[Paper Review] Superconducting phase in the BCS model with imaginary magnetic field. II. Multi-scale infrared analysis

Yohei Kashima|arXiv (Cornell University)|Sep 20, 2017

Physics of Superconductivity and Magnetism15 references3 citations

TL;DR

This paper establishes the existence of spontaneous U(1)-symmetry breaking and off-diagonal long-range order (ODLRO) in the reduced BCS model with an imaginary magnetic field using a multi-scale infrared (IR) analysis of Grassmann Gaussian integrals. The analysis proves superconducting order in a large parameter domain at positive temperatures, including arbitrarily low temperatures, while showing the order vanishes in the zero-temperature limit, despite the interaction strength being fixed and independent of temperature and field.

ABSTRACT

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.

Motivation & Objective

• To rigorously establish the existence of spontaneous U(1)-symmetry breaking (SSB) and off-diagonal long-range order (ODLRO) in the reduced BCS model under an imaginary magnetic field.
• To extend previous results by removing the dependence of the interaction strength on temperature and magnetic field, fixing it independently.
• To analyze the model in a large domain of temperature and imaginary magnetic field, including arbitrarily low temperatures.
• To demonstrate that SSB and ODLRO persist down to arbitrarily low positive temperatures but vanish in the zero-temperature limit.
• To develop and apply a multi-scale infrared integration scheme based on Grassmann Gaussian formulations, building on prior work with a double-scale integration approach.

Proposed method

• Formulates the reduced BCS model with an imaginary magnetic field using Grassmann Gaussian integrals, enabling rigorous analysis of non-Hermitian Hamiltonians.
• Implements a multi-scale infrared (IR) integration scheme with a double-scale structure to control the flow of correlations across energy scales.
• Introduces an artificial term to regularize the integration and ensure convergence, later removed via analytic continuation.
• Applies generalized covariances and scale-dependent bounds to control the growth of interaction terms across successive integration steps.
• Uses a decomposition of the covariance into low- and high-momentum components to isolate infrared contributions and control long-range order.
• Employs a rigorous infinite-volume limit procedure based on uniform bounds and analyticity in the coupling parameters.

Experimental results

Research questions

• RQ1 Does spontaneous U(1)-symmetry breaking and ODLRO emerge in the reduced BCS model with an imaginary magnetic field at positive temperatures, even when the interaction strength is fixed and independent of temperature and field?
• RQ2 Can the superconducting phase be rigorously established in a large domain of temperature and imaginary magnetic field, including arbitrarily low temperatures?
• RQ3 What happens to the superconducting order in the zero-temperature limit, despite its presence at all positive temperatures?
• RQ4 How does the multi-scale IR analysis with double-scale integration ensure convergence and control of the Grassmann integral in the presence of long-range correlations?
• RQ5 Can the model be rigorously analyzed beyond the BCS functional approach, particularly when the Gibbs state is not well-defined due to non-Hermiticity?

Key findings

• Spontaneous U(1)-symmetry breaking and off-diagonal long-range order (ODLRO) are rigorously proven to occur in a large domain of the temperature and imaginary magnetic field parameter space.
• The superconducting phase persists at arbitrarily low positive temperatures, confirming the stability of the order in the low-temperature regime.
• Despite the persistence at all positive temperatures, the SSB and ODLRO vanish in the zero-temperature limit, indicating a non-analytic behavior at T=0.
• The interaction strength is fixed independently of temperature and imaginary magnetic field, overcoming a key limitation of prior work where the interaction had to shrink as temperature decreased or field approached criticality.
• The multi-scale infrared analysis successfully controls the Grassmann integral through a double-scale integration scheme, ensuring convergence and enabling the proof of long-range order.
• The results apply to physically relevant models such as nearest-neighbor hopping on 3D/4D cubic lattices with degenerate Fermi surfaces and the honeycomb lattice at zero chemical potential.