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arXiv:1111.6148 · Physical Review B

Correlated singlet phase in the one-dimensional Hubbard-Holstein model

Sahinur Reja, Sudhakar Yarlagadda, P. B. Littlewood

Published Jul 12, 2012• 16 citations
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Abstract

We show that a nearest-neighbor singlet phase results (from an effective Hamiltonian) for the one-dimensional Hubbard-Holstein model in the regime of strong electron-electron and electron-phonon interactions and under nonadiabatic conditions ($t/{\ensuremath{\omega}}_{0}\ensuremath{\leqslant}1$). By mapping the system of nearest-neighbor singlets at a filling ${N}_{p}/N$ onto a hard-core-boson (HCB) $t$-$V$ model at a filling ${N}_{p}/(N\ensuremath{-}{N}_{p})$, we demonstrate explicitly that superfluidity and charge density wave (CDW) occur mutually exclusively with the diagonal long range order manifesting itself only at one-third filling. Furthermore, we also show that the Bose-Einstein condensate (BEC) occupation number ${n}_{0}$ for the singlet phase, similar to the ${n}_{0}$ for a HCB tight binding model, scales as $\sqrt{N}$; however, the coefficient of $\sqrt{N}$ in the ${n}_{0}$ for the interacting singlet phase is numerically demonstrated to be smaller.

DOI
10.1103/physrevb.86.045110

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We show that a nearest-neighbor singlet phase results (from an effective Hamiltonian) for the one-dimensional Hubbard-Holstein model in the regime of strong electron-electron and electron-phonon interactions and under nonadiabatic conditions (t/\ensuremathω0\ensuremath⩽1t/{\ensuremath{\omega}}_{0}\ensuremath{\leqslant}1t/\ensuremathω0​\ensuremath⩽1). By mapping the system of nearest-neighbor singlets at a filling Np/N{N}_{p}/NNp​/N onto a hard-core-boson (HCB) ttt-VVV model at a filling Np/(N\ensuremath−Np){N}_{p}/(N\ensuremath{-}{N}_{p})Np​/(N\ensuremath−Np​), we demonstrate explicitly that superfluidity and charge density wave (CDW) occur mutually exclusively with the diagonal long range order manifesting itself only at one-third filling. Furthermore, we also show that the Bose-Einstein condensate (BEC) occupation number n0{n}_{0}n0​ for the singlet phase, similar to the n0{n}_{0}n0​ for a HCB tight binding model, scales as N\sqrt{N}N​; however, the coefficient of N\sqrt{N}N​ in the n0{n}_{0}n0​ for the interacting singlet phase is numerically demonstrated to be smaller.
We derive an effective electronic Hamiltonian for the square lattice Hubbard-Holstein model (HHM) in the strong electron-electron $(e\text{\ensuremath{-}}e)$ and electron-phonon $(e\text{\ensuremath{-}}ph)$ coupling regime and under nonadiabatic conditions $(t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1)$, ttt and ${\ensuremath{\omega}}_{0}$ being the electron hopping and phonon frequency respectively. Using the density matrix renormalization-group method, we simulate this effective electronic model on a four-leg cylinder system at quarter filling and present a phase diagram in the $g\text{\ensuremath{-}}U$ plane where ggg and UUU are the $e\text{\ensuremath{-}}ph$ coupling constant and Hubbard on-site interaction respectively. For larger ggg, we find that a cluster of spins, i.e., phase separation (PS), gives way to a charge density wave (CDW) phase made of nearest-neighbor singlets which abruptly goes to another CDW phase as we increase UUU. But for smaller ggg, we find a metallic phase sandwiched between PS and the singlet CDW phase. This phase is characterized by a vanishing charge gap but a finite spin gap, suggesting a singlet superconducting phase.
We derive an effective Hamiltonian for the one-dimensional Hubbard-Holstein model, valid in a regime of both strong electron-electron (e-e) and electron-phonon (e-ph) interactions and in the nonadiabatic limit ($t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1$), by using a nonperturbative approach. We obtain the phase diagram at quarter filling by employing a modified Lanczos method and studying various density-density correlations. The spin-spin AF (antiferromagnetic) interactions and nearest-neighbor repulsion, resulting from the e-e and the e-ph interactions, respectively, are the dominant terms (compared to hopping) and compete to determine the various correlated phases. As e-e interaction (U/t)(U/t)(U/t) is increased, the system transits from an AF cluster to a correlated singlet phase through a discontinuous transition at all strong e-ph couplings $2\ensuremath{\le}g\ensuremath{\le}3$ considered. At higher values of U/tU/tU/t and moderately strong e-ph interactions ($2\ensuremath{\le}g\ensuremath{\le}2.6$), the singlets break up to form an AF order and then to a paramagnetic order all in a single sublattice; whereas at larger values of ggg (>>>2.62.62.6), the system jumps directly to the spin disordered charge-density-wave (CDW) phase.
We derive an effective d-dimensional Hamiltonian for a system of hard-core-bosons coupled to optical phonons in a lattice. At non-half-fillings, a superfluid-supersolid transition occurs at intermediate boson-phonon couplings, while at strong-couplings the system phase separates. We demonstrate explicitly that the presence of next-nearest-neighbor hopping and nearest-neighbor repulsion leads to supersolidity. Thus we present a microscopic mechanism for the homogeneous coexistence of charge-density-wave and superfluid orders.
Exploring supersolidity in naturally occurring and artificially designed systems has been and will continue to be an area of immense interest. Here, we study how superfluid and charge-density-wave (CDW) states cooperate or compete in a minimal model for hard-core-bosons (HCBs) coupled locally to optical phonons: a two-dimensional Bose-Holstein model. Our study is restricted to the parameter regimes of strong HCB-phonon coupling and non-adiabaticity. We use Quantum Monte Carlo simulation (involving stochastic-series-expansion technique) to study phase transitions and to investigate whether we have homogeneous or phase-separated coexistence. The effective Hamiltonian involves, besides a nearest-neighbor hopping and a nearest-neighbor repulsion, sizeable double-hopping terms (obtained from second-order perturbation). At densities not far from half-filling, in the parameter regime where the double-hopping terms are non-negligible (negligible) compared to the nearest-neighbor hopping, we get checkerboard-supersolidity (phase separation) with CDW being characterized by ordering wavevector Q⃗=(π,π)\vec{Q}=(π,π)Q​=(π,π).
We derive an effective d-dimensional Hamiltonian for a system of hard-core-bosons coupled to optical phonons in a lattice. Away from half-filling, we show that the presence of next-nearest-neighbor hopping in the effective Hamiltonian leads to a superfluid-to-supersolid transition at intermediate boson-phonon (b-p) couplings, while at strong-couplings the system phase separates. However, at half-filling and at a critical b-p coupling (as in the xxz-model), the system undergoes a superfluid-to-charge-density-wave transition without any signature of supersolidity. Our analyses is based on extensive calculations of the structure factor, the superfluid fraction, the Bose-Einstein condensate fraction, and the system energy at various fillings. We present a phase diagram for this system and compare it to that of the xxz-model. We also demonstrate explicitly that the next-nearest-neighbor hopping (in the absence of nearest-neighbor hopping) in the effective Hamiltonian leads only to a single transition, i.e., a first-order superfluid-to-supersolid transition.
Phase diagram of the two-dimensional Hubbard-Holstein model: enhancement of sss-wave pairing between charge and magnetic orders
We investigate the role of electron-electron and electron-phonon interactions in strongly correlated systems by performing unbiased quantum Monte Carlo simulations in the square lattice Hubbard-Holstein model at half-filling. We study the competition and interplay between antiferromagnetism (AFM) and charge-density wave (CDW), establishing its very rich phase diagram. In the region between AFM and CDW phases, we have found an enhancement of superconducting pairing correlations, favoring (nonlocal) sss-wave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.