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Convexity of the Phase Boundary in the BCS Model with Imaginary Magnetic Field

Yohei Kashima

Published Jul 1, 2024• 0 citations
OpenAlex

Abstract

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 $\sqrt{9-4\sqrt{5}}$. 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 $\sqrt{9-4\sqrt{5}}$. 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.

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Quantum phase transition occurs at a quantum critical value of a control parameter such as the magnetic field in the Ising model in a transverse magnetic field (ITF). Recently, it is shown that ramping across the quantum critical point generates non-analytic behaviors in the time evolution of a closed quantum system in the thermodynamic limit at zero temperature. The mentioned phenomenon is called the dynamical quantum phase transition (DQPT). Here, we consider the one-dimensional ITF model with added the Dzyaloshinskii-Moriya interaction (DMI). Using the fermionization technique, the Hamiltonian is exactly diagonalized. Although the DMI induces chiral phase in the ground state phase diagram of the model, the study of the rate function of the return probability has proven that the DMI does not affect the DQPT. We conclude accordingly that the ramping across the quantum critical point is not a necessary and sufficient condition for DQPT.

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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 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 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.
In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit (Δ≪μΔ\ll μΔ≪μ). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature (∂Δ/∂T<0\partial Δ/ \partial T < 0∂Δ/∂T<0), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable u=e−4Ku=e^{-4K}u=e−4K, where K=J/(kBT)K=J/(k_BT)K=J/(kB​T), for nonzero external magnetic field, HHH. Exact results are given for the phase diagram in the uuu plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real h=H/(kBT)h=H/(k_BT)h=H/(kB​T), these results provide new insights into features of our earlier study of this case. We also consider complex h=H/(kBT)h=H/(k_BT)h=H/(kB​T) and μ=e−2hμ=e^{-2h}μ=e−2h. Calculations of complex-uuu zeros of the partition function on sections of the square lattice are presented. For the case of imaginary hhh, i.e., μ=eiθμ=e^{iθ}μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the uuu plane. We find that in this case, the phase boundary Bu{\cal B}_uBu​ contains a real line segment extending through part of the physical ferromagnetic interval 0≤u≤10 \le u \le 10≤u≤1, with a right-hand endpoint urheu_{rhe}urhe​ at the temperature for which the Yang-Lee edge singularity occurs at μ=e±iθμ=e^{\pm iθ}μ=e±iθ. Conformal field theory arguments are used to relate the singularities at urheu_{rhe}urhe​ and the Yang-Lee edge.
Quantum phase transition occurs at a quantum critical value of a control parameter such as the magnetic field in the Ising model in a transverse magnetic field (ITF). Recently, it is shown that ramping across the quantum critical point generates non-analytic behaviors in the time evolution of a closed quantum system in the thermodynamic limit at zero temperature. The mentioned phenomenon is called the dynamical quantum phase transition (DQPT). Here, we consider the one-dimensional ITF model with added the Dzyaloshinskii-Moriya interaction (DMI). Using the fermionization technique, the Hamiltonian is exactly diagonalized. Although the DMI induces chiral phase in the ground state phase diagram of the model, the study of the rate function of the return probability has proven that the DMI does not affect the DQPT. We conclude accordingly that the ramping across the quantum critical point is not a necessary and sufficient condition for DQPT.