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arXiv:1008.3406 · Physical Review B

Quantum phase diagram of the integrable px+ipy fermionic superfluid

Stefan Rombouts, J. Dukelsky, Gerardo Ortíz

Published Dec 13, 2010• 99 citations
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Abstract

We determine the zero-temperature quantum phase diagram of a ${p}_{x}+i{p}_{y}$ pairing model based on the exactly solvable hyperbolic Richardson-Gaudin model. We present analytical and large-scale numerical results for this model. In the continuum limit, the exact solution exhibits a third-order quantum phase transition, separating a strong-pairing from a weak-pairing phase. The mean-field solution allows to connect these results to other models with ${p}_{x}+i{p}_{y}$ pairing order. We define an experimentally accessible characteristic length scale, associated with the size of the Cooper pairs, that diverges at the transition point, indicating that the phase transition is of a confinement-deconfinement type without local order parameter. We propose an experimental measurement to detect the transition. We show that this phase transition is not limited to the ${p}_{x}+i{p}_{y}$ pairing model but can be found in any representation of the hyperbolic Richardson-Gaudin model and is related to a symmetry that is absent in the rational Richardson-Gaudin model.

DOI
10.1103/physrevb.82.224510

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We determine the zero-temperature quantum phase diagram of a px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing model based on the exactly solvable hyperbolic Richardson-Gaudin model. We present analytical and large-scale numerical results for this model. In the continuum limit, the exact solution exhibits a third-order quantum phase transition, separating a strong-pairing from a weak-pairing phase. The mean-field solution allows to connect these results to other models with px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing order. We define an experimentally accessible characteristic length scale, associated with the size of the Cooper pairs, that diverges at the transition point, indicating that the phase transition is of a confinement-deconfinement type without local order parameter. We propose an experimental measurement to detect the transition. We show that this phase transition is not limited to the px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing model but can be found in any representation of the hyperbolic Richardson-Gaudin model and is related to a symmetry that is absent in the rational Richardson-Gaudin model.
Recently, interest has increased in the hyperbolic family of integrable Richardson-Gaudin (RG) models. It was pointed out that a particular linear combination of the integrals of motion of the hyperbolic RG model leads to a Hamiltonian that describes ppp-wave pairing in a two-dimensional system. Such an interaction is found to be present in fermionic superfluids (3He{}^{3}\text{He}3He), ultracold atomic gases, and ppp-wave superconductivity. Furthermore the phase diagram is intriguing, with the presence of the Moore-Read and Read-Green lines. At the Read-Green line a rare third-order quantum phase transition occurs. The present paper makes a connection between collective bosonic states and the exact solutions of the px+ipy{p}_{x}+i{p}_{y}px​+ipy​ pairing Hamiltonian. This makes it possible to investigate the effects of the Pauli principle on the energy spectrum, by gradually reintroducing the Pauli principle. It also introduces an efficient and stable numerical method to probe all the eigenstates of this class of Hamiltonians. We extend the phase diagram to repulsive interactions, an area that was not previously explored due to the lack of a proper mean-field solution in this region. We found a connection between the point in the phase diagram where the ground state connects to the bosonic state with the highest collectivity, and the Moore-Read line where all the Richardson-Gaudin (RG) variables collapse to zero. In contrast with the reduced BCS case, the overlap between the ground state and the highest collective state at the Moore-Read line is not the largest. In fact it shows a minimum when most other bosonic states show a maximum of the overlap. We found remnants of the Read-Green line for finite systems, by investigating the total spectrum. A symmetry was found between the Hamiltonian with and without single-particle part. When the interaction was repulsive we found four different classes of trajectories of the RG variables.
We introduce a family of integrable spin-boson models, focusing on a particular realization describing a system of spinless fermions interacting through a ppp-wave Feshbach resonance. Unlike the BCS-BEC crossover of the sss-wave case, the ppp-wave model has a third-order quantum phase transition. The critical point coincides with the deconfinement of a single molecule within a BEC of bound dipolar molecules. The exact many-body wave function yields a unique perspective of the quantum critical region, suggesting that the size of the condensate wave function that diverges logarithmically with the chemical potential could be used as an experimental indicator of the phase transition.
We explore the zero-temperature phase diagram of bosons interacting via Feshbach resonant pairing interactions in one dimension. Using DMRG (density matrix renormalization group) and field theory techniques we characterize the phases and quantum phase transitions in this low-dimensional setting. We provide a broad range of evidence in support of an Ising quantum phase transition separating distinct paired superfluids, including results for the energy gaps, correlation functions, and entanglement entropy. In particular, we show that the Ising correlation length, order parameter and critical properties are directly accessible from a ratio of the atomic and molecular two-point functions. We further demonstrate that both the zero-momentum occupation numbers and the visibility are in accordance with the absence of a purely atomic superfluid phase. We comment on the connection to recent studies of boson pairing in a generalized classical XY\mathit{XY}XY model.
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.