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arXiv:1609.06121

Superconducting phase in the BCS model with imaginary magnetic field

Yohei Kashima

Published Sep 20, 2016• 3 citations
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Abstract

We prove that in the reduced quartic BCS model with an imaginary external magnetic field a spontaneous U(1)-symmetry breaking (SSB) and an off-diagonal long range order (ODLRO) occur. The system is defined on a hyper-cubic lattice with periodic boundary conditions at positive temperature. In the free part of the Hamiltonian we assume the nearest-neighbor hopping. The chemical potential is fixed so that the free Fermi surface does not degenerate. The term representing the interaction between electrons' spin and the imaginary external magnetic field is the z-component of the spin operator multiplied by a pure imaginary parameter. The SSB and the ODLRO are shown in the infinite-volume limit of the thermal average over the full Fermionic Fock space. The magnitude of the negative coupling constant must be larger than a certain value so that the gap equation is solvable. The gap equation is different from that of the conventional mean field BCS model because of the presence of the imaginary magnetic field. By adjusting the imaginary magnetic field this model shows the SSB and the ODLRO in high temperature, weak coupling regimes where the conventional reduced BCS model does not show these phenomena. The proof is based on Grassmann Gaussian integral formulations and a double-scale integration scheme to analytically control the formulations.

DOI
10.48550/arxiv.1609.06121

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We prove that in the reduced quartic BCS model with an imaginary external magnetic field a spontaneous U(1)-symmetry breaking (SSB) and an off-diagonal long range order (ODLRO) occur. The system is defined on a hyper-cubic lattice with periodic boundary conditions at positive temperature. In the free part of the Hamiltonian we assume the nearest-neighbor hopping. The chemical potential is fixed so that the free Fermi surface does not degenerate. The term representing the interaction between electrons' spin and the imaginary external magnetic field is the z-component of the spin operator multiplied by a pure imaginary parameter. The SSB and the ODLRO are shown in the infinite-volume limit of the thermal average over the full Fermionic Fock space. The magnitude of the negative coupling constant must be larger than a certain value so that the gap equation is solvable. The gap equation is different from that of the conventional mean field BCS model because of the presence of the imaginary magnetic field. By adjusting the imaginary magnetic field this model shows the SSB and the ODLRO in high temperature, weak coupling regimes where the conventional reduced BCS model does not show these phenomena. The proof is based on Grassmann Gaussian integral formulations and a double-scale integration scheme to analytically control the formulations.
We analyze the reduced BCS model with an imaginary magnetic field in a large domain of the temperature and the imaginary magnetic field. The magnitude of the attractive reduced BCS interaction is fixed to be small but independent of the temperature and the imaginary magnetic field unless the temperature is high. We impose a series of conditions on the free dispersion relation. These conditions are typically satisfied by free electron models with degenerate Fermi surface. For example, our theory applies to the model with nearest-neighbor hopping on 3 or 4-dimensional (hyper-)cubic lattice having degenerate free Fermi surface or the model with nearest-neighbor hopping on the honeycomb lattice with zero chemical potential. We prove that a spontaneous U(1)-symmetry breaking (SSB) and an off-diagonal long range order (ODLRO) occur in many areas of the parameter space. The SSB and the ODLRO are proved to occur in low temperatures arbitrarily close to zero in particular. However, it turns out that the SSB and the ODLRO are not present in the zero-temperature limit. The proof is based on Grassmann Gaussian integral formulations and a multi-scale infrared analysis of the formulations. We keep using notations and lemmas of our previous work [Y. Kashima, accepted for publication in J. Math. Sci. Univ. Tokyo, arXiv:1609.06121] implementing the double-scale integration scheme. So the multi-scale analysis this paper presents is a continuation of the previous work.
We analyze a class of the BCS model, whose free dispersion relation is non-vanishing, under the influence of imaginary magnetic field at positive temperature. The magnitude of the negative coupling constant must be small but is allowed to be independent of the temperature and the imaginary magnetic field. The infinite-volume limit of the free energy density is characterized. A spontaneous symmetry breaking and an off-diagonal long range order are proved to occur only in high temperatures. This is because the gap equation in this model has a positive solution only if the temperature is higher than a critical value. The proof is based on a double-scale integration of the Grassmann integral formulation. In this scheme we integrate with the infrared covariance first and with the ultra-violet covariance afterwards. As the other focus, we study geometric properties of the phase boundaries, which are periodic copies of a closed curve in the two-dimensional space of the temperature and the real time variable. Here we adopt the real time variable in place of the temperature times the imaginary magnetic field by considering its relevance within contemporary physics of dynamical phase transition at positive temperature. As the main result, we show that for any choice of a non-vanishing free dispersion relation the representative curve of the phase boundaries has only one local minimum point, or in other words the phase boundaries do not oscillate with temperature, if and only if the minimum of the magnitude of the free dispersion relation over the maximum is larger than the critical value 17−122\sqrt{17-12\sqrt{2}}17−122​​.
We show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑
We study geometric properties of the domain of the two parameters (inverse temperature, imaginary magnetic field) where the gap equation of the BCS model with imaginary magnetic field has a positive solution. If the interaction is weak and the free dispersion relation is non-vanishing, the domain is a disjoint union of periodic copies of one representative set in the plane of (inverse temperature, imaginary magnetic field). In this paper we provide a necessary and sufficient condition for the representative set to be convex as the main result. More precisely we prove the following. The representative set is convex for any weak coupling and non-vanishing free dispersion relation if and only if the minimum of the magnitude of the free dispersion relation over the maximum is larger than the critical value 9−45\sqrt{9-4\sqrt{5}}9−45​​. In the context of dynamical quantum phase transition (DQPT) the imaginary magnetic field is considered as the real time variable. So this is an analysis of the phase boundary of a DQPT in the plane of (inverse temperature, real time). In particular convexity of the representative phase boundary is characterized by the critical constant 9−45\sqrt{9-4\sqrt{5}}9−45​​. The gap equation rigorously derived in the preceding paper [Y. Kashima, J. Math. Sci. Univ. Tokyo {\bf 28} (2021), 399-556] is at the core of our analysis.
We study the BCS--BEC crossover of the large NNN attractive Fermi-Hubbard model on a one-dimensional lattice using the mean field approximation in the presence of an imaginary chemical potential. We show that the crossover is governed by three parameters. The imaginary chemical potential iθiθiθ, the temperature via a thermal kernel g(βEk,βθ)g(βE_k,βθ)g(βEk​,βθ) and the parameter δuδ_uδu​ whose sign controls the weak and strong coupling regimes. At the unitarity point (U=UcU=U_cU=Uc​), we find a thermal window ϕ=βθ=2π/3,4π/3ϕ=βθ=2π/3,4π/3ϕ=βθ=2π/3,4π/3 where the gap vanishes while the fermion number NfN_fNf​, which quantifies the balance between particle-like and hole-like excitations, has a local maximum/minimum. Inside this thermal window BCS and BEC physics are await changes in the coupling to be selected as the dominant regime. We expect that our results will unveil a better understanding of pairing correlations in lattice many-body physics.
We analyze a mean-field model of electrons on a square lattice with two types of interaction: forward scattering favoring a ddd-wave Pomeranchuk instability and a BCS pairing interaction driving ddd-wave superconductivity. By tuning the interaction parameters a rich variety of phase diagrams is obtained. If the BCS interaction is not too strong, Fermi surface symmetry breaking is stabilized around van Hove filling, and coexists with superconductivity at low temperatures. For pure forward scattering Fermi surface symmetry breaking occurs typically via a first order transition at low temperatures. The presence of superconductivity reduces the first order character of this transition and, if strong enough, can turn it into a continuous one. This gives rise to a quantum critical point within the superconducting phase. The superconducting gap tends to suppress Fermi surface symmetry breaking. For a relatively strong BCS interaction, Fermi surface symmetry breaking can be limited to intermediate temperatures, or can be suppressed completely by pairing.