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arXiv:1903.09780

Superconducting phase in the BCS model with imaginary magnetic field. III. Non-vanishing free dispersion relations

Yohei Kashima

Published Mar 23, 2019• 2 citations
arXivHTMLPDF

Abstract

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 $\sqrt{17-12\sqrt{2}}$.

DOI
10.48550/arxiv.1903.09780

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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 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 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 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.
Renormalization group analysis for multi-band many-electron systems at half-filling at positive temperature is presented. The analysis includes the Matsubara ultra-violet integration and the infrared integration around the zero set of the dispersion relation. The multi-scale integration schemes are implemented in a finite-dimensional Grassmann algebra indexed by discrete position-time variables. In order that the multi-scale integrations are justified inductively, various scale-dependent estimates on Grassmann polynomials are established. We apply these theories in practice to prove that for the half-filled Hubbard model with nearest-neighbor hopping on a square lattice the infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette and 0 (mod 2 pi) through the large circles around the periodic lattice. Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev. Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy density of the flux phase problem converges to an analytic function of the coupling constant in the infinite-volume, zero-temperature limit. The proof of the theorem is based on a four-band formulation of the model Hamiltonian and an extension of Giuliani-Mastropietro's renormalization designed for the half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V. Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).
We solve the problem of fermionic pairing mediated by a massless boson in the limit of large coupling constant. At weak coupling, the transition temperature is exponentially small and superconductivity is robust against phase fluctuation. In the strong coupling limit, the pair formation occurs at a temperature of the order of the Fermi energy, however, the actual transition temperature is much smaller due to phase and amplitude fluctuations of the pairing gap. Our model calculations describe superconductivity due to color magnetic interactions in quark matter and in systems close to a ferromagnetic quantum critical point with Ising symmetry. Our strong coupling results are, however, more general and can be applied to other systems as well, including the antiferromagnetic exchange in 2D used for description of the cuprates.
We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.