Superconducting Correlation Functions Research Articles

Ab initio low-energy effective Hamiltonians for the high-temperature superconducting cuprates Bi2Sr2CuO6, Bi2Sr2CaCu2O8, HgBa2CuO4, and CaCuO2

We derive $ab$ $initio$ low-energy effective Hamiltonians (LEH) for high-temperature superconducting (SC) copper oxides Bi$_2$Sr$_2$CuO$_6$ (Bi2201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 10$ K), Bi$_2$Sr$_2$CaCu$_2$O$_8$ (Bi2212, $N_{\ell}=2$, $T_c^{\rm exp} \sim 84$ K), HgBa$_2$CuO$_4$ (Hg1201, $N_{\ell}=1$, $T_c^{\rm exp} \sim 90$ K) and CaCuO$_2$ (Ca11, $N_{\ell}=\infty$, $T_c^{\rm exp} \sim 110$ K), with different experimental optimal SC transition temperature $T_c^{\rm exp}$ and number $N_{\ell}$ of laminated CuO$_2$ planes between the two neighboring block layers. We apply the latest methodology of the multiscale $ab$ $initio$ scheme for correlated electron systems (MACE), and focus on the LEH consisting of one antibonding (AB) Cu$3d_{x^2-y^2}$/O$2p_{\sigma}$ orbital centered on each Cu atom. We discuss prominent features of this LEH: (1) The ratio $U/|t_1|$ between the onsite effective Coulomb repulsion (ECR) $U$ and amplitude of nearest neighbour hopping $t_1$ increases with $T^{\rm exp}_c$ and $N_{\ell}$, consistently with the expected increase in $d$-wave SC correlation function $P_{dd}$ with $U/|t_1|$. One possible cause of the increase of $U/|t_1|$ is the replacement of apical O atoms by Cu atoms from neighbouring CuO$_2$ planes when $N_{\ell}$ increases. Furthermore, we show that the increase in distance between Cu and apical O atoms decreases the effective screening (ES) by electrons outside of the LEH and increases $U/|t_1|$. (2) For Hg1201 and Ca11, we show that $U/|t_1|$ decreases when hole doping per AB orbital $\delta$ increases, which may partly account for the disappearance of SC when $\delta$ exceeds the optimal value in experiment. (3) For $N_{\ell} \geq 2$, off-site inter-CuO$_2$ plane ECR is comparable to off-site intra-CuO$_2$ plane ECR. We discuss contributions of inter-CuO$_2$ plane ECR to both $P_{dd}$ and the stability of the SC state.

Tunable stripe order and weak superconductivity in the Moiré Hubbard model

The moir\'e Hubbard model describes correlations in certain homobilayer twisted transition metal dichalcogenides. Using exact diagonalization and density matrix renormalization group methods, we find magnetic Mott insulating and metallic phases, which, upon doping exhibit intertwined charge and spin ordering and, in some regimes, pair binding of holes. The phases are highly tunable via an interlayer potential difference. Remarkably, the hole binding energy is found to be highly tunable revealing an experimentally accessible regime where holes become attractive. In this attractive regime, we study the superconducting correlation function and point out the possibility of weak superconductivity.

Role of long-range interaction on the photon-excited η-pairing of electrons

We report studies on the influence of a time-dependent external driving fields of the photons on a quantum system described by Hubbard model with long-range interactions. Through the exact diagonalization method to calculate the superconducting pair correlation functions, we show that a weak repulsive long-range Coulomb interaction can help enhance superconducting pair correlation function in some parameter space, which may appear in cuprates, while the number of doublons is increased by the attractive long-range interaction. Out of the parameter space, long-range interaction can be harmful to η-pairing and on-site pairing of electrons. Our results may help people find and design new materials with significant enhancement of transient superconductivity induced by light pulse irradiations.

Long-range order in one-dimensional spinless Fermi gas with attractive dipole–dipole interaction

One-dimensional spinless Fermi gas with attractive dipole–dipole interaction is investigated. Results obtained show that when the interaction is weak, the excitation spectrum is linear and the superconducting correlation function decays as a power law, indicating the validity of the Tomonaga–Luttinger (TL) liquid picture. However, when the interaction reaches a critical value, the excitation spectrum is nonlinear and the superconducting correlation function remains finite for infinity separation, indicating that real long–range order has been established and the breakdown of the TL liquid picture. We prove that the existence of long–range order is not in contradiction with the Hohenberg theorem and show that this system is related to the Kitaev toy model, therefore, it has potential applications for future topological quantum computation.

Crossover between BCS Superconductor and Doped Mott Insulator ofd-Wave Pairing State in Two-Dimensional Hubbard Model

With high-Tc cuprates in mind, properties of correlated dx2-y2-wave superconducting (SC) and antiferromagnetic (AF) states are studied for the Hubbard (t-t'-U) model on square lattices, using a variational Monte Carlo method. We employ simple trial wave functions including only crucial parameters, such as a doublon-holon binding factor indispensable to describe correlated SC and normal states as doped Mott insulators. U/t, t'/t and \delta (doping rate) dependence of relevant quantities are systematically calculated. As U/t increases, a sharp crossover of SC properties occurs at U_co/t \sim 10 from a conventional BCS type to a kinetic-energy-driven type for any t'/t. As \delta decreases, U_co/t is smoothly connected to the Mott transition point at half filling. For U/t\lsim 5, steady superconductivity corresponding to the cuprates is not found, whereas the d-wave SC correlation function Pd^\infty rapidly increases for U/t\gsim 6 and becomes maximum at U=U_co. Comparing the \delta dependence of Pd^\infty with experimentally observed dome-shaped Tc and condensation energy, we find that the effective value of $U$ for the cuprates should be larger than the band width, for which the t-J model is valid. Analyzing the kinetic energy, we reveal that for U>U_co only doped holes (electrons) become charge carriers, which will make a small Fermi surface (hole pocket), but for U<U_co all the electrons (holes) contribute to conduction and will make an ordinary large Fermi surface, which is contradictory to the feature of cuprates. By introducing a proper negative (positive) t'/t, the SC (AF) state is stabilized. In the underdoped regime, the strength of SC for U>U_co is determined by two factors, i.e., the AF spin correlation, which creates singlet pairs (pseudogap), and the charge mobility dominated by Mott physics.

Variational Study of Mott Transition by Means of Drude Weight and Superfluid Density

Abstract To distinguish a metal from an insulator, the Drude weight ( D , zero-frequency component of conductivity) is a useful measure, as Kohn pointed out long ago: D > 0(=0) for a metal (insulator). Later, Millis and Coppersmith showed that a variational wave function Ψ Q in which the key ingredient for a Mott transition (a doublon-holon binding effect) is introduced exhibits D > 0 (metallic) even for sufficiently large correlation strength, namely, a Mott transition is absent from. Ψ Q In contrast, variational Monte Carlo studies using Ψ Q confirmed, by studying relevant quantities such as doublon density d and a superconducting correlation function, P d ∞ that Ψ Q undoubtedly raises a Mott transition. This contradiction has been a long-standing perplexing problem. We definitely settle this problem by adding Ψ Q to a configuration-dependent phase factor P θ , which has been hitherto overlooked. This factor appropriately picks out a negative counterpart in D for insulators, so that D vanishes. Because P θ does not affect the quantities such as d and P d ∞ , the previous results of Ψ Q on the Mott transition remain intact for P θ Ψ Q .

Variational Monte Carlo Study of Spin-Gapped Normal State and BCS–BEC Crossover in Two-Dimensional Attractive Hubbard Model

We study properties of normal, superconducting (SC) and CDW states for an attractive Hubbard model on the square lattice, using a variational Monte Carlo method. In trial wave functions, we introduce an interspinon binding factor, indispensable to induce a spin-gap transition in the normal state, in addition to the onsite attractive and intersite repulsive factors. It is found that, in the normal state, as the interaction strength $|U|/t$ increases, a first-order spin-gap transition arises at $|U_{\rm c}|\sim W$ ($W$: band width) from a Fermi liquid to a spin-gapped state, which is conductive through hopping of doublons. In the SC state, we confirm by analysis of various quantities that the mechanism of superconductivity undergoes a smooth crossover at around $|U_{\ma{co}}|\sim |U_{\rm c}|$ from a BCS type to a Bose-Einstein condensation (BEC) type, as $|U|/t$ increases. For $|U|<|U_{\ma{co}}|$, quantities such as the condensation energy, a SC correlation function and the condensate fraction of onsite pairs exhibit behavior of $\sim \exp(-t/|U|)$, as expected from the BCS theory. For $|U|>|U_{\ma{co}}|$, quantities such as the energy gain in the SC transition and superfluid stiffness, which is related to the cost of phase coherence, behave as $\sim t^2/|U|\propto T_{\rm c}$, as expected in a bosonic scheme. In this regime, the SC transition is induced by a gain in kinetic energy, in contrast with the BCS theory. We refer to the relevance to the pseudogap in cuprate superconductors.

Renormalization group and the superconducting susceptibility of a Fermi liquid

A free Fermi gas has, famously, a superconducting susceptibility that diverges logarithmically at zero temperature. In this paper we ask whether this is still true for a Fermi liquid and find that the answer is that it does not. From the perspective of the renormalization group for interacting fermions, the question arises because a repulsive interaction in the Cooper channel is a marginally irrelevant operator at the Fermi liquid fixed point and thus is also expected to infect various physical quantities with logarithms. Somewhat surprisingly, at least from the renormalization group viewpoint, the result for the superconducting susceptibility is that two logarithms are not better than one. In the course of this investigation we derive a Callan-Symanzik equation for the repulsive Fermi liquid using the momentum-shell renormalization group, and use it to compute the long-wavelength behavior of the superconducting correlation function in the emergent low-energy theory. We expect this technique to be of broader interest.

Close relation between antinodal Fermi-surface effect and superconductivity in cuprates

A strongly correlated Hubbard (t–t′–U) model is studied using a variational Monte Carlo method. The magnitude of momentum distribution function n(k) varies slightly near the nodal quasi-Fermi surface [∼(π/2,π/2)], but varies outstandingly in an antinodal part [near (π,0)], as the value of t′/t varies, which is sensitive to the strength of superconductivity. Furthermore, the behavior of the slope of n(k) around (π,0) coincides well with that of the d-wave superconducting correlation function. It follows the electrons near the antinode play a leading role to control the strength of superconductivity.

Supersymmetric valence bond solid states

In this work we investigate the supersymmetric version of the valence bond solid (SVBS) state. In one dimension, the SVBS states continuously interpolate between the valence bond states for integer and half-integer spin chains, and they generally describe superconducting valence bond liquid states. Spin and superconducting correlation functions can be computed exactly for these states and their correlation lengths are equal at the supersymmetric point. In higher dimensions, the wave function for the SVBS states can describe resonating valence bond states. The SVBS states for the spin models are shown to be precisely analogous to the bosonic Pfaffian states of the quantum Hall effect. We also give microscopic Hamiltonians for which the SVBS state is the exact ground state.

Influence of Electronic Structure on the Phase Diagram of Weakly Doped Superconducting Cuprates

Motivated by the difference between the experimental phase diagram of cuprates and expectations of the t−J model, we analyze the influence of the electronic structure on superconducting state generation for small levels of doping. Following some theoretical studies of the Fermi surface for hole doped superconducting cuprates we base our calculations on the t−t′−t′′−J model. We construct the spin polaron model, which is an effective model for the t−J model, and we expand it by adding the new terms related to the next neighbor hopping integrals t′ and t′′. In the framework of this model we study evolutions of superconducting correlation functions with doping. As a result of numerical calculations we find out that superconducting state vanishes for small levels of doping and finite values of t′ and t′′. This is in qualitative agreement with experimental results.

Superconducting Fluctuation and Pseudogap in Disordered Short Coherence Length Superconductor

We investigate the role of disorder on the superconducting (SC) fluctuation in short coherence length d-wave superconductors. The particular intetest is focused on the disorder-induced microscopic inhomogeneity of SC fluctuation and its effect on the pseudogap phenomena. We formulate the self-consistent 1-loop order theory for the SC fluctuation in inhomogeneous systems and analyze the disordered $t$-$t'$-$V$ model. The SC correlation function, electronic DOS and the critical temperature are estimated. The SC fluctuation is localized like a nanoscale granular structure when the coherence length is short, namely the transition temperature is high. This is contrasted to the long coherence length superconductors where the order parameter is almost uniform in the microscopic scale. In the former case, the SC fluctuation is enhanced by the disorder in contrast to the Abrikosov-Gorkov theory. These results are consistent with the STM, NMR and transport measurements in high-$T_{\rm c}$ cuprates and illuminate the essential role of the microscopic inhomogeneity. We calculate the spacial dependence of DOS around the single impurity and discuss the consistency with the NMR measurements.

Broad Peak in thedx2−y2Superconducting Correlation Length as a Function of Hole Concentration in the Two-Dimensionalt−JModel

We have calculated high temperature series to 12th order in inverse temperature for singlet superconducting correlation functions of the 2D t-J model with s, dx2-y2, and dxy symmetry pairs. Our calculations differ from previous work by removing disconnected pieces from the original four-point correlator and by treating the resulting pairing correlator as a matrix. We find the correlation length for dx2-y2 pairing grows significantly with decreasing temperature and develops a broad peak as a function of doping around delta=0.25 for T/J=0.25 at J/t=0.4. The correlation lengths for s and dxy symmetry remain small and do not display peaks. Antiferromagnetic spin correlations at low doping act to suppress the dx2-y2 and dxy superconducting correlation lengths.

Construction of a microscopic model forf-electron systems on the basis of aj−jcoupling scheme

We construct a microscopic model for f-electron systems, composed of f-electron hopping, Coulomb interaction, and crystalline electric field (CEF) terms. In order to clarify the meaning of one f-electron state, here the j-j coupling scheme is considered, since the spin-orbit interaction is generally large in f-electron systems. Thus, the f-electron state at each site is labelled by $\mu$, namely, the z-component of total angular momentum j. By paying due attention to f-orbital symmetry, the hopping amplitudes between f-electron states are expressed using Slater's integrals. The Coulomb interaction terms among the $\mu$-states are written by Slater-Condon or Racah parameters. Finally, the CEF terms are obtained from the table of Hutchings. The constructed Hamiltonian is regarded as an orbital degenerate Hubbard model, since it includes two pseudo-spin and three pseudo-orbital degrees of freedom. For practical purposes, it is further simplified into a couple of two-orbital models by discarding one of the three orbitals. One of those simplified models is here analyzed using the exact diagonalization method to clarify ground-state properties by evaluating several kinds of correlation functions. Especially, the superconducting pair correlation function in orbital degenerate systems is carefully calculated based on the concept of off-diagonal long-range order. We attempt to discuss a possible relation of the present results with experimental observations for recently discovered heavy fermion superconductors CeMIn$_5$ (M=Ir, Co, and Rh), and a comprehensive scenario to understand superconducting and antiferromagnetic tendencies in the so-called ``115'' materials such as CeMIn$_5$, UMGa$_5$, and PuCoGa$_5$ from the microscopic viewpoint.

Effect of the Medium-Range Transfer Energies to the Superconductivity in the Two-Chain Hubbard Model

We investigate the two-chain Hubbard model including the next nearest-neighbor transfer energies both in the interchain and intrachain directions by use of the exact-diagonalization and the variational Monte Carlo methods. We find that superconducting correlation functions are enhanced in such a situation where the density of states of the antibonding band in the neighborhood of the Fermi level enlarges due to the above-mentioned transfer energies, especially, in the situation where the Fermi level is located a little above the bottom of the antibonding band. In such a situation we find from the calculated momentum distribution function that this system is in almost one-band situation close to the two-band one. The relevance to ladder-type superconductor Sr 14- x Ca x Cu 24 O 41 is asserted.

Finite Size Scaling in the tt'-Hubbard Model and in BCS-Reduced Hubbard Models

With the projector quantum Monte Carlo algorithm and the stochastic diagonalization it is possible to calculate the ground state of the Hubbard model for small finite clusters. Nevertheless the usual finite size scaling of the Hubbard model has problems of deducing the behavior of the infinite system correctly from the numerical data of small system sizes. Therefore we study the finite size scaling of the superconducting correlation functions in superconducting BCS-reduced Hubbard models to analyze the finite size behavior in small finite clusters. The ground state of the BCS-reduced Hubbard models is calculated with the stochastic diagonalization without any approximations. As result of these analyses we propose a new finite size scaling ansatz for the Hubbard model, which is able describe the finite size effects in a consistent way taking the corrections to scaling into account, which are dominant for weak interaction strength and small clusters. With this new finite size scaling ansatz it is possible to give evidence for superconductivity for all interaction strengths for both the attractive tt'-Hubbard model (with s-wave symmetry) and the repulsive tt'-Hubbard model (with dx2-y2-wave symmetry).

Application of the SD Technique for Solving a BCS-Reduced Hubbard like Hamiltonian

We present exact and stochastic diagonalization results for a BCS-reduced Hubbard model. The kinetic Hamiltonian is the same as in the single band Hubbard model with additional next nearest neighbor hopping. The interaction of this model is designed to inhibit superconductivity in the d x2-y2 channel. The ground state of this model is studied by exact and stochastic diagonalization technique. We present a review of the technical details of the application of the stochastic diagonalization algorithm on this problem. To verify our results obtained with the stochastic diagonalization, they are compared with the exact diagonalization results. In order to show the convergence of the stochastic diagonalization we give a detailed analysis of the behavior of physical properties with increasing number of states. Finally we study superconductivity in this BCS-reduced Hubbard model. As an indicator of superconductivity we use the occurrence of Off Diagonal Long Range Order. We study the scaling behavior of this model for various attractive interactions and in addition the dependence of the superconducting correlation functions from the filling of the system.

Parallelization of the exact diagonalization of the t− t′-Hubbard model

We present a new parallel algorithm for the exact diagonalization of the t−t′-Hubbard model with the Lanczos method. By invoking a new scheme of labeling the states we were able to obtain a speedup of up to four on 16 nodes of an IBM SP2 for the calculation of the ground state energy and an almost linear speedup for the calculation of the correlation functions. Using this algorithm we performed an extensive study of the influence of the next-nearest hopping parameter t′ in the t−t′-Hubbard model on ground state energy and the superconducting correlation functions for both attractive and repulsive interaction.

Indications for RVB superconductivity in a generalized t- J model for the cuprates

We evaluate numerically several superconducting correlation functions in a generalized t - J model derived for hole-doped CuO 2 planes. The model includes a three-site term t ″ similar to that obtained in the large- U limit of the Hubbard model but of opposite sign for realistic OO hopping. For parameters close to realistic values, we obtain strong evidence of superconductivity of predominantly d x 2 − y 2 character. The ground state has a large overlap with a very simple resonating-valence-bond (RVB) wave function with off-diagonal long-range order. This function reproduces the main features of the magnetic and superconducting correlation functions.

Superconducting, magnetic, and charge correlations in the doped two-chain Hubbard model.

We have studied the superconducting, magnetic, and charge correlation functions and the spin excitation spectrum in the doped two-chain Hubbard model by projector Monte Carlo and Lanczos diagonalization methods. The exponent of the interchain singlet superconducting correlation function, \ensuremath{\gamma}, is found to be close to 2.0 as long as two distinct noninteracting bands cross the Fermi level. Magnetic and charge correlation functions decay more rapidly than or as fast as the interchain singlet superconducting correlation function along the chains. The superconducting correlation in the doped two-chain Hubbard model is the most long-range correlation studied here. Implications of the results for the possible universality class of the doped two-chain Hubbard model are discussed.