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arXiv:1110.6752 · Journal of Physics A Mathematical and Theoretical

Exactly solvable pairing models in two dimensions

Lian-Ao Wu

Published May 8, 2017• 1 citation
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Abstract

Abstract The BCS theory models electron correlations with pure zero-momentum pairs. Here we consider a family of pairing Hamiltonians, where electron correlations are modelled by pure arbitrary-momentum pairs. These models are two-dimensional, defined by S z which is the only conserved component of the total electron spin. We find that all models in this family are exactly solvable, and present the solutions. It is interesting to note that the d -wave pair condensate state in high T c superconductivity could be a ground state of one of model Hamiltonians, which has analytical solutions. Significantly, it is shown that there is a sudden change in ground state energy versus pairing strength, suggesting a quantum phase transition from an independent-particle picture to the d -wave pair condensate.

DOI
10.1088/1751-8121/aa7177

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Abstract The BCS theory models electron correlations with pure zero-momentum pairs. Here we consider a family of pairing Hamiltonians, where electron correlations are modelled by pure arbitrary-momentum pairs. These models are two-dimensional, defined by S z which is the only conserved component of the total electron spin. We find that all models in this family are exactly solvable, and present the solutions. It is interesting to note that the d -wave pair condensate state in high T c superconductivity could be a ground state of one of model Hamiltonians, which has analytical solutions. Significantly, it is shown that there is a sudden change in ground state energy versus pairing strength, suggesting a quantum phase transition from an independent-particle picture to the d -wave pair condensate.
We present a new model describing strongly correlated electrons on a general d-dimensional lattice. It is an extended Hubbard model and it contains the t-J model as a special case. The model naturally describes local electron pairs, which can move coherently at arbitrary momentum. By using an \ensuremath{\eta}-pairing mechanism we can construct eigenstates of the Hamiltonian with off-diagonal long-range order. In the attractive case the exact ground state is superconducting in any number of dimensions. On a one-dimensional lattice, the model is exactly solvable by Bethe ansatz.
The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson's early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-NNN limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.
We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with p_x + i p_y symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parameterized by the filling fraction and the coupling constant. It consists of three phases denoted weak coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are dressed versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.
We present an exactly solvable Hamiltonian which consists of nearest neighbor hopping and long-range interaction. The ground state and the thermodynamic quantities are analytically obtained in any dimensions. In a repulsive case, the Luttinger's theorem breaks down and the hypothesis of the adiabatic continuation no longer holds. At half filling, we show the existence of a metal-insulator transition as a function of the strength of the interaction. The zero energy excitation disappears at the critical value of the interaction when we increase the interaction. In this case, the system is an insulator. The charge compressibility behaves as κ≈|1-ρ| 1-2/ d near half filling in d dimensions where ρ is the density of electrons. At other fillings, there is always a zero energy excitation and the system is metallic. When the interaction is attractive, the ground state is composed of pairs of electrons with opposite spins. The low energy excitations, however, is not a particle-hole type of a non-interacting case but a pairwise excitation across the Fermi surface.
We present the exact Bethe ansatz solution for the two-dimensional BCS pairing Hamiltonian with px+ipy{p}_{x}+i{p}_{y}px​+ipy​ symmetry. Using both mean-field theory and the exact solution we obtain the ground-state phase diagram parametrized by the filling fraction and the coupling constant. It consists of three phases that are denoted weak-coupling BCS, weak pairing, and strong pairing. The first two phases are separated by a topologically protected line where the exact ground state is given by the Moore-Read pfaffian state. In the thermodynamic limit the ground-state energy is discontinuous on this line. The other two phases are separated by the critical line, also topologically protected, previously found by Read and Green. We establish a duality relation between the weak and strong pairing phases, whereby ground states of the weak phase are ``dressed'' versions of the ground states of the strong phase by zero energy (Moore-Read) pairs and characterized by a topological order parameter.
We consider an exactly solvable two-band model of electrons moving in one dimension and interacting with a $\ensuremath{\delta}$-function spin-exchange potential. The relative population of the bands is determined by the band splitting, the magnetic field, and the temperature. The interaction leads to the formation of spin-triplet bound states of the Cooper type (hard-core bosons that do not condensate and exist at all temperatures). In the zero-field ground state the spin-triplet states are effectively free bosons. A threshold band splitting is required to break up a triplet pair. The low-temperature and finite size properties are discussed in the context of anomalous superconducting and antiferromagnetic fluctuations.