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arXiv:1512.06688 · Physical review. B./Physical review. B

Coexistence of nematic order and superconductivity in the Hubbard model

Jan Kaczmarczyk, Tobias Schickling, Jörg Bünemann

Published Aug 30, 2016• 16 citations
arXivHTMLPDF

Abstract

We study the interplay of nematic and superconducting order in the two-dimensional Hubbard model and show that they can coexist, especially when superconductivity is not the energetically dominant phase. Due to a breaking of the ${C}_{4}$ symmetry, the coexisting phase inherently contains admixture of the $s$-wave pairing components. As a result, the superconducting gap exhibits nonstandard features including changed nodal directions. Our results also show that in the optimally doped regime the pure superconducting phase is typically unstable towards developing nematicity (breaking of the ${C}_{4}$ symmetry). This has implications for the cuprate high-${T}_{c}$ superconductors, for which in this regime the so-called intertwined orders have recently been observed. Namely, the coexisting phase may be viewed as a precursor to such more involved patterns of symmetry breaking.

DOI
10.1103/physrevb.94.085152

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We study the interplay of nematic and superconducting order in the two-dimensional Hubbard model and show that they can coexist, especially when superconductivity is not the energetically dominant phase. Due to a breaking of the C4{C}_{4}C4​ symmetry, the coexisting phase inherently contains admixture of the sss-wave pairing components. As a result, the superconducting gap exhibits nonstandard features including changed nodal directions. Our results also show that in the optimally doped regime the pure superconducting phase is typically unstable towards developing nematicity (breaking of the C4{C}_{4}C4​ symmetry). This has implications for the cuprate high-Tc{T}_{c}Tc​ superconductors, for which in this regime the so-called intertwined orders have recently been observed. Namely, the coexisting phase may be viewed as a precursor to such more involved patterns of symmetry breaking.
We investigate the interplay between superconductivity and nematic order in a two-dimensional electron gas with competing pairing and quadrupolar forward-scattering interactions. The model includes both sss-wave and ddd-wave superconducting channels. We compute the mean-field free energy density and determine the phase diagrams as functions of interaction strengths and temperature by solving a set of coupled self-consistent equations. At zero-temperature, we find that the nematic order competes strongly with ddd-wave superconductivity, leading to a direct first-order phase transition, while its interplay with sss-wave pairing allows for a coexistence phase characterized by an anisotropic Fermi surface with a uniform superconducting gap. At finite-temperatures, quadrupolar interactions promote the emergence of additional superconducting components, giving rise to regimes where sss-wave, ddd-wave, and nematic orders coexist. Our results highlight the role of symmetry and interaction strength in shaping the phase structure and provide a minimal framework to describe intertwined nematic and superconducting phases in correlated electron systems.
We analyze the effect of intersite interactions on the stability of the coexisting superconducting-nematic phase (SC+N) within the extended Hubbard and t – J – U models on the square lattice. In order to take into account the correlation effects to a proper precision, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which goes beyond the renormalized mean-field theory (RMFT) in a systematic manner. As a starting point of our analysis we discuss the SC+N phase stability as a function of the intrasite Coulomb repulsion and hole doping for the case of the Hubbard model. Next, we show that the exchange interaction term enhances superconductivity while suppresses the nematicity, whereas the intersite Coulomb repulsion acts in the opposite manner. The competing character of the SC and N phases interplay is clearly visible throughout the analysis. A universal conclusion is that the nematic phase does not survive within the t – J – U model for the value of J integral typical for the high- T C cuprates ( J ≈ 0.1 eV). This result is helpful in providing the understanding of the fundamental role of the nodal direction. For the sake of completeness, the effect of the correlated hopping term is also analyzed. Thus the present discussion contains all relevant two-site interactions which appear in the parametrized single-band model of correlated fermions. At the end, the influence of the higher-order terms of the DE on the rotational symmetry breaking is also shown by comparing the DE-GWF results with those of the RMFT.
The Hubbard model is an iconic model in quantum many-body physics and has been intensely studied, especially since the discovery of high-temperature cuprate superconductors. Combining the complementary capabilities of two computational methods, we found superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity was weaker and was accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexisted with stripe order, which persisted into the overdoped region with weaker hole-density modulation. These stripe orders varied in fillings between 0.6 and 0.8. Our results suggest the applicability of the Hubbard model with next-nearest hopping for describing cuprate high–transition temperature ( T c ) superconductivity.
Nematicity is a well-known property of liquid crystals and has been recently discussed in the context of strongly interacting electrons. An electronic nematic phase has been seen in many experiments in certain strongly correlated materials, in particular, in the pseudogap phase generic to many hole-doped cuprate superconductors. Recent measurements in high Tc superconductors have shown that even if the lattice is perfectly rotationally symmetric, the ground state can still have strongly nematic local properties. Our study of the two-dimensional one-band Hubbard model provides strong support for the recent experimental results on local rotational C4 symmetry breaking. The variational cluster approach is used here to show the possibility of an electronic nematic state and the proximity of the underlying symmetry-breaking ground state within the Hubbard model. We identify this nematic phase in the overdoped region and show that the local nematicity decreases with increasing electron filling. Our results also indicate that strong Coulomb interaction may drive the nematic phase into a phase similar to the stripe structure. The calculated spin (magnetic) correlation function in momentum space shows the effects resulting from real-space nematicity.
Using a dynamic cluster quantum Monte Carlo approximation, we study a two-dimensional Hubbard model with a small orthorhombic distortion in the nearest-neighbor hopping integrals. We find a large nematic response in the low-frequency single-particle scattering rate which develops with decreasing temperature and doping as the pseudogap region is entered. At the same time, the ddd-wave superconducting gap function develops an sss-wave component and its amplitude becomes anisotropic. The strength of the pairing correlations, however, is found to be unaffected by the strong anisotropy, indicating that ddd-wave superconductivity can coexist with strong nematicity in the system.
In view of cuprate superconductors, interplay between antiferromagnetism (AF) and superconductivity (SC) is studied for the square-lattice Hubbard model with a diagonal transfer t , using a variational Monte Carlo method.In the trial wave function, essential factors of Mott physics and band renormalization are introduced.It is found that whether AF and SC become coexistent or competitive is sensitive to the values of model parameters, U/t, t /t, and δ (doping rate).Furthermore, in case AF gap is finite, namely, in pure AF or coexisting cases, the state is unstable against phase separation.