Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1711.11047 · Physical review. B./Physical review. B

p -wave superconductivity in weakly repulsive 2D Hubbard model with Zeeman splitting and weak Rashba spin-orbit coupling

Henning G. Hugdal, Asle Sudbø

Published Jan 24, 2018• 3 citations
arXivHTMLPDF

Abstract

We study the superconducting order in a two-dimensional square lattice Hubbard model with weak repulsive interactions, subject to a Zeeman field and weak Rashba spin-orbit interactions. Diagonalizing the noninteracting Hamiltonian leads to two separate bands, and by deriving an effective low-energy interaction we find the mean field gap equations for the superconducting order parameter on the bands. Solving the gap equations just below the critical temperature, we find that superconductivity is caused by Kohn-Luttinger-type interaction, while the pairing symmetry of the bands is indirectly affected by the spin-orbit coupling. The dominating attractive momentum channel of the Kohn-Luttinger term depends on the filling fraction $n$ of the system, and it is therefore possible to change the momentum dependence of the order parameter by tuning $n$. Moreover, $n$ also determines which band has the highest critical temperature. Rotating the magnetic field changes the momentum dependence from states that for small momenta reduce to a chiral ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ type state for out-of-plane fields, to a nodal $p$-wave-type state for purely in-plane fields.

DOI
10.1103/physrevb.97.024515

Related papers

Rashba versus Kohn-Luttinger: Evolution of p -wave superconductivity in magnetized two-dimensional Fermi gases subject to spin-orbit interactions

Ethan Lake, Caleb Webb, D. A. Pesin +1 more

We study how the Rashba spin-orbit interaction influences unconventional superconductivity in a two-dimensional electron gas partially spin polarized by a magnetic field. Somewhat surprisingly, we find that for all field orientations, only the larger Fermi surface is superconducting. When the magnetic field is oriented out of plane, the system realizes a topological $p+ip$ pairing state. When the field is rotated in plane, the order parameter develops nodes along the field direction and finite center-of-mass-momentum pairing is realized. We demonstrate that the pairing symmetry of the system can be easily probed experimentally due to the dependence of various thermodynamic quantities on the magnetic field geometry, and calculate the electronic specific heat as an example.

arXiv:1602.02741•Physical review. B./Physical review. B•Jun 24, 2016•11 citations

Revisiting superconductivity in the extended one-band Hubbard model: Pairing via spin and charge fluctuations

Mercè Roig, Astrid Tranum Rømer, P. J. Hirschfeld +1 more

The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while ${B}_{1g}$ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and ${B}_{2g}$ (${d}_{xy}$) pairing dominates in the absence of repulsive longer-range Coulomb interactions ${V}_{\mathrm{NN}}$, the latter induces pairing in other symmetry channels, including, e.g., ${A}_{2g}$ ($g$-wave), nodal ${A}_{1g}$ (extended $s$-wave), or nodal ${E}_{u}$ ($p$-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, ${V}_{\mathrm{NN}}<0$, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts ${E}_{u}$ ($p$-wave) spin-triplet or ${B}_{1g}$ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.

arXiv:2211.07423•Physical review. B./Physical review. B•Dec 30, 2022•16 citations

Kohn-Luttinger instability of the t−t′ Hubbard model in two dimensions: Variational approach

J. Mráz, R. Hlubina

An effective Hamiltonian for the Kohn-Luttinger superconductor is constructed and solved in the BCS approximation. The method is applied to the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model in two dimensions with the following results: (i) The superconducting phase diagram at half filling is shown to provide a weak-coupling analog of the recently proposed spin liquid state in the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. (ii) In the parameter region relevant for the cuprates, we have found a nontrivial energy dependence of the gap function in the dominant d-wave pairing sector. The hot spot effect in the angular dependence of the superconducting gap is shown to be quite weak.

arXiv:cond-mat/0212176•Physical review. B, Condensed matter•May 22, 2003•16 citations

Phase diagram of thet–U2Hamiltonian of the weak coupling Hubbard model

Takashi Yanagisawa

We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.

arXiv:0803.1739•New Journal of Physics•Feb 12, 2008•31 citations

Unconventional superconductivity in the extended Hubbard model: Weak-coupling renormalization group

Sebastian Wolf, Thomas L. Schmidt, Stephan Rachel

In superconducting phases, repulsive interactions between the electrons usually lead to higher-angular momentum Cooper pairing. Here, the authors employ the weak-coupling renormalization group approach to study unconventional superconducting phases emerging in the extended, repulsive Hubbard model on paradigmatic two-dimensional lattices. Among their findings are extended regions of chiral topological superconductivity on the square, triangular, and honeycomb lattices. They provide detailed superconducting phase diagrams as well as the coupling strengths that quantify the corresponding critical temperatures depending on the band-structure parameters, band filling, and interaction parameters. The importance of the resolution and accuracy of the numerical integration for the method is discussed and a way to higher numerical efficiency provided.

arXiv:1808.02589•Physical review. B./Physical review. B•Nov 28, 2018•28 citations

Kohn-Luttinger effect and anomalous pairing in repulsive Fermi-systems at low density (Review Article)

M. Yu. Kagan, Dmitri V. Efremov, M. S. Mar’enko +1 more

We demonstrate the possibility of triplet p-wave pairing at low electron density a large number of models such as 3D and 2D Fermi-gas models with hard-core repulsion, 3D and 2D Hubbard models, and the Shubin-Vonsovsky model. The critical temperature for p-wave pairing can be considerably higher in the spin-polarized case or even in a two-band situation at low density and can reach experimentally observable values of 1–5 K. We also discuss briefly the d-wave pairing and high-Tc superconductivity with Tc ∼ 100 K which arise in the extended Hubbard model and in the generalized t-J model when close to half-filling.

Low Temperature Physics•Sep 1, 2012•1 citation
We study the superconducting order in a two-dimensional square lattice Hubbard model with weak repulsive interactions, subject to a Zeeman field and weak Rashba spin-orbit interactions. Diagonalizing the noninteracting Hamiltonian leads to two separate bands, and by deriving an effective low-energy interaction we find the mean field gap equations for the superconducting order parameter on the bands. Solving the gap equations just below the critical temperature, we find that superconductivity is caused by Kohn-Luttinger-type interaction, while the pairing symmetry of the bands is indirectly affected by the spin-orbit coupling. The dominating attractive momentum channel of the Kohn-Luttinger term depends on the filling fraction nnn of the system, and it is therefore possible to change the momentum dependence of the order parameter by tuning nnn. Moreover, nnn also determines which band has the highest critical temperature. Rotating the magnetic field changes the momentum dependence from states that for small momenta reduce to a chiral px\ifmmode±\else\textpm\fiipy{p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}px​\ifmmode±\else\textpm\fiipy​ type state for out-of-plane fields, to a nodal ppp-wave-type state for purely in-plane fields.
We study how the Rashba spin-orbit interaction influences unconventional superconductivity in a two-dimensional electron gas partially spin polarized by a magnetic field. Somewhat surprisingly, we find that for all field orientations, only the larger Fermi surface is superconducting. When the magnetic field is oriented out of plane, the system realizes a topological p+ipp+ipp+ip pairing state. When the field is rotated in plane, the order parameter develops nodes along the field direction and finite center-of-mass-momentum pairing is realized. We demonstrate that the pairing symmetry of the system can be easily probed experimentally due to the dependence of various thermodynamic quantities on the magnetic field geometry, and calculate the electronic specific heat as an example.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g{B}_{2g}B2g​ (dxy{d}_{xy}dxy​) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN{V}_{\mathrm{NN}}VNN​, the latter induces pairing in other symmetry channels, including, e.g., A2g{A}_{2g}A2g​ (ggg-wave), nodal A1g{A}_{1g}A1g​ (extended sss-wave), or nodal Eu{E}_{u}Eu​ (ppp-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0{V}_{\mathrm{NN}}<0VNN​<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu{E}_{u}Eu​ (ppp-wave) spin-triplet or B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
An effective Hamiltonian for the Kohn-Luttinger superconductor is constructed and solved in the BCS approximation. The method is applied to the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model in two dimensions with the following results: (i) The superconducting phase diagram at half filling is shown to provide a weak-coupling analog of the recently proposed spin liquid state in the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model. (ii) In the parameter region relevant for the cuprates, we have found a nontrivial energy dependence of the gap function in the dominant d-wave pairing sector. The hot spot effect in the angular dependence of the superconducting gap is shown to be quite weak.
We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.
In superconducting phases, repulsive interactions between the electrons usually lead to higher-angular momentum Cooper pairing. Here, the authors employ the weak-coupling renormalization group approach to study unconventional superconducting phases emerging in the extended, repulsive Hubbard model on paradigmatic two-dimensional lattices. Among their findings are extended regions of chiral topological superconductivity on the square, triangular, and honeycomb lattices. They provide detailed superconducting phase diagrams as well as the coupling strengths that quantify the corresponding critical temperatures depending on the band-structure parameters, band filling, and interaction parameters. The importance of the resolution and accuracy of the numerical integration for the method is discussed and a way to higher numerical efficiency provided.
We demonstrate the possibility of triplet p-wave pairing at low electron density a large number of models such as 3D and 2D Fermi-gas models with hard-core repulsion, 3D and 2D Hubbard models, and the Shubin-Vonsovsky model. The critical temperature for p-wave pairing can be considerably higher in the spin-polarized case or even in a two-band situation at low density and can reach experimentally observable values of 1–5 K. We also discuss briefly the d-wave pairing and high-Tc superconductivity with Tc ∼ 100 K which arise in the extended Hubbard model and in the generalized t-J model when close to half-filling.