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We introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and report benchmark results for the Holstein model on the square lattice. The determinant quantum Monte Carlo (DQMC) method is a widely used tool for simulating simple electron-phonon models at finite temperatures, but incurs a computational cost that scales cubically with system size. Alternatively, near-linear scaling with system size can be achieved with the hybrid Monte Carlo (HMC) method and an integral representation of the Fermion determinant. Here, we introduce a collection of methodologies that make such simulations even faster. To combat "stiffness" arising from the bosonic action, we review how Fourier acceleration can be combined with time-step splitting. To overcome phonon sampling barriers associated with strongly-bound bipolaron formation, we design global Monte Carlo updates that approximately respect particle-hole symmetry. To accelerate the iterative linear solver, we introduce a preconditioner that becomes exact in the adiabatic limit of infinite atomic mass. Finally, we demonstrate how stochastic measurements can be accelerated using fast Fourier transforms. These methods are all complementary and, combined, may produce multiple orders of magnitude speedup, depending on model details.

The possibility of "orbitally selective Mott transitions" within a multiband Hubbard model, in which one orbital with large on-site electron-electron repulsion $U_1$ is insulating and another orbital, to which it is hybridized, with small $U_{-1}$, is metallic, is a problem of long-standing debate and investigation. In this paper we study an analogous phenomenon, the co-existence of metallic and insulating bands in a system of orbitals with different electron-phonon coupling (EPC). To this end, we examine two variants of the bilayer Holstein model: a uniform bilayer and a "Holstein-Metal interface" where the electron-phonon coupling, $\lambda$, is zero in the "metallic" layer. In the uniform bilayer Holstein model, charge density wave (CDW) order dominates at small interlayer hybridization $t_3$, but decreases and eventually vanishes as $t_3$ grows, providing a charge analog of singlet (spin liquid) physics. In the interface case, we show that CDW order penetrates into the metal layer and forms long-range CDW order at intermediate ratio of inter- to intra-layer hopping strengths, $1.4 \lesssim t_3/t \lesssim 3.4$. This is consistent with the occurrence of an "orbitally selective CDW" regime at weak $t_3$ in which the layer with $\lambda_{1} \neq 0$ exhibits long-range charge order, but the "metallic layer" with $\lambda_{-1}=0$, to which it is hybridized, does not., Comment: 10 pages, 11 figures

The Hubbard and Su-Schrieffer-Heeger Hamiltonians (SSH) are iconic models for understanding the qualitative effects of electron-electron and electron-phonon interactions respectively. In the two-dimensional square lattice Hubbard model at half filling, the on-site Coulomb repulsion, $U$, between up and down electrons induces antiferromagnetic (AF) order and a Mott insulating phase. On the other hand, for the SSH model, there is an AF phase when the electron-phonon coupling $\lambda$ is less than a critical value $\lambda_c$ and a bond order wave when $\lambda > \lambda_c$. In this work, we perform numerical studies on the square lattice optical Su-Schrieffer-Heeger-Hubbard Hamiltonian (SSHH), which combines both interactions. We use the determinant quantum Monte Carlo (DQMC) method which does not suffer from the fermionic sign problem at half filling. We map out the phase diagram and find that it exhibits a direct first-order transition between an antiferromagnetic phase and a bond-ordered wave as $\lambda$ increases. The AF phase is characterized by two different regions. At smaller $\lambda$ the behavior is similar to that of the pure Hubbard model; the other region, while maintaining long range AF order, exhibits larger kinetic energies and double occupancy, i.e. larger quantum fluctuations, similar to the AF phase found in the pure SSH model.

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 $T_{\rm cdw}$ in two dimensions. Quantum Monte Carlo calculations have determined $T_{\rm cdw}$ 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 (KT) universality class, and is much less well characterized. In this paper we determine $T_{\rm sc}$ for the square lattice, for several values of the density $\rho$ and phonon frequency $\omega_0$. We find that quasi-long range order sets in at $T_{\rm sc} \lesssim t/20$, where $t$ is the near neighbor hopping amplitude, consistent with previous rough estimates from simulations which only extrapolated to the temperatures we reach from considerably higher $T$. We also show evidence for a discontinuous evolution of the density as the CDW transition is approached at half-filling., Comment: 9 pages and 7 figures

Over the last several years, a new generation of quantum simulations has greatly expanded our understanding of charge density wave phase transitions in Hamiltonians with coupling between local phonon modes and the on-site charge density. A quite different, and interesting, case is one in which the phonons live on the bonds, and hence modulate the electron hopping. This situation, described by the Su-Schrieffer-Heeger (SSH) Hamiltonian, has so far only been studied with quantum Monte Carlo in one dimension. Here we present results for the 2D SSH model, and show that a bond ordered wave (BOW) insulator is present in the ground state at half-filling, and argue that a critical value of the electron-phonon coupling is required for its onset, in contradistinction with the 1D case where BOW exists for any nonzero coupling. We determine the precise nature of the bond ordering pattern, which has hitherto been controversial, and the critical transition temperature, which is associated with a spontaneous breaking of ${\cal Z}_4$ symmetry., Comment: 5 pages, 7 figures

The interplay between electron-electron correlations and disorder has been a central theme of condensed matter physics over the last several decades, with particular interest in the possibility that interactions might cause delocalization of an Anderson insulator into a metallic state, and the disrupting effects of randomness on magnetic order and the Mott phase. Here we extend this physics to explore electron-phonon interactions and show, via exact quantum Monte Carlo simulations, that the suppression of the charge density wave correlations in the half-filled Holstein model by disorder can stabilize a superconducting phase. Our simulations thus capture qualitatively the suppression of charge ordered phases and emergent superconductivity recently seen experimentally.

Self-learning Monte Carlo method (SLMC), using a trained effective model to guide Monte Carlo sampling processes, is a powerful general-purpose numerical method recently introduced to speed up simulations in (quantum) many-body systems. In this work, we further improve the efficiency of SLMC by enforcing physical symmetries on the effective model. We demonstrate its effectiveness in the Holstein Hamiltonian, one of the most fundamental many-body descriptions of electron-phonon coupling. Simulations of the Holstein model are notoriously difficult due to the combination of the typical cubic scaling of fermionic Monte Carlo and the presence of extremely long autocorrelation times. Our method addresses both bottlenecks. This enables simulations on large lattices in the most difficult parameter regions, and evaluation of the critical point for the charge density wave transition at half-filling with high precision. We argue that our work opens a new research area of quantum Monte Carlo (QMC), providing a general procedure to deal with ergodicity in situations involving Hamiltonians with multiple, distinct low energy states., Comment: 4 pages, 3 figures with 2 pages supplemental material

The one-dimensional extended bosonic Hubbard model has been shown to exhibit a variety of phases ranging from Mott insulator and superfluid to exotic supersolids and Haldane insulators depending on the filling and the relative value of the contact ($U$) and near neighbor ($V$) interaction strengths. In this paper we use the density matrix renormalization group and the time evolving block decimation numerical methods to study in detail the dynamics and excitation spectra of this model in its various phases. In particular, we study in detail the behavior of the charge and neutral gaps which characterize the Mott, charge density and Haldane insulating phases. We also show that in addition to the gapless modes at $k=0$, the supersolid phase exhibits gapless modes at a finite $k$ which depends on the filling., Comment: 11 pages, 17 figures

We consider soft-core bosons with onsite interaction loaded in the honeycomb lattice with different site energies for the two sublattices. Using both a mean-field approach and quantum Monte-Carlo simulations, we show that the topology of the honeycomb lattice results in a non-vanishing Berry curvature for the band structure of the single-particle excitations of the system. This Berry curvature induces an anomalous Hall effect. It is seen by studying the time evolution of a wavepacket, namely a superfluid ground state in a harmonic trap, subjected either to a constant force (Bloch oscillations) or to a sudden shift of the trap center., Comment: 10 pages, 11 figures

Advances in pure optical trapping techniques now allow the creation of degenerate Bose gases with internal degrees of freedom. Systems such as ${}^{87}$Rb, $^{39}$K or ${}^{23}$Na in the $F=1$ hyperfine state offer an ideal platform for studying the interplay of superfluidity and quantum magnetism. Motivated by the experimental developments, we study ground state phases of a two-component Bose gas loaded on an optical lattice. The system is described effectively by the Bose-Hubbard Hamiltonian with onsite and near neighbor spin-spin interactions. An important feature of our investigation is the inclusion of interconversion (spin flip) terms between the two species, which has been observed in optical lattice experiments. Using mean-field theory and quantum Monte Carlo simulations, we map out the phase diagram of the system. A rich variety of phases is identified, including antiferromagnetic (AF) Mott insulators, ferromagnetic and AF superfluids., Comment: 11 pages, 10 figures, revised version

Determinant Quantum Monte Carlo (DQMC) is used to study the effect of non-zero hopping t_f in the localized f-band of the periodic Anderson model (PAM) in two dimensions. The low temperature properties are determined in the plane of interband hybridization V and t_f at fixed U_f and half-filling, including the case when the sign of t_f is opposite to that of the conduction band t_d. For t_f and t_d of the same sign, and when t_f=t_d > (V =4_td)^2, the non-interacting system is metallic. We show that a remnant of the band insulator to metal line at U_f = 0 persists in the interacting system, manifesting itself as a maximal tendency toward antiferromagnetic correlations at low temperature. In this optimal t_f region, short range (e.g. near-neighbor) and long-range spin correlations develop at similar temperatures and have comparable magnitude. Both observations are in stark contrast with the situation in the widely studied PAM (t_f = 0) and single band Hubbard model, where short range correlations are stronger and develop at higher temperature. The effect that finite t_f has on Kondo screening is investigated by considering the evolution of the local density of states for selected t_f as a function of V . We use mean field theory as a tool to discriminate those aspects of the physics that are genuinely many-body in character., Comment: 8 pages, 7 figures

The finite temperature phase diagram for the 2D attractive fermion Hubbard model with spin-dependent disorder is considered within Bogoliubov-de Gennes mean field theory. Three types of disorder are studied. In the first, only one species is coupled to a random site energy; in the second, the two species both move in random site energy landscapes which are of the same amplitude, but different realizations; and finally, in the third, the disorder is in the hopping rather than the site energy. For all three cases we find that, unlike the case of spin-symmetric randomness, where the energy gap and average order parameter do not vanish as the disorder strength increases, a critical disorder strength exists separating distinct phases. In fact, the energy gap and the average order parameter vanish at distinct transitions, $V_{c}^{\rm gap}$ and $V_{c}^{\rm op}$, allowing for a gapless superconducting (gSC) phase. The gSC phase becomes smaller with increasing temperature, until it vanishes at a temperature $T^{\ast}$., Comment: 9 pages, 7 figures

The effect of on-site electron-electron repulsion $U$ in a band insulator is explored for a bilayer Hubbard Hamiltonian with opposite sign hopping in the two sheets. The ground state phase diagram is determined at half-filling in the plane of $U$ and the interplanar hybridization $V$ through a computation of the antiferromagnetic (AF) structure factor, local moments, single particle and spin wave spectra, and spin correlations. Unlike the case of the ionic Hubbard model, no evidence is found for a metallic phase intervening between the Mott and band insulators. Instead, upon increase of $U$ at large $V$, the behavior of the local moments and of single-particle spectra give quantitative evidence of a crossover to a Mott insulator state preceding the onset of magnetic order. Our conclusions generalize those of single-site dynamical mean field theory, and show that including interlayer correlations results in an increase of the single particle gap with $U$., Comment: 5 pages, 5 figures

Studies of systems with two fermionic bands with repulsive interaction strength U have a long history, with the Periodic Anderson Model (PAM) being one of the most frequently considered Hamiltonians. In this paper, we use Quantum Monte Carlo to study analogous issues for attractive interactions. As in the Periodic Anderson Model, we focus on a case where one band is uncorrelated (U=0), and focus on the effect of hybridization V between the bands on the pairing correlations. A key difference with the PAM is that there is no sign problem, so that we are able to explore the physics of doped multi-band attractive systems at low temperatures whereas ground state properties of repulsive models can be determined only at half-filling. For small V, pairing in the U<0 layer induces pairing in the U=0 layer. At larger V the ground state of the coupled system loses its superconducting character. The Quantum Monte Carlo data are complemented by results obtained with the Bogoliubov-de Gennes approximation., Comment: 6 pages, 5 figures

We study the conductivity, density of states, and magnetic correlations of a two dimensional, two band fermion Hubbard model using determinant Quantum Monte Carlo (DQMC) simulations. We show that an orbitally selective Mott transition (OSMT) occurs in which the more weakly interacting band can be metallic despite complete localization of the strongly interacting band. The DQMC method allows us to test the validity of the use of a momentum independent self-energy which has been a central approximation in previous OSMT studies. In addition, we show that long range antiferromagnetic order (LRAFO) is established in the insulating phase, similar to the single band, square lattice Hubbard Hamiltonian. Because the critical interaction strengths for the onset of insulating behavior are much less than the bandwidth of the itinerant orbital, we suggest that the development of LRAFO plays a key role in the transitions., Comment: 4 pages, 5 figures

We use quantum Monte Carlo simulations to obtain zero-temperature state diagrams for strongly correlated lattice bosons in one and two dimensions under the influence of a harmonic confining potential. Since harmonic traps generate a coexistence of superfluid and Mott insulating domains, we use local quantities such as the quantum fluctuations of the density and a local compressibility to identify the phases present in the inhomogeneous density profiles. We emphasize the use of the "characteristic density" to produce a state diagram that is relevant to experimental optical lattice systems, regardless of the number of bosons or trap curvature and of the validity of the local-density approximation. We show that the critical value of U/t at which Mott insulating domains appear in the trap depends on the filling in the system, and it is in general greater than the value in the homogeneous system. Recent experimental results by Spielman et al. [Phys. Rev. Lett. 100, 120402 (2008)] are analyzed in the context of our two-dimensional state diagram, and shown to exhibit a value for the critical point in good agreement with simulations. We also study the effects of finite, but low (T

Using quantum Monte Carlo simulations, we study a mixture of bosons and fermions loaded on an optical lattice. With simple on-site repulsive interactions, this system can be driven into a solid phase. We dope this phase and, in analogy with pure bosonic systems, identify the conditions under which the bosons enter a supersolid phase, i.e., exhibiting at the same time charge density wave and superfluid order. We perform finite size scaling analysis to confirm the presence of a supersolid phase and discuss its properties, showing that it is a collective phase that also involve phase coherence of the fermions.

We discuss the link between uncorrelated noise and Hurst exponent for one and two-dimensional interfaces. We show that long range correlations cannot be observed using one-dimensional cuts through two-dimensional self-affine surfaces whose height distributions are characterized by a Hurst exponent lower than -1/2. In this domain, fractional and white noise are not distinguishable. A method analysing the correlations in two dimensions is necessary. For Hurst exponents larger than -1/2, a crossover regime leads to a systematic over estimate of the Hurst exponent., Comment: 3 pages RevTeX, 4 Postscript figures