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Register a new userRelaxation dynamics of an exactly solvable electron-phonon model
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We address the question whether observables of an exactly solvable model of electrons coupled to (optical) phonons relax into large time stationary state values and investigate if the asymptotic expectation values can be computed using a stationary density matrix. Two initial nonequilibrium situations are considered. A sudden quench of the electron-phonon coupling, starting from the noninteracting canonical equilibrium at temperature T in the electron as well as in the phonon subsystems, leads to a rather simple dynamics. A richer time evolution emerges if the initial state is taken as the product of the phonon vacuum and the filled Fermi sea supplemented by a highly excited additional electron. Our model has a natural set of constants of motion, with as many elements as degrees of freedom. In accordance with earlier studies of such type of models we find that expectation values which become stationary can be described by the density matrix of a generalized Gibbs ensemble which differs from that of a canonical ensemble. For the model at hand it appears to be evident that the eigenmode occupancy operators should be used in the construction of the stationary density matrix.
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We have proposed an exactly solvable quantum spin-3/2 model on a square lattice. Its ground state is a quantum spin liquid with a half integer spin per unit cell. The fermionic excitations are gapless with a linear dispersion, while the topological vison excitations are gapped. Moreover, the massless Dirac fermions are stable. Thus, this model is, to the best of our knowledge, the first exactly solvable model of half-integer spins whose ground state is an algebraic spin liquid.
We introduce in this paper an exact solvable BCS-Hubbard model in arbitrary dimensions. The model describes a p-wave BCS superconductor with equal spin pairing moving on a bipartite (cubic, square etc.) lattice with on site Hubbard interaction $U$. We show that the model becomes exactly solvable for arbitrary $U$ when the BCS pairing amplitude $Delta$ equals the hopping amplitude $t$. The nature of the solution is described in detail in this paper. The construction of the exact solution is parallel to the exactly solvable Kitaev honeycomb model for $S=1/2$ quantum spins and can be viewed as a generalization of Kitaevs construction to $S=1/2$ interacting lattice fermions. The BCS-Hubbard model discussed in this paper is just an example of a large class of exactly solvable lattice fermion models that can be constructed similarly.
We present an exactly solvable model for synthetic anyons carrying non-Abelian flux. The model corresponds to a two-dimensional electron gas in a magnetic field with a specific spin interaction term, which allows only fully aligned spin states in the ground state; the ground state subspace is thus two-fold degenerate. This system is perturbed with identical solenoids carrying a non-Abelian gauge potential. We explore dynamics of the ground state as these solenoids are adiabatically braided and show they behave as anyons with a non-Abelian flux. Such a system represents a middle ground between the ordinary Abelian anyons and the fully non-Abelian anyons.
We study entanglement in the Hatsugai-Kohmoto model, which exhibits a continuous interaction-driven Mott transition. By virtue of the all-to-all nature of its center-of-mass conserving interactions, the model lacks dynamical spectral weight transfer, which is the key to intractability of the Hubbard model for $d>1$. In order to maintain a non-trivial Mott-like electron propagator, SU(2) symmetry is preserved in the Hamiltonian, leading to a ground state that is mixed on both sides of the phase transition. Because of this mixture, even the metal in this model is unentangled between any pair of sites, unlike free fermions whose ground state carries a filling-dependent site-site entanglement. We focus on the scaling behavior of the one- and two-site entropies $s_1$ and $s_2$, as well as the entropy density $s$, of the ground state near the Mott transition. At low temperatures in the two-dimensional Hubbard model, it was observed numerically (Walsh et al., 2018, arXiv:1807.10409) that $s_1$ and $s$ increase continuously into the metal, across a first-order Mott transition. In the Hatsugai-Kohmoto model, $s_1$ acquires the constant value $ln4$ even at the Mott transition. The ground states non-trivial entanglement structure is manifest in $s_2$ and $s$ which decrease into the metal, and thereby act as sharp signals of the Mott transition in any dimension. Specifically, we find that in one dimension, $s_2$ and $s$ exhibit kinks at the transition while in $d=2$, only $s$ exhibits a kink.
In an attempt to regularize a previously known exactly solvable model [Yang and Zhang, Eur. J. Phys. textbf{40}, 035401 (2019)], we find yet another exactly solvable toy model. The interesting point is that while the Hamiltonian of the model is parameterized by a function $f(x)$ defined on $[0, infty )$, its spectrum depends only on the end values of $f$, i.e., $f(0)$ and $f(infty )$. This model can serve as a good exercise in quantum mechanics at the undergraduate level.
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