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arXiv:2302.08367 · Physical review. B./Physical review. B

Density matrix renormalization group study of superconducting pairing near the quarter-filled Wigner crystal

Rudolf Torsten Clay, Beau A. Thompson

Published Apr 3, 2025• 1 citation
arXivHTMLPDF

Abstract

Charge ordering is often found in the phase diagram of unconventional superconductors in close proximity to the superconducting state. This has led to the suggestion that fluctuations of charge order can mediate superconducting pairing. While several mechanisms can lead to charge order, one common mechanism is the long-range Coulomb interaction, resulting in a Wigner crystal charge ordered state. For an electron density of 0.5 per site, we investigate the extended Hubbard model on the two-dimensional square lattice using exact diagonalization and density matrix renormalization group methods. Our results show that the strength of pairing decreases with the nearest-neighbor Coulomb interaction strength $V$ and remains weaker than the tendency of pairing for noninteracting electrons.

DOI
10.1103/physrevb.111.155108

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Charge ordering is often found in the phase diagram of unconventional superconductors in close proximity to the superconducting state. This has led to the suggestion that fluctuations of charge order can mediate superconducting pairing. While several mechanisms can lead to charge order, one common mechanism is the long-range Coulomb interaction, resulting in a Wigner crystal charge ordered state. For an electron density of 0.5 per site, we investigate the extended Hubbard model on the two-dimensional square lattice using exact diagonalization and density matrix renormalization group methods. Our results show that the strength of pairing decreases with the nearest-neighbor Coulomb interaction strength VVV and remains weaker than the tendency of pairing for noninteracting electrons.
In a Fermi system near or at half-filling, a specific superconducting pairing channel, if not explicitly included in the Hamiltonian, can be boosted by suppressing a competing pairing channel; this is exemplified by the enhancement of extended sss-wave correlations upon suppressing sss-wave Cooper pairing. This phenomenon, originally found by the use of generalized uncertainty relations is referred to as order by projection. The case of zero on-site Coulomb interaction in the thermodynamic limit, confirms this mechanism through the analytical solution. In this study, we go further and systematically investigate this mechanism for a strongly correlated fermionic Hubbard model, now with finite on-site interaction, on a square lattice with an extended set of hopping parameters. We explore the behaviors of different pairing channels when one of them is suppressed, utilizing density matrix renormalization group calculations. Our findings provide numerical evidence supporting the existence of order by projection in the strongly correlated system we studied. We also investigate the effect of the strength of Hubbard UUU, next-nearest-neighbor ${t}^{\ensuremath{'}}$, hole doping, as well as finite-size scaling approaching the thermodynamic limit.
The magnetic phase diagram of the quarter-filled generalized Wigner lattice with nearest-neighbor and next-nearest-neighbor hoppings, t1{t}_{1}t1​ and t2{t}_{2}t2​, is explored. We find a region at negative t2{t}_{2}t2​ with fully saturated ferromagnetic ground states that we attribute to kinetic exchange. Such interaction disfavors antiferromagnetism at t2<0{t}_{2}<0t2​<0 and stems from virtual excitations across the charge gap of the Wigner lattice, which is much smaller than the Mott-Hubbard gap $\ensuremath{\propto}U$. Remarkably, we find a strong dependence of the charge structure factor on magnetism even in the limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$, in contrast to the expectation that charge ordering in the Wigner lattice regime should be well described by spinless fermions. Our results, obtained using the density-matrix renormalization group and exact diagonalization, can be transparently explained by means of an effective low-energy Hamiltonian.
We present a generalization of the recently proposed variational cluster perturbation theory to extended Hubbard models at half-filling with repulsive nearest neighbor interaction. The method takes into account short-range correlations correctly by the exact diagonalization of clusters of finite size, whereas long-range order beyond the size of the clusters is treated on a mean-field level. For one dimension, we show that quantum Monte Carlo and density-matrix renormalization-group results can be reproduced with very good accuracy. Moreover we apply the method to the two-dimensional extended Hubbard model on a square lattice. In contrast to the one-dimensional case, a first order phase transition between spin density wave phase and charge density wave phase is found as function of the nearest-neighbor interaction at onsite interactions $U\ensuremath{\geqslant}3t$. The single-particle spectral function is calculated for both the one-dimensional and the two-dimensional system.
The authors employ the density-matrix renormalization group to investigate the doped Hubbard model on six-leg square cylinders. It uncovers a rich quantum phase diagram, intricately sensitive to next-nearest-neighbor electron hopping ($t\ensuremath{'}$). The positive-ttt\ensuremath{'} region shows a robust ddd-wave superconducting phase with intertwined superconducting and charge-density-wave orders. In contrast, the negative-ttt\ensuremath{'} side remains insulating, where doped holes form either long-range charge stripe order at small ttt\ensuremath{'} or a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin correlations at larger ttt'.
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Abstract Using the singular-mode functional renormalization group method, we studied the doping-dependent phase diagram of the two-dimensional extended Hubbard model on square lattice. At half-filling, a variety of states strongly compete, including spin-density wave, charge-density wave, phase separation, as well as d -wave and s -wave superconductivity (SC). Upon electron doping, the system exhibits a strong preference for superconducting states. At slight dopings, the spin-density wave phase is strongly suppressed, and d -wave SC region expands into positive U , V region. At moderate dopings, a region of degenerated p -wave superconductivities emerge for positive U and negative V . Our results demonstrate how doping tunes the dominant fluctuations, suppressing density waves and stabilizing unconventional superconducting states in two dimensions.