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Annalen der Physik

Quantum Integrability of 1D Ionic Hubbard Model

Abolfath Hosseinzadeh, S. A. Jafari

Published Feb 16, 2020• 4 citations
Publisher

Abstract

Abstract The quantum integrability of the 1D ionic Hubbard model (IHM) is established using two independent numerical methods, namely i) energy level spacing statistics and ii) occupation profile of one‐particle density matrix (OPDM) eigen‐values. Both methods suggest that the 1D IHM is integrable. The calculations of energy level statistics reproduce the known results for the standard Hubbard model. Upon turning on the the ionic term, the energy level spacing distribution of this model continues to obey the Poissonian distribution. Occupation patterns as extracted from OPDM indicate that quasi‐particles are sharpened upon increasing the ionic potential. This is evidenced by a larger jump in the occupation number distribution.

DOI
10.1002/andp.201900601

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Abstract The quantum integrability of the 1D ionic Hubbard model (IHM) is established using two independent numerical methods, namely i) energy level spacing statistics and ii) occupation profile of one‐particle density matrix (OPDM) eigen‐values. Both methods suggest that the 1D IHM is integrable. The calculations of energy level statistics reproduce the known results for the standard Hubbard model. Upon turning on the the ionic term, the energy level spacing distribution of this model continues to obey the Poissonian distribution. Occupation patterns as extracted from OPDM indicate that quasi‐particles are sharpened upon increasing the ionic potential. This is evidenced by a larger jump in the occupation number distribution.
In this paper we analyze the spectral level statistics of the one-dimensional ionic Hubbard model, the Hubbard model with an alternating on-site potential. In particular, we focus on the statistics of the gap ratios between consecutive energy levels. This quantity is often used in order to signal whether a many-body system is integrable or chaotic. A chaotic system has typically the statistics of a Gaussian ensemble of random matrices while the spectral properties of the integrable system follow a Poisson statistics. We find that whereas the Hubbard model without alternating potential is known to be integrable and its spectral properties follow a Poissonian statistics, the presence of an alternating potential causes a drastic change in the spectral properties, which resemble the one of a Gaussian ensemble of random matrices. However, to uncover this behavior one has to separately consider the blocks of all symmetries of the ionic Hubbard model.
Strongly correlated electron systems are one of the most fascinating problems in current physics. The strong electron-electron interaction in these materials leads to the emergence of nontrivial elementary excitations (quasiparticles, QPs) above the ground state ranging from fractional spins in quasi-one-dimensional materials to magnetic monopole in the pyrochlore lattice. The condensation of these quasiparticles upon changing some external parameters may stabilize new exotic states of matter. Experimental measurements such as inelastic neutron scattering provide us with valuable information about the excitation spectrum of such systems which require microscopic models to be described. This thesis is devoted to a detailed analysis of the excitation spectrum and of the quantum phase transitions in the one-dimensional (1D) ionic Hubbard model (IHM). The IHM consists of a nearest-neighbor (n.n.) hopping, onsite Hubbard interaction, and an ionic (staggered) potential separating the odd and even sites energetically. The model exhibits two continuous phase transitions on increasing the Hubbard interaction identified by a low-energy effective field theory and confirmed by a rigorous density matrix renormalization group (DMRG) analysis after several attempts. The first transition occurs from band insulator (BI) phase to the 2-fold degenerate spontaneously dimerized insulator (SDI) phase. The transition is in the Ising universality class as is plausible from symmetry considerations. The SDI phase becomes unstable towards a quasi-long-range order Mott insulator (MI) phase at a second transition point resembling the Kosterlitz-Thouless (KT) transition in the frustrated Heisenberg chain. We employ continuous unitary transformations (CUT) to systematically map the IHM to effective Hamiltonians (almost) conserving the number of QPs in the system. Using an analysis in the BI regime where electrons and holes define QPs, the low-energy excitation spectrum of the model is quantitatively determined in the BI phase almost up to the first transition point. The transition from the BI to the SDI phase is signaled by the vanishing of an S=0 exciton mode at the total momentum K=\pi. The condensation of these excitons beyond the first transition point is described by a BCS-type-theory showing the stabilization of the SDI phase. The mean-field solution indicates no second phase transition to the quasi-long-range order MI phase. This is interpreted as the effect of strong quantum fluctuations in one dimension. We consider the IHM in the dimer limit where the uniform chain is separated into independent dimers. The different phases of the IHM are studied by increasing the interdimer hopping and reaching the uniform limit. This dimer limit satisfactorily produces the excitation spectrum of the BI phase confirming the vanishing of an S=0 exciton mode at the first transition point. It is found that the SDI-to-MI transition takes place by softening of a magnetic S=1 excitation, i.e., a triplon. We report rigorous results for the gapless triplon dispersion in the MI phase and discuss the binding effects in the 2-triplon sector.
We study the zero-temperature phase diagram of the half-filled one-dimensional ionic Hubbard model. This model is governed by the interplay of the on-site Coulomb repulsion and an alternating one-particle potential. Various many-body energy gaps, the charge-density-wave and bond-order parameters, the electric as well as the bond-order susceptibilities, and the density-density correlation function are calculated using the density-matrix renormalization group method. In order to obtain a comprehensive picture, we investigate systems with open as well as periodic boundary conditions and study the physical properties in different sectors of the phase diagram. A careful finite-size scaling analysis leads to results which give evidence in favor of a scenario with two quantum critical points and an intermediate spontaneously dimerized phase. Our results indicate that the phase transitions are continuous. Using a scaling ansatz we are able to read off critical exponents at the first critical point. In contrast to a bosonization approach, we do not find Ising critical exponents. We show that the low-energy physics of the strong-coupling phase can only partly be understood in terms of the strong-coupling behavior of the ordinary Hubbard model.
In our earlier work [M. Hafez et al., Phys. Lett. A 373, 4479 (2009)] we employed the flow equation method to obtain a classical effective model from a quantum mechanical parent Hamiltonian called, the ionic Hubbard model. The classical ionic Hubbard model (CIHM) obtained in this way contains solely Fermionic occupation numbers of two species corresponding to particles with $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ spin, respectively. In this paper, we employ the transfer matrix method to analytically solve the CIHM at finite temperature in one dimension. In the limit of zero temperature, we find two insulating phases at large and small Coulomb interaction strength, UUU, mediated with a gapless phase, resulting in two continuous metal-insulator transitions. Our results are further supported with Monte Carlo simulations.
The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction UUU, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization VVV and two types of intralayer staggered potentials $\mathrm{\ensuremath{\Delta}}$: one with the same (in-phase) and the other with a $\ensuremath{\pi}$-phase shift (antiphase) potential between layers. Our determinant quantum Monte Carlo (DQMC) simulations at lowest accessible temperatures demonstrate that the interaction-driven metallic phase between Mott and band insulators expands in the $\mathrm{\ensuremath{\Delta}}\text{\ensuremath{-}}V$ phase diagram of bilayer IHM only for in-phase ionic potentials; while antiphase potential always induces an insulator with charge density order. This implies possible further extension of the ionic Hubbard model from the bilayer case here to a realistic three-dimensional model.
We study an ionic Hubbard model in one dimension by using a quantum Monte Carlo method. The ionic Hubbard model we study is a strongly correlated model for electrons in alternating potentials with a lattice period of 2. By modifying the Hirsch-Fye algorithm, we incorporate the local variation of the potential in the weight of the partition function at each imaginary-time segment. For a given alternating potential of strength Δ = 1.0, as we change the local Coulomb repulsion, U , between spin-up and spin-down electrons, we calculate various physical quantities, including kinetic energies, correlation energies, and band energies. We try to find the signature of the phase transitions from the behavior of the kinetic energy and that of double occupancies at finite temperatures.