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arXiv:2103.13763 · Physica Scripta

Absence of superconductivity in the repulsive Hubbard model on a square lattice in the regime of strong coupling

A. Sherman

Published May 21, 2021• 4 citations

Abstract

Abstract The possibility of the superconducting transition in the repulsive Hubbard model on a square lattice is studied using the ladder approximation of the strong coupling diagram technique. The t – U and t – t ′– t ″– U models are considered in the regime of strong correlations, for the on-site Coulomb repulsion U = 8 t , in the range of temperatures 0.02 t ≲ T ≲ 0.3 t . To avoid the influence of the phase separation and size effects the calculations are performed in an infinite crystal, in the part of the phase diagram without inhomogeneities, for the electron concentration n ¯ = 0.92 . Solutions of the Eliashberg equation for singlet and triplet pairing, which are transformed according to one-dimensional representations of the lattice point group D 4 , are considered. For both models and all considered symmetries, eigenvalues of the Eliashberg equation are less than unity and demonstrate no upward trend with decreasing temperature. This result points to the absence of superconductivity in the Hubbard model in the strong-coupling regime. We discuss the reason for the small eigenvalue of the Eliashberg equation in the case of the singlet d x 2 − y 2 pairing.

DOI: 10.1088/1402-4896/ac03e2

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Abstract The possibility of the superconducting transition in the repulsive Hubbard model on a square lattice is studied using the ladder approximation of the strong coupling diagram technique. The t – U and t – t ′– t ″– U models are considered in the regime of strong correlations, for the on-site Coulomb repulsion U = 8 t , in the range of temperatures 0.02 t ≲ T ≲ 0.3 t . To avoid the influence of the phase separation and size effects the calculations are performed in an infinite crystal, in the part of the phase diagram without inhomogeneities, for the electron concentration n ¯ = 0.92 . Solutions of the Eliashberg equation for singlet and triplet pairing, which are transformed according to one-dimensional representations of the lattice point group D 4 , are considered. For both models and all considered symmetries, eigenvalues of the Eliashberg equation are less than unity and demonstrate no upward trend with decreasing temperature. This result points to the absence of superconductivity in the Hubbard model in the strong-coupling regime. We discuss the reason for the small eigenvalue of the Eliashberg equation in the case of the singlet d x 2 − y 2 pairing.

We address the problem of superconductivity in a system where the only many-body interactions are repulsive. The Eliashberg equations have been generalized to account for possible pairing correlations in the two-dimensional Hubbard model. Details of the two-dimensional band structure have been explicitly taken into account when considering the symmetry of the superconducting state. The pairing kernels have been discussed at low doping and at temperatures close to the superconducting transition temperature. We have proved that local Coulomb repulsion leads to attractive pairing correlations in the

d

-wave channel. Extended

s

-wave superconductivity is less likely to occur within this purely electronic model, at least in the physically interesting region of doping.

We analyze superconducting instabilities in 3D and 2D extended Hubbard model with Coulomb repulsion between electrons on neighboring sites in the limit of low electron density (n el → 0) on simple cubic (square) lattice. We show that in a realistic strong-coupling case U ≫ V ≫ W (U and V are the onsite and the intersite Coulomb repulsions, respectively, and W the bandwidth) the main SC instability corresponds to the p-wave pairing and in the leading order is correctly described by the equations obtained earlier in the absence of the intersite Coulomb interaction V = 0.

We study the phase diagram of the Hubbard model in the limit where

U

, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for

{T}_{c}

, the superconducting transition temperature, in terms of the correlation functions of the noninteracting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing

{T}_{c}

involves first integrating out all degrees of freedom having energy higher than an unphysical initial cutoff ${\ensuremath{\Omega}}_{0}$. Then, the renormalization group (RG) flows of the resulting effective action are computed and

{T}_{c}

is obtained by determining the scale below which the RG flows in the Cooper channel diverge. We prove that

{T}_{c}

is independent of ${\ensuremath{\Omega}}_{0}$. Using this method, we find a variety of unconventional superconducting ground states in two- and three-dimensional lattice systems, and present explicit results for

{T}_{c}

and pairing symmetries as a function of the electron concentration.

We study superconductivity in the extended Hubbard model with a strong on-site repulsion U and weak attractive density-density interaction

{\mathit{V}}_{\mathit{ij}}

. The electron correlation effects, beyond the weak-coupling regime of U, are included approximately by employing Hubbard decoupling approximations. The formulas for superconducting transition temperature

{\mathit{T}}_{\mathit{c}}

of the present approach are compared to those obtained with the strong-coupling expansion and slave-boson techniques. The methods share some common features. We discuss behavior of

{\mathit{T}}_{\mathit{c}}

for superconducting phases with s- and d-wave symmetry of the superconducting order parameter. In contrast to the mean-field treatment of U, we find that the strong on-site repulsion does not destroy superconductivity induced by ${\mathit{V}}_{\mathit{ij}}$0 even when U\ensuremath{\gg}|${\mathit{V}}_{\mathit{ij}}$|. \textcopyright{} 1996 The American Physical Society.

A new method for estimating the parameter ensuring the fulfillment of the Mermin-Wagner theorem in the strong coupling diagram technique (SCDT) for the two-dimensional Hubbard model is suggested. With the precise parameter value, calculated magnetic quantities are in good agreement with the results of numeric and optical-lattice experiments. Obtained spin and charge vertices are used for investigating superconductivity in the

t

-

U

and

t

-

t'

-

t"

-

U

Hubbard models in the regime of strong correlations. We found no superconducting transition in the

t

-

U

model. In the

t

-

t'

-

t"

-

U

model, the transition occurs for the singlet

d_{x^2-y^2}

pairing at

T_c\approx0.016t

. The difference between the two models is in the renormalized hopping describing electron motion in SCDT. In the

t

-

U

model, it vanishes for momenta

t"

,

(0,π)

of extrema of the

d

-wave order parameter, while the hopping is finite in the

t

-

t'

-

t"

-

U

model. In the one-band model, there are optimal values of

t'

and

t"

ensuring the highest

T_c

.