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arXiv:nucl-th/9805039 · Physical Review Letters

Exactly Solvable Model of a Superconducting to Rotational Phase Transition

DAVID J. ROWE, C. Bahri, Wasantha Wijesundera

Published May 18, 1998• 47 citations
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Abstract

We consider a many-fermion model which exhibits a transition from a superconducting to a rotational phase with variation of a parameter in its Hamiltonian. The model has analytical solutions in its two limits due to the presence of dynamical symmetries. However, the symmetries are basically incompatible with one another; no simple solution exists in intermediate situations. Exact (numerical) solutions are possible and enable one to study the behavior of competing but incompatible symmetries and the phase transitions that result in a semirealistic situation. The results are remarkably simple and shed light on the nature of phase transitions.

DOI
10.1103/physrevlett.80.4394

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We consider a many-fermion model which exhibits a transition from a superconducting to a rotational phase with variation of a parameter in its Hamiltonian. The model has analytical solutions in its two limits due to the presence of dynamical symmetries. However, the symmetries are basically incompatible with one another; no simple solution exists in intermediate situations. Exact (numerical) solutions are possible and enable one to study the behavior of competing but incompatible symmetries and the phase transitions that result in a semirealistic situation. The results are remarkably simple and shed light on the nature of phase transitions.
We consider the pairing-plus-quadrupole model Hamiltonian within the framework of a many-fermion shell model. The Hamiltonian exhibits a transition from a superconducting to a rotational phase with variation of a parameter. The model has analytical solutions in its two limits due to the presence of dynamical symmetries. However, the symmetries are basically incompatible with one another which means that no simple solution exists in intermediate situations. Exact (numerical) solutions are nevertheless possible on a computer and enable one to study the behavior of competing but incompatible symmetries and the phase transitions that result in a semirealistic situation. This mixed-symmetry dynamics generates a rich structure which gives new insight for the study of phase transitions.
Many-body models undergoing a quantum phase transition to a broken-symmetry phase that survives up to a critical temperature must possess, in the ordered phase, symmetric as well as nonsymmetric eigenstates. We predict, and explicitly show in the fully connected Ising model in a transverse field, that these two classes of eigenstates do not overlap in energy, and therefore that an energy edge exists separating low-energy symmetry-breaking eigenstates from high-energy symmetry-invariant ones. This energy is actually responsible, as we show, for the dynamical phase transition displayed by this model under a sudden large increase of the transverse field. A second situation we consider is the opposite, where the symmetry-breaking eigenstates are those in the high-energy sector of the spectrum, whereas the low-energy eigenstates are symmetric. In that case too a special energy must exist marking the boundary and leading to unexpected out-of-equilibrium dynamical behavior. An example is the fermonic repulsive Hubbard model Hamiltonian H\mathcal{H}H. Exploiting the trivial fact that the high-energy spectrum of H\mathcal{H}H is also the low-energy one of $\ensuremath{-}\mathcal{H}$, we conclude that the high-energy eigenstates of the Hubbard model are superfluid. Simulating in a time-dependent Gutzwiller approximation the time evolution of a high-energy BCS-like trial wave function, we show that a small superconducting order parameter will actually grow in spite of the repulsive nature of the interaction.
Girardeau has shown that a particular class of variational approximations for the free energy of a Fermi-Dirac system may be expressed in terms of the exact free energy of a certain soluble "model Hamiltonian." This exact formula for the partition function associated with the model system involves the solution of a certain nonlinear integral equation, and in the present work it is shown in detail that there is a correspondence between phase transitions in the system and the existence of multiple solutions of this equation. The connection between the existence of phase transitions and the properties of the interaction is studied, and it is shown that these phase transitions may occur for a very wide class of interactions. It is established that there is an upper bound to the temperature, above which no phase transitions will occur at any density. The extension of these results to a classical and a Bose-Einstein system is indicated, and the lattice-gas model previously studied by Mermin is re-examined in the light of the present results.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].