Temperature Dependence of Paramagnetic Critical Magnetic Field in Disordered Attractive Hubbard Model

Abstract

Within the generalized DMFT+Σ approach, we study disorder effects in the temperature dependence of paramagnetic critical magnetic field H_cp(T) for Hubbard model with attractive interaction. We consider the wide range of attraction potentials U—from the weak coupling limit, when superconductivity is described by BCS model, up to the limit of very strong coupling, when superconducting transition is related to Bose–Einstein condensation (BEC) of compact Cooper pairs. The growth of the coupling strength leads to the rapid growth of H_cp(T) at all temperatures. However, at low temperatures, paramagnetic critical magnetic field H_cp grows with U much more slowly, than the orbital critical field, and in BCS limit, the main contribution to the upper critical magnetic filed is of paramagnetic origin. The growth of the coupling strength also leads to the disappearance of the low temperature region of instability towards type I phase transition and Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) phase, characteristic of BCS weak coupling limit. Disordering leads to the rapid drop of H_cp(T) in BCS weak coupling limit, while in BCS–BEC crossover region and BEC limit H_cp(T → 0) dependence on disorder is rather weak. Within DMFT+Σ approach, disorder influence on H_cp(T) is of universal nature at any coupling strength and is related only to disorder widening of the conduction band. In particular, this leads to the drop of the effective coupling strength with disorder, so that disordering restores the region of type I transition in the intermediate coupling region.

APPENDIX

EQUATION FOR PARAMAGNETIC CRITICAL MAGNETIC FIELD

In general case the Noziers–Schmitt-Rink approach [1] assumes, that corrections from strong pairing interaction significantly change the chemical potential of the system, but possible vertex corrections from this interaction in Cooper channel are irrelevant, so that to analyze Cooper instability we can use the weak coupling approximation (ladder approximation). In this approximation the condition of Cooper instability in disordered attractive Hubbard model is written as:

$$1 = U{{\chi }_{0}}(q = 0,{{\omega }_{m}} = 0),$$

((7))

where

$${{\chi }_{0}}(q = 0,{{\omega }_{m}} = 0) = T\sum\limits_n^{} {\sum\limits_{{\mathbf{pp}}'}^{} {{{\Phi }_{{{\mathbf{pp}}'}}}({{\varepsilon }_{n}})} } $$

((8))

is two-particle loop in Cooper channel “dressed” only by impurity scattering, while Φ_pp'(ε_n) is the averaged over impurities two-particle Green’s function in Cooper channel at Matsubara frequencies ε_n = πT(2n + 1).

To obtain $\sum\nolimits_{{\mathbf{pp}}'}^{} {{{\Phi }_{{{\mathbf{pp}}'}}}}$ (ε_n) we use an exact Ward identity, derived by us in [8]:

$$\begin{gathered} {{G}_{ \uparrow }}({{\varepsilon }_{n}},{\mathbf{p}}) - {{G}_{ \downarrow }}( - {{\varepsilon }_{n}}, - {\mathbf{p}}) \\ = - \sum\limits_{\mathbf{p}}^{} {{{\Phi }_{{{\mathbf{pp}}'}}}({{\varepsilon }_{n}})(G_{{0 \uparrow }}^{{ - 1}}({{\varepsilon }_{n}},{\mathbf{p}}{\text{'}}) - G_{{0 \downarrow }}^{{ - 1}}( - {{\varepsilon }_{n}}, - {\mathbf{p}})).} \\ \end{gathered} $$

((9))

Here G_0↑,↓(ε_n, p) = (iε_n + μ – ε(p) ± μ_BH)^–1 is the “bare” Green’s function and G_↑,↓(ε_n, p) is averaged over impurities (but not “dressed” by Hubbard interaction!) single-particle Green’s function. Using the symmetry ε(p) = ε(–p) we obtain from Ward identity (9):

$$\sum\limits_{{\mathbf{pp}}'}^{} {{{\Phi }_{{{\mathbf{pp}}'}}}({{\varepsilon }_{n}}) = - \frac{{\sum\limits_{\mathbf{p}}^{} {{{G}_{ \uparrow }}({{\varepsilon }_{n}},{\mathbf{p}}) - \sum\limits_{\mathbf{p}}^{} {{{G}_{ \downarrow }}( - {{\varepsilon }_{n}},{\mathbf{p}})} } }}{{2i{{\varepsilon }_{n}} + 2{{\mu }_{{\text{B}}}}H}},} $$

((10))

so that for Cooper susceptibility (8) we get:

$$\begin{gathered} {{\chi }_{0}}(q = 0,{{\omega }_{m}} = 0) \\ = - \frac{T}{2}\sum\limits_n^{} {\frac{{\sum\limits_{\mathbf{p}}^{} {{{G}_{ \uparrow }}({{\varepsilon }_{n}},{\mathbf{p}}) - \sum\limits_{\mathbf{p}}^{} {{{G}_{ \downarrow }}( - {{\varepsilon }_{n}},{\mathbf{p}})} } }}{{i{{\varepsilon }_{n}} + {{\mu }_{{\text{B}}}}H}}} \\ = - \frac{T}{2}\sum\limits_n^{} {\left( {\frac{{\sum\limits_{\mathbf{p}}^{} {{{G}_{ \uparrow }}({{\varepsilon }_{n}},{\mathbf{p}})} }}{{i{{\varepsilon }_{n}} + {{\mu }_{{\text{B}}}}H}} + \frac{{\sum\limits_{\mathbf{p}}^{} {{{G}_{ \downarrow }}({{\varepsilon }_{n}},{\mathbf{p}})} }}{{i{{\varepsilon }_{n}} - {{\mu }_{{\text{B}}}}H}}} \right)} . \\ \end{gathered} $$

((11))

Performing the standard summation over Fermion Matsubara frequencies, we obtain:

$$\begin{gathered} {{\chi }_{0}} = - \frac{1}{{8\pi i}}\int\limits_{ - \infty }^\infty {d\varepsilon \left( {\frac{{\sum\limits_{\mathbf{p}}^{} {G_{ \uparrow }^{R}(\varepsilon ,{\mathbf{p}}) - \sum\limits_{\mathbf{p}}^{} {G_{ \downarrow }^{A}(\varepsilon ,{\mathbf{p}})} } }}{{\varepsilon + {{\mu }_{{\text{B}}}}H}}} \right.} \\ \left. { + \frac{{\sum\limits_{\mathbf{p}}^{} {G_{ \downarrow }^{R}(\varepsilon ,{\mathbf{p}}) - \sum\limits_{\mathbf{p}}^{} {G_{ \downarrow }^{A}(\varepsilon ,{\mathbf{p}})} } }}{{\varepsilon - {{\mu }_{{\text{B}}}}H}}} \right)\tanh \frac{\varepsilon }{{2T}} \\ = \frac{1}{2}\int\limits_{ - \infty }^\infty {d\varepsilon \left( {\frac{{{{{\tilde {N}}}_{{0 \uparrow }}}(\varepsilon )}}{{\varepsilon + {{\mu }_{{\text{B}}}}H}} + \frac{{{{{\tilde {N}}}_{{0 \downarrow }}}(\varepsilon )}}{{\varepsilon - {{\mu }_{{\text{B}}}}H}}} \right)\tanh \frac{\varepsilon }{{2T}}} , \\ \end{gathered} $$

((12))

where ${{\tilde {N}}_{{0 \uparrow , \downarrow }}}$(ε) is the “bare” (U = 0) density of states for different spin projections, “dressed” by impurity scattering. Spin splitting can be considered as an addition to chemical potential, so that introducing the “bare” density of states “dressed” by disorder in the absence of external magnetic field ${{\tilde {N}}_{0}}$(ε), we obtain the final result for Cooper susceptibility:

$${{\chi }_{0}}\, = \,\frac{1}{4}\int\limits_{ - \infty }^\infty {d\varepsilon \frac{{{{{\tilde {N}}}_{0}}(\varepsilon )}}{\varepsilon }\left( {{\text{tanh}}\frac{{\varepsilon + {{\mu }_{{\text{B}}}}H}}{{2T}}\, + \,\tanh \frac{{\varepsilon - {{\mu }_{{\text{B}}}}H}}{{2T}}} \right)} \,.$$

((13))

In Eq. (13) energy ε is counted from the chemical potential level. If we count it from the middle of the conduction band we have to replace ε → ε – μ and the condition of Cooper instability (7) leads to the equation defining critical temperature depending on the external magnetic field, which gives the equation for paramagnetic critical magnetic filed (6). The chemical potential for different values of U and Δ should be determined from DMFT+Σ calculations, i.e. from the standard equation for electron number (band filling), which allows us to find H_cp for the wide range of model parameters, including the region of BCS–BEC crossover and the limit of strong coupling at different levels of disorder. This reflects the physical meaning of Nozieres–Scmitt-Rink approximation—in the weak coupling region the temperature of superconducting transition is controlled by the equation for Cooper instability (6), while in the strong coupling limit it is defined as the temperature of BEC, which is controlled by chemical potential. The joint solution of Eq. (6) and the equation for the chemical potential guarantees the correct interpolation for H_cp in the region of BCS–BEC crossover.