using a semiclassical spin wave treatment, we calculate the dynamical spin structure factor for weakly coupled stripes. We find a characteristic hourglass magnetic excitation spectrum with high-energy peaks rotated by 45 degrees compared to the incommensurate (IC) low-energy peaks in good agreement with the experimental data.The similarity at high energy between this semiclassical treatment and quantum fluctuations in spin ladders may be attributed to the proximity of a quantum critical point with a small critical exponent $\eta $. We also find that the low energy intensity is strongly peaked on the inner branches of the spin wave cones when coupling across the stripes is weak, so that the entire spin wave cone is not likely to be resolvable experimentally. (Phys. Rev. Lett. 97, 017003 (2006), Phys. Rev. B 73, 224525 (2006))   

s-Wave Superconductivity Phase Diagram for the Two Dimensional Inhomogeneous Attractive Hubbard Model

We study s-wave superconductivity in the two-dimensional square lattice attractive Hubbard Hamiltonian for various inhomogeneous patterns of interacting sites at different concentration values $f$. Using the Bogoliubov-de Gennes (BdG) mean field approximation, we find the phase diagram for inhomogeneous interaction patterns in which the on-site interaction $U_i$ takes on two values, $U_i=0,U/(1-f)$ as a function of electron occupation per site $n$ and study the evolution of the phase diagram as $f$ varies. In certain regions of the phase diagram, inhomogeneity results in a larger zero temperature averaged pairing amplitude and also the superconducting phase transition temperature $T_c$, relative to a uniform system with $U_i=U$ on all sites. These effects are observed for stripe, checkerboard, and even random patterns of the attractive centers, suggesting that the pattern of inhomogeneity is unimportant. The phase diagrams also include regions where superconductivity is obliterated due to the formation of various charge ordered phases. We show that for certain regular inhomogeneous patterns, increasing temperature works against the formation of these charge ordered phases and as a result, can enhance superconductivity.   

Competition between charge order and superconductivity in La$_{7/8}$Ba$_{1/8}$CuO$_{4}$

Understanding the role of stripe physics in cuprate superconductors is believed to be essential in elucidating the superconducting mechanism of the cuprates. Despite the fundamental importance of charge ordering in the cuprates, a comprehensive examination of the relationship between charge stripes and superconductivity is still lacking. We have carried out a detailed investigation of temperature and magnetic field dependence of charge order in La$_{7/8}$Ba$_{1/8}$CuO$_{4}$ utilizing high-resolution x-ray scattering. We find that the correlation length of the charge order exhibits unusual temperature and magnetic field dependence. Specifically, at zero field the correlation length decreases as the sample is cooled below $\sim $12K, while it increases as magnetic field is applied in the superconducting phase. These observations suggest that the size of the charge ordered region seems to be inversely correlated with superconductivity. This finding clearly shows that static charge order competes with the superconducting ground state, and supports the microscopic phase separation picture discussed by the recent $\mu $SR work.   

On the dimensionality of spin and charge modulations in 1/8 doped lanthanum cuprates

I compare the standard one-dimensional stripe interpretation of elastic scattering experiments in 1/8 doped lanthanum cuprates with two two-dimensional interpretations. One of them is known as grid[1,2] and the other one is the lattice of magnetic vortices[3]. Both can induce a 4x4 charge modulation similar to the one detected by scanning tunneling spectroscopy. The case of magnetic vortices, however, is favored against grid by a recent spin polarized neutron scattering experiment. \newline [1] B.V. Fine, Phys. Rev. B, v. 70, p. 224508 (2004) \newline [2] B.V. Fine, cond-mat/0606300 \newline [3] B.V. Fine, cond-mat/0610748   

Fluctuating Cu-O-Cu Bond model of high temperature superconductivity in cuprates

Twenty years of research have yet to produce a consensus on the origin of high temperature superconductivity (HTS). However, several generic characteristics of cuprate superconductors have emerged as the essential ingredients of and/or constraints on any viable microscopic model of HTS. Besides a $T_{c}$ of order $100$ K, they include a $d$-wave superconducting (SC) gap with Fermi liquid nodal excitations, a pseudogap with $d$-symmetry and the characteristic temperature scale $T^{\ast}$, an anomalous doping-dependent oxygen isotope shift, nanometer-scale gap inhomogeneity, etc.. The isotope shift implies a key role for oxygen vibrations, but conventional BCS single-phonon coupling is essentially forbidden by symmetry and by the on-site Coulomb interaction $U$. Hence the present work invokes nonlinear coupling of planar oxygen vibrations to the Cu-Cu hopping integral $t$. A dominant Fluctuating Bond field emerges involving oxygen vibrational square amplitudes - and associated Cu-Cu $t$'s - in a pattern of quadrupolar symmetry around a given Cu site. Such fluctuations in Cu-Cu bonds mediate $d$-wave pairing, leading to a $d$-wave SC gap, and an explanation of the anomalous isotope shift. A quadrupolar CDW generates a $d$-wave pseudogap related to $T^{\ast}$. Other salient features of HTS are also explained by our model. This work is to appear in Nature Physics.   

Pair Binding in Small Hubbard Clusters

One of the key issues in high-T$_{c}$ superconductors is how (and whether) high temperature pairing can arise in an electronic system with only repulsive interactions. Here, we report the results of analytic and numerical exact diagonalization studies of small Hubbard clusters (up to 16 sites). Taking the N-electron ground-state as the ``vacuum state'' of the cluster, we define the effective interaction between two added electrons to be V$^{eff}$(N) = E(N+2)+E(N)--2E(N+1), where E(N) is the ground-state energy with N electrons on the cluster. Not surprisingly, for most clusters and most values of N, V$^{eff}$ is repulsive (V$^{eff}>$0), but there exist special clusters in which, for special N and in an appropriate range of U/t, there is an effective attraction, V$^{eff}<$0. In the weak coupling limit (U/t$<<$1), the results can be understood within perturbation theory, and the effective attraction, where it is occurs, is associated with the existence of an anomalous ``resonantly entangled'' groundstate. In the strong coupling limit, V$^{eff}$ is always positive (or zero) due to Nagaoka physics. In some sense, the optimal cluster is the Hubbard-tetrahedron, for which V$^{eff}$ is negative for all U/t. Finally, by studying the dependence of V$^{eff}$ on the patterns of inhomogeneous couplings within a single cluster, we obtain some insight into the issue of whether there exists an optimal inhomogeneity for high-T$_{c}$ superconductivity.   

Searching for orbital currents in the pseudo-gap state of $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$

Among the many outstanding riddles involving the cuprate materials is the microscopic nature of the so-called `pseudo-gap state'. Several theories have been put forth over the years, including pre-formed pairs, superconducting fluctuations and several brands of unconventional order. An example of the latter which has been getting particular attention of late is the idea that the pseudo-gap corresponds to an ordering of orbital currents. This renewed debate is mostly due to recent polarized neutron data on $\mathrm{YBa_{2}Cu_{3}O_{7-\delta}}$, which claims to support a current ordered state which does not break translational invariance [PRL 96, 197001 (2006)]. These neutron results are not universally accepted, however, and clarifying experiments are necessary. In this spirit, we performed zero-field $\mu SR$ on $\mathrm{La_{2-x}Sr_{x}CuO_{4}}$ crystals with a wide range of $T^{*}$ values, and searched for the sponteous magnetic fields that would necessarily be associated with current order. We present the results of this search and discuss the implications our data for the interpretation of past and future experiments.   

The effect of inhomogeneous pairing amplitude on superfluid stiffness in a $d$-wave superconductor

To explain the disparity between $T_c$ and $T^*$ in optimally doped and underdoped cuprates, we propose that $T_c$ is related to superconducting gap amplitude standard deviation ($\sigma$); while $T^*$ is related to average gap amplitude. We calculate the superfluid stiffness ($D_s$) using BdG formalism for a $d$- wave superconductor. The calculations show that $D_s$ decreases as ($\sigma$) increases, suggesting lower $T_c$ for more inhomogeneous gap distribution. The theoretic result is consistent with recent STM experiments which study the electronic inhomogeneities due to out-of-plane disorder.   

The Ground State of the Pseudogap in Cuprates

In conventional superconductors, the appearance of an energy gap in the electronic spectrum indicates pairing of electrons into Cooper pairs and a simultaneous transition into a macroscopic superconducting state. In contrast, in the underdoped high temperature superconductors, an energy gap is already present in the normal state. An understanding of this normal state gap or `pseudogap' has proven elusive, because its ground state electronic structure was unknown. Here, we present studies of electronic structure in La$_{2-x}$Ba$_{x}$CuO$_{4}$, a unique system where the superconductivity is strongly suppressed and static spin and charge orders or `stripes' develop near a doping level of $x=$1/8. Using angle-resolved photoemission and scanning tunneling microscopy, we detect an energy gap at the Fermi surface with magnitude consistent with $d$-wave symmetry and with linear density of states, vanishing only at four nodal points, even when superconductivity disappears at $x=$1/8. Thus, the non-superconducting, `striped' state at $x=$1/8 is consistent with a phase incoherent $d$-wave superconductor whose Cooper pairs form spin/charge ordered structures instead of becoming superconducting.   

The effects of local inhomogeneities on the phonon modulated DOS in Bi2212

Recent scanning tunneling microscopy experiments on Bi2212 have reveled microscopic inhomogeneities in the local density of states and anomalous signatures of coupling to a bosonic mode. Gap referenced estimates for the mode energy are negatively correlated with the local gap size and the distribution of the mode estimates shows a clear isotope shift upon $^{18}$O substitution. Motivated by the clear isotope effect we examine electron-phonon coupling to the 55 meV apical oxygen mode in Bi2212 within the framework of Migdal-Eliashberg theory. The interplay of this interaction with local inhomogeneous broadening effects are also considered. The effects of the local dopant atoms on the electron-phonon interaction strength are examined using the Ewald summation technique.