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arXiv:1710.01171 · Physical review. B./Physical review. B

Generalized Schrieffer-Wolff transformation of multiflavor Hubbard models

Seung‐Sup B. Lee, Jan von Delft, Andreas Weichselbaum

Published Dec 5, 2017• 14 citations
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Abstract

We give a self-contained derivation of the low-energy effective interactions of the $\mathrm{SU}(N)$ Hubbard model, a multiflavor generalization of the one-band Hubbard model, by using a generalized Schrieffer-Wolff transformation (SWT). The effective interaction of doublons and holons, which has been largely ignored in previous SWT studies (e.g., the $t\text{\ensuremath{-}}J$ model), leads to distinct peaks in the local density of states. As shown by Lee et al. [Phys. Rev. Lett. 119, 236402 (2017)], this underlying effective doublon-holon interaction explains the numerical observation of the subpeaks at the inner edges of the Hubbard bands in the metallic phase close to the Mott transition.

DOI
10.1103/physrevb.96.245106

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We give a self-contained derivation of the low-energy effective interactions of the SU(N)\mathrm{SU}(N)SU(N) Hubbard model, a multiflavor generalization of the one-band Hubbard model, by using a generalized Schrieffer-Wolff transformation (SWT). The effective interaction of doublons and holons, which has been largely ignored in previous SWT studies (e.g., the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model), leads to distinct peaks in the local density of states. As shown by Lee et al. [Phys. Rev. Lett. 119, 236402 (2017)], this underlying effective doublon-holon interaction explains the numerical observation of the subpeaks at the inner edges of the Hubbard bands in the metallic phase close to the Mott transition.
Dynamical mean-field theory (DMFT) studies frequently observe a fine structure in the local spectral function of the SU(2) Fermi-Hubbard model at half filling: In the metallic phase close to the Mott transition, subpeaks emerge at the inner edges of the Hubbard bands. Here we demonstrate that these subpeaks originate from the low-energy effective interaction of doublon-holon pairs, by investigating how the correlation functions of doublon and holon operators contribute to the subpeaks. A mean-field analysis of the low-energy effective Hamiltonian provides results consistent with our DMFT calculation using the numerical renormalization group as an impurity solver. In the SU(3) and SU(4) Hubbard models, the subpeaks become more pronounced due to the increased degeneracy of doublon-holon pair excitations.
Schrieffer-Wolff transformation (SWT) has been extensively used in quantum many-body physics to calculate the low energy effective Hamiltonian. It provides a perturbative method to comprehend the renormalization effects of strong correlations in the quantum many-body models. The generator for Schrieffer-Wolff transformation is calculated usually by heuristic methods. Recently, a systematic and elegant method for the calculation of this extremely significant transformation has been reported [1]. Given the huge significance of SWT for many areas including quantum condensed matter physics, quantum optics and quantum cavity electrodynamics, it is imperative to develop quantum algorithm for carrying out SWT on quantum computer. In this paper, we put forward this quantum algorithm and demonstrate it for single impurity Anderson model (SIAM), thereby arriving at Kondo model as effective Hamiltonian. We implement our quantum algorithm in QisKit and carry out SWT for SIAM on IBM Quantum computers. To the best of our knowledge, this work is the first of its kind to obtain Kondo model from Anderson impurity model using a quantum algorithm.
We use the Schrieffer-Wolff transformation (SWT) to analyze Josephson junctions between superconducting leads described by the charge-conserving BCS theory. Starting from the single-electron tunneling terms, we directly recover the conventional effective Hamiltonian, −EJcos⁡φ^-E_J\cos\hatφ−EJ​cosφ^​, with an operator-valued phase bias φ^\hatφφ^​. The SWT approach has the advantage that it can be systematically extended to more complex scenarios. We show that if a Bogoliubov quasiparticle is present its motion couples to that of Cooper pairs, introducing correlated dynamics that reshape the energy spectrum of the junction. Furthermore, higher-order terms in the SWT naturally describe Josephson harmonics, whose amplitudes are directly related to the microscopic properties of the superconducting leads and the junction. We derive expressions that could facilitate tuning the ratio between the different harmonics in a controlled way.
We generalize the Schrieffer-Wolff transformation to periodically driven systems using Floquet theory. The method is applied to the periodically driven, strongly interacting Fermi-Hubbard model, for which we identify two regimes resulting in different effective low-energy Hamiltonians. In the nonresonant regime, we realize an interacting spin model coupled to a static gauge field with a nonzero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, we derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the amplitude of the periodic drive.
In a partially filled flat Bloch band electrons do not have a well defined Fermi surface and hence the low-energy theory is not a Fermi liquid. Nevertheless, under the influence of an attractive interaction, a superconductor well described by the Bardeen-Cooper-Schrieffer (BCS) wave function can arise. Here we study the low-energy effective Hamiltonian of a generic Hubbard model with a flat band. We obtain an effective Hamiltonian for the flat band physics by eliminating higher-lying bands via the perturbative Schrieffer-Wolff transformation. At first order in the interaction energy we recover the usual procedure of projecting the interaction term onto the flat band Wannier functions. We show that the BCS wave function is the exact ground state of the projected interaction Hamiltonian, if a simple uniform pairing condition on the single-particle states is satisfied, and that the compressibility is diverging as a consequence of an emergent SU(2)\text{SU}(2)SU(2) symmetry. This symmetry is broken by second-order interband transitions resulting in a finite compressibility, which we illustrate for a one-dimensional ladder with two perfectly flat bands. These results motivate a further approximation leading to an effective ferromagnetic Heisenberg model. The gauge-invariant result for the superfluid weight of a flat band can be obtained from the ferromagnetic Heisenberg model only if the maximally localized Wannier functions in the Marzari-Vanderbilt sense are used. Finally, we prove an important inequality $D\ensuremath{\ge}{\mathcal{W}}^{2}$ between the Drude weight DDD and the winding number W\mathcal{W}W, which guarantees ballistic transport for topologically nontrivial flat bands in one dimension.
Recent experiments revealed the importance of higher-band effects for the Mott-insulator(MI)--superfluid transition(SF) of ultracold bosonic atoms or mixtures of bosons and fermions in deep optical lattices [Best et al., Phys. Rev. Lett. 102, 030408 (2009); Will et al., Nature (London) 465, 197 (2010)]. In the present work we derive an effective lowest-band Hamiltonian in three dimensions that generalizes the standard Bose-Fermi-Hubbard model taking these effects as well as nonlinear corrections of the tunneling amplitudes mediated by interspecies interactions into account. It is shown that a correct description of the lattice states in terms of the bare-lattice Wannier functions, rather than approximations such as harmonic-oscillator states, is essential. In contrast to self-consistent approaches based on effective Wannier functions, our approach captures the observed reduction of the superfluid phase for repulsive interspecies interactions.