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Physical review. B, Condensed matter

Reduced-density-matrix analysis of superconducting correlation in two-dimensional and two-chain Hubbard models

Yoshihiro Asai

Published Sep 1, 1994• 11 citations
Publisher

Abstract

We have calculated the largest eigenvalues of the reduced density matrix of the singlet superconducting correlation function and pair-field susceptibility in two-dimensional and two-chain Hubbard models with quantum Monte Carlo methods. A singlet superconducting channel opens upon doping but is otherwise closed at half-filling when U/t=4. The interaction vertex contribution to the largest eigenvalue of the singlet pair-field susceptibility grows with a power law as temperature is decreased. With these two observations, we revise our previous understanding of the interaction vertex in the two-dimensional Hubbard model. We observe that some phases reside in the two-chain Hubbard model. An analysis of the 2${\mathit{k}}_{\mathit{F}}$ charge correlation up to 32\ifmmode\times\else\texttimes\fi{}2 sites suggests that the 2${\mathit{k}}_{\mathit{F}}$ correlation is dominant over the superconducting correlation (0.25\ensuremath{\le}${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$\ensuremath{\le}1.00) in the two-component Luttinger-liquid phase (U/t=2 and ${\mathit{t}}_{\mathit{v}}$/t=1.4). We conclude the reduced-density-matrix analysis discussed here is needed to unveil the nature of Hubbard and related models.

DOI
10.1103/physrevb.50.6519

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We have calculated the largest eigenvalues of the reduced density matrix of the singlet superconducting correlation function and pair-field susceptibility in two-dimensional and two-chain Hubbard models with quantum Monte Carlo methods. A singlet superconducting channel opens upon doping but is otherwise closed at half-filling when U/t=4. The interaction vertex contribution to the largest eigenvalue of the singlet pair-field susceptibility grows with a power law as temperature is decreased. With these two observations, we revise our previous understanding of the interaction vertex in the two-dimensional Hubbard model. We observe that some phases reside in the two-chain Hubbard model. An analysis of the 2kF{\mathit{k}}_{\mathit{F}}kF​ charge correlation up to 32\ifmmode\times\else\texttimes\fi{}2 sites suggests that the 2kF{\mathit{k}}_{\mathit{F}}kF​ correlation is dominant over the superconducting correlation (0.25\ensuremath{\le}K\ensuremathρ{\mathit{K}}_{\mathrm{\ensuremath{\rho}}}K\ensuremathρ​\ensuremath{\le}1.00) in the two-component Luttinger-liquid phase (U/t=2 and tv{\mathit{t}}_{\mathit{v}}tv​/t=1.4). We conclude the reduced-density-matrix analysis discussed here is needed to unveil the nature of Hubbard and related models.
In this study we have computed the pair correlation functions in the two-dimensional Hubbard model using a quantum Monte Carlo method. We employ a new diagonalization algorithm in the quantum Monte Carlo method which is free from the negative sign problem. We show that the d-wave pairing correlation function is indeed enhanced slightly for the positive on-site Coulomb interaction U when doping away from the half-filling. When the system size becomes large, the pair correlation function P d increases for U > 0 compared to the non-interacting case, while P d is suppressed for U > 0 when the system size is small. The enhancement ratio P d [ U ]/ P d [ U = 0] will give a criterion on the existence of superconductivity. The ratio P d [ U ]/ P d [ U = 0] increases almost linearly ∝ L when the system size L × L is increased. This increase is a good indication of the existence of a superconducting phase in the two-dimensional Hubbard model. There is, however, no enhancement of pair correlation functions in the half-filled case, which indicates the absence of superconductivity without hole doping.
Using the recently introduced multiloop extension of the functional renormalization group, we compute the magnetic, density, and superconducting susceptibilities of the two-dimensional Hubbard model at weak coupling and present a detailed analysis of their evolution with temperature, interaction strength, and loop order. By breaking down the susceptibilities into contributions from the bare susceptibility and the individual channels, we investigate their relative importance as well as the channel interplay. In particular, we trace the influence of antiferromagnetic fluctuations on the ddd-wave superconductivity and provide an analytical understanding for the observed behavior.
We study the ground state properties of the one-dimensional extended Hubbard model at half filling from the perspective of its particle reduced density matrix. We focus on the reduced density matrix of two fermions and perform an analysis of its quantum correlations and coherence along the different phases of the model. Specifically, we study its (i) entanglement entropy, (ii) ${\ensuremath{\ell}}_{1}$ norm of coherence, (iii) irreducible two-body cumulant matrix, and (iv) entanglement spectrum. Our results show that these different properties are complementary to each other depending on the phase of the system, exhibiting peculiar behaviors such as discontinuities and maximum or minimum values at the quantum phase transitions, thus providing a qualitative view of the phase diagram of the model. In particular, in the superconducting region, we obtain that the entanglement spectrum signals a transition from a dominant singlet (SS) to triplet (TS) pairing ordering in the system. Moreover, from the analysis of the dominant eigenvector in the reduced state, we can relate the SS-TS transition to the spatial separation between the fermion pairs in the two different pairing orderings. The entanglement gap is also able to highlight a transition---at a few-body level---in the ground state wave function, not discussed previously in the literature. While other quantifiers are less sensitive to few-body defects in the wave function, the entanglement gap can work as a magnifying glass for these, capturing such small fluctuations.
In a recent publication [Chen et al., Phys. Rev. B 86, 165136 (2012)], we identified a line of Lifshitz transition points separating the Fermi liquid and pseudogap regions in the hole-doped two-dimensional Hubbard model. Here, we extend the study to further determine the superconducting transition temperature in the phase diagram. By means of large-scale dynamical cluster quantum Monte Carlo simulations, we are able to identify the evolution of the ddd-wave superconducting dome in the hole-dope side of the phase diagram, with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. To obtain the superconducting transition temperature Tc{T}_{c}Tc​, we employ two-particle measurements of the pairing susceptibilities. As ${t}^{\ensuremath{'}}$ goes from positive to negative values, we find the ddd-wave projected irreducible pairing vertex function is enhanced, and the curvature of its doping dependence changes from convex to concave, which fixes the position of the maximum superconducting temperature at the same filling ($n\ensuremath{\approx}0.85$) and constraints the dome from precisely following the Lifshitz line. We furthermore decompose the irreducible vertex function into fully irreducible, charge and spin components via the parquet equations, and consistently find that the spin component dominates the pairing vertex function in the doping range where the dome is located. Our investigations deepen the understanding of the phase diagram of the two-dimensional Hubbard model and, more importantly, pose new questions to the field. For example, we found as ${t}^{\ensuremath{'}}$ goes from positive to negative values, the curvature of the pairing strength as a function of doping changes from convex to concave, and the nature of the dominant fluctuations changes from charge degree of freedom to spin degree of freedom. The study of these issues will lead to further understanding of the phase diagram of the two-dimensional Hubbard model and also the physics of the hole-doped cuprate high-temperature superconductors.
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.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.