The Holstein model on a square lattice at half-filling has a well-established finite temperature phase transition to an insulating state with long range charge density wave (CDW) order. Because this CDW formation suppresses pairing, a superconducting (SC) phase emerges only with doping. In this work, we study the effects of dilution of the local phonon degrees of freedom in the Holstein model while keeping the system at half filling. We find not only that the CDW remains present up to a dilution fraction $f\phantom{\rule{0.16em}{0ex}}\ensuremath{\sim}\phantom{\rule{0.16em}{0ex}}0.15$, but also that long range pairing is stabilized with increasing $f$, resulting in a supersolid regime centered at $f\ensuremath{\approx}0.10$, where long range diagonal and off-diagonal correlations coexist. Further dilution results in a purely SC phase, and ultimately in a normal metal. Our results provide a new route to the supersolid phase via the introduction of impurities at fixed positions which both increase quantum fluctuations and also are immune to the competing tendency to phase separation often observed in the doped case.
The Holstein model on a square lattice at half-filling has a well-established finite temperature phase transition to an insulating state with long range charge density wave (CDW) order. Because this CDW formation suppresses pairing, a superconducting (SC) phase emerges only with doping. In this work, we study the effects of dilution of the local phonon degrees of freedom in the Holstein model while keeping the system at half filling. We find not only that the CDW remains present up to a dilution fraction f\ensuremath∼0.15, but also that long range pairing is stabilized with increasing f, resulting in a supersolid regime centered at f\ensuremath≈0.10, where long range diagonal and off-diagonal correlations coexist. Further dilution results in a purely SC phase, and ultimately in a normal metal. Our results provide a new route to the supersolid phase via the introduction of impurities at fixed positions which both increase quantum fluctuations and also are immune to the competing tendency to phase separation often observed in the doped case.
The Holstein model describes the interaction between fermions and a collection of local (dispersionless) phonon modes. In the dilute limit, the phonon degrees of freedom dress the fermions, giving rise to polaron and bipolaron formation. At higher densities, the phonons mediate collective superconducting (SC) and charge-density wave (CDW) phases. Quantum Monte Carlo (QMC) simulations have considered both these limits but have not yet focused on the physics of more general phonon spectra. Here we report QMC studies of the role of phonon dispersion on SC and CDW order in such models. We quantify the effect of finite phonon bandwidth and curvature on the critical temperature T_{cdw} for CDW order and also uncover several novel features of diagonal long-range order in the phase diagram, including a competition between charge patterns at momenta q=(π,π) and q=(0,π) which lends insight into the relationship between Fermi surface nesting and the wave vector at which charge order occurs. We also demonstrate SC order at half filling in situations where a nonzero bandwidth sufficiently suppresses T_{cdw}.
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low-density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low-temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second-order (Ising) transition and therefore at a finite Tcdw in two dimensions. Quantum Monte Carlo calculations have determined Tcdw for a variety of geometries, including square, honeycomb, and Lieb lattices. The superconducting transition, on the other hand, in d=2 is in the Kosterlitz-Thouless universality class and is much less well characterized. In this paper we determine Tsc for the square lattice for several values of the density $\ensuremath{\rho}$ and phonon frequency ${\ensuremath{\omega}}_{0}$. We find that quasilong-range order sets in at ${T}_{\mathrm{sc}}\ensuremath{\lesssim}t/20$, where t is the near-neighbor hopping amplitude, consistent with previous rough estimates from simulations which extrapolated to only the temperatures we reach from considerably higher T. We also show evidence of a discontinuous evolution of the density as the CDW transition is approached at half filling.
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 Q=(π,π).
Phase diagram of the two-dimensional Hubbard-Holstein model: enhancement of s-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) s-wave pairs. Our study sheds light over past inconsistencies in the literature, in particular the emergence of CDW in the pure Holstein model case.
By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping t, the on-site electron-electron interaction U, the phonon energy ${\ensuremath{\omega}}_{0}$, and the electron-phonon coupling g. At half filling, the ground state is an antiferromagnetic insulator for $U\ensuremath{\gtrsim}2{g}^{2}/{\ensuremath{\omega}}_{0}$, while it is a charge-density-wave (or bipolaronic) insulator for $U\ensuremath{\lesssim}2{g}^{2}/{\ensuremath{\omega}}_{0}$. In addition to these phases, we find a superconducting phase that intrudes between them. For ${\ensuremath{\omega}}_{0}/t=1$, superconductivity emerges when both U/t and $2{g}^{2}/t{\ensuremath{\omega}}_{0}$ are small; then, by increasing the value of the phonon energy ${\ensuremath{\omega}}_{0}$, it extends along the transition line between antiferromagnetic and charge-density-wave insulators. Away from half filling, phase separation occurs when doping the charge-density-wave insulator, while a uniform (superconducting) ground state is found when doping the superconducting phase. In the analysis of finite-size effects, it is extremely important to average over twisted boundary conditions, especially in the weak-coupling limit and in the doped case.
Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large-U and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction V or a JP term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with V or JP coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and d-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and V or JP, we establish two quantum phase diagrams at 121 doping ratio. In the SC phases, the increased V or JP coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased V or JP. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although V and JP couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.