Determinant Quantum Monte Carlo Research Articles

Enhanced Pair-Density-Wave Vertices in a Bilayer Hubbard Model at Half Filling.

Motivated by the pair-density-wave (PDW) state found in the one-dimensional Kondo-Heisenberg chain, we report on a determinant quantum MonteCarlo study of pair fields for a two-dimensional half-filled Hubbard layer coupled to an itinerant, noninteracting layer with one electron per site. In a specific range of interlayer hopping, the pairing vertex associated with PDW order becomes more attractive than that for uniform d-wave pairing, although both remain subdominant to the leading antiferromagnetic correlations at half filling. Our result sheds light on where one potentially may find a PDW state in such a model.

Robust Fermi-Liquid Instabilities in Sign Problem-Free Models.

Determinant quantum MonteCarlo (DQMC) is a powerful numerical technique to study many-body fermionic systems. In recent years, several classes of sign-free (SF) models have been discovered, where the notorious sign problem can be circumvented. However, it is not clear what the inherent physical characteristics and limitations of SF models are. In particular, which zero-temperature quantum phases of matter are accessible within such models, and which are fundamentally inaccessible? Here, we show that a model belonging to any of the known SF classes within DQMC cannot have a stable Fermi-liquid ground state in spatial dimension d≥2, unless the antiunitary symmetry that prevents the sign problem is spontaneously broken (for which there are currently no known examples in SF models). For SF models belonging to one of the symmetry classes (where the absence of the sign problem follows from a combination of nonunitary symmetries of the fermionic action), any putative Fermi liquid fixed point generically includes an attractive Cooper-like interaction that destabilizes it. In the recently discovered lower-symmetry classes of SF models, the Fermi surface (FS) is generically unstable even at the level of the quadratic action. Our results suggest a fundamental link between Fermi liquids and the fermion sign problem. Interestingly, our results do not rule out a non-Fermi-liquid ground state with a FS in a sign-free model.

Tangent Space Approach for Thermal Tensor Network Simulations of the 2D Hubbard Model.

Accurate simulations of the two-dimensional (2D) Hubbard model constitute one of the most challenging problems in condensed matter and quantum physics. Here we develop a tangent space tensor renormalization group (tanTRG) approach for the calculations of the 2D Hubbard model at finite temperature. An optimal evolution of the density operator is achieved in tanTRG with a mild O(D^{3}) complexity, where the bond dimension D controls the accuracy. With the tanTRG approach we boost the low-temperature calculations of large-scale 2D Hubbard systems on up to a width-8 cylinder and 10×10 square lattice. For the half-filled Hubbard model, the obtained results are in excellent agreement with those of determinant quantum MonteCarlo (DQMC). Moreover, tanTRG can be used to explore the low-temperature, finite-doping regime inaccessible for DQMC. The calculated charge compressibility and Matsubara Green's function are found to reflect the strange metal and pseudogap behaviors, respectively. The superconductive pairing susceptibility is computed down to a low temperature of approximately 1/24 of the hopping energy, where we find d-wave pairing responses are most significant near the optimal doping. Equipped with the tangent-space technique, tanTRG constitutes a well-controlled, highly efficient and accurate tensor network method for strongly correlated 2D lattice models at finite temperature.

Phase diagram of the Su-Schrieffer-Heeger-Hubbard model on a square lattice

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.

Thermodynamics of correlated electrons in a magnetic field

The Hofstadter–Hubbard model captures the physics of strongly correlated electrons in an applied magnetic field, which is relevant to many recent experiments on Moiré materials. Few large-scale, numerically exact simulations exists for this model. In this work, we simulate the Hubbard–Hofstadter model using the determinant quantum Monte Carlo (DQMC) algorithm. We report the field and Hubbard interaction strength dependence of charge compressibility, fermion sign, local moment, magnetic structure factor, and specific heat. The gross structure of magnetic Bloch bands and band gaps determined by the non-interacting Hofstadter spectrum is preserved in the presence of U. Incompressible regions of the phase diagram have improved fermion sign. At half filling and intermediate and larger couplings, a strong orbital magnetic field delocalizes electrons and reduces the effect of Hubbard U on thermodynamic properties of the system.

Quantum phase transitions from competing short- and long-range interactions on a π -flux lattice

Quantum phase transitions from the cluster-charge interaction, which is composed of competing short- and long-range interactions, are investigated on a $\ensuremath{\pi}$-flux lattice by using the mean-field theory and determinant quantum Monte Carlo (DQMC) simulations. Both methods identify a plaquette-dimer phase, which develops from a finite interaction strength. While its signature in DQMC is relatively weak, an obvious antiferromagnetic transition is revealed in the spin structure factor instead. The corresponding critical interaction and exponents are readily obtained by finite-size scalings, with the plaquette-dimer structure factor that can also be well scaled. These results suggest a possible deconfined quantum critical point between the plaquette-dimer and antiferromagnetic phases driven by the cluster-charge interaction on a $\ensuremath{\pi}$-flux lattice.

Fast and scalable quantum Monte Carlo simulations of electron-phonon models.

We introduce methodologies for highly scalable quantum Monte Carlo simulations of electron-phonon models, and we 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 it 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.

Two-dimensional attractive Hubbard model and the BCS-BEC crossover

Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature ${T}_{c}$ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength $U$ from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ ($t$ is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing ${T}_{p}$ and degeneracy ${T}_{d}$ have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.

Nematic antiferromagnetism and deconfined criticality from the interplay between electron-phonon and electron-electron interactions

Systems with strong electron-phonon couplings typically exhibit various forms of charge order, while strong electron-electron interactions lead to magnetism. We use determinant quantum Monte Carlo (DQMC) calculations to solve a model on a square lattice with a caricature of these interactions. In the limit where electron-electron interactions dominate it has antiferromagnetic (AF) order, while where electron-phonon coupling dominates there is columnar valence-bond solid (VBS) order. We find a novel intervening phase that hosts coexisting nematic and antiferromagnetic orders. We have also found evidence of a Landau-forbidden continuous quantum phase transition with an emergent $O(4)$ symmetry between the VBS and the nematic antiferromagnetic phases.

Quantum many-body simulations of the two-dimensional Fermi-Hubbard model in ultracold optical lattices

Understanding quantum many-body states of correlated electrons is one main theme in modern condensed matter physics. Given that the Fermi-Hubbard model, the prototype of correlated electrons, has been recently realized in ultracold optical lattices, it is highly desirable to have controlled numerical methodology to provide precise finite-temperature results upon doping, to directly compare with experiments. Here, we demonstrate the exponential tensor renormalization group (XTRG) algorithm [Phys. Rev. X 8, 031082 (2018)], complemented with independent determinant quantum Monte Carlo (DQMC) offer a powerful combination of tools for this purpose. XTRG provides full and accurate access to the density matrix and thus various spin and charge correlations, down to unprecedented low temperature of few percents of the fermion tunneling energy scale. We observe excellent agreement with ultracold fermion measurements at both half-filling and finite-doping, including the sign-reversal behavior in spin correlations due to formation of magnetic polarons, and the attractive hole-doublon and repulsive hole-hole pairs that are responsible for the peculiar bunching and antibunching behavior of the antimoments.

Quantum Monte Carlo study of the Hubbard model with next-nearest-neighbor hopping t′: pairing and magnetism

Using the finite-temperature determinant quantum Monte Carlo (DQMC) algorithm, we study the pairing symmetries of the Hubbard Hamiltonian with next-nearest-neighbor (NNN) hopping t′ on square lattices. By varying the value of t′, we find that the d-wave pairing is suppressed by the onset of t′, while the p + ip-wave pairing tends to emerge for low electron density and t′ around −0.7. Together with the calculation of the anti-ferromagnetic and ferromagnetic spin correlation function, we explore the relationship between anti-ferromagnetic order and the d-wave pairing symmetry, and the relationship between ferromagnetic order and the p + ip-wave pairing symmetry. Our results may be useful for the exploration of the mechanism of the electron pairing symmetries, and for the realization of the exotic p + ip-wave superconductivity.

Antiferromagnetic transitions of Dirac fermions in three dimensions

We use determinant quantum Monte Carlo (DQMC) simulations to study the role of electron-electron interactions on three-dimensional (3D) Dirac fermions based on the $\pi$-flux model on a cubic lattice. We show that the Hubbard interaction drives the 3D Dirac semimetal to an antiferromagnetic (AF) insulator only above a finite critical interaction strength and the long-ranged AF order persists up to a finite temperature. We evaluate the critical interaction strength and temperatures using finite-size scaling of the spin structure factor. The critical behaviors are consistent with the (3+1)d Gross-Neveu universality class for the quantum critical point and 3D Heisenberg universality class for the thermal phase transitions. We further investigate correlation effects in birefringent Dirac fermion system. It is found that the critical interaction strength $U_c$ is decreased by reducing the velocity of the Dirac cone, quantifying the effect of velocity on the critical interaction strength in 3D Dirac fermion systems. Our findings unambiguously uncover correlation effects in 3D Dirac fermions, and may be observed using ultracold atoms in an optical lattice.

Fast and stable determinant quantum Monte Carlo

We assess numerical stabilization methods employed in fermion many-body quantum Monte Carlo simulations. In particular, we empirically compare various matrix decomposition and inversion schemes to gain control over numerical instabilities arising in the computation of equal-time and time-displaced Green's functions within the determinant quantum Monte Carlo (DQMC) framework. Based on this comparison, we identify a procedure based on pivoted QR decompositions which is both efficient and accurate to machine precision. The Julia programming language is used for the assessment and implementations of all discussed algorithms are provided in the open-source software library StableDQMC.jl.

Charge density wave order on a π -flux square lattice

The effect of electron-phonon coupling (EPC) on Dirac fermions has recently been explored numerically on a honeycomb lattice, leading to precise quantitative values for the finite temperature and quantum critical points. In this paper, we use the unbiased determinant Quantum Monte Carlo (DQMC) method to study the Holstein model on a half-filled staggered-flux square lattice, and compare with the honeycomb lattice geometry, presenting results for a range of phonon frequencies $0.1 \leqslant \omega \leqslant 2.0$. We find that the interactions give rise to charge-density wave (CDW) order, but only above a finite coupling strength $\lambda_{\rm crit}$. The transition temperature is evaluated and presented in a $T_c$-$\lambda$ phase diagram. An accompanying mean-field theory (MFT) calculation also predicts the existence of quantum phase transition (QPT), but at a substantially smaller coupling strength.

Charge density waves on a half-filled decorated honeycomb lattice

Tight binding models like the Hubbard Hamiltonian are most often explored in the context of uniform intersite hopping $t$. The electron-electron interactions, if sufficiently large compared to this translationally invariant $t$, can give rise to ordered magnetic phases and Mott insulator transitions, especially at commensurate filling. The more complex situation of non-uniform $t$ has been studied within a number of situations, perhaps most prominently in multi-band geometries where there is a natural distinction of hopping between orbitals of different degree of overlap. In this paper we explore related questions arising from the interplay of multiple kinetic energy scales and electron-phonon interactions. Specifically, we use Determinant Quantum Monte Carlo (DQMC) to solve the half-filled Holstein Hamiltonian on a `decorated honeycomb lattice', consisting of hexagons with internal hopping $t$ coupled together by $t^{\,\prime}$. This modulation of the hopping introduces a gap in the Dirac spectrum and affects the nature of the topological phases. We determine the range of $t/t^{\,\prime}$ values which support a charge density wave (CDW) phase about the Dirac point of uniform hopping $t=t^{\,\prime}$, as well as the critical transition temperature $T_c$. The QMC simulations are compared with the results of Mean Field Theory (MFT).

Biexciton Condensation in Electron-Hole-Doped Hubbard Bilayers: A Sign-Problem-Free Quantum MonteCarlo Study.

The bilayer Hubbard model with electron-hole doping is an ideal platform to study excitonic orders due to suppressed recombination via spatial separation of electrons and holes. However, suffering from the sign problem, previous quantum MonteCarlo studies could not arrive at an unequivocal conclusion regarding the presence of phases with clear signatures of excitonic condensation in bilayer Hubbard models. Here, we develop a determinant quantum MonteCarlo algorithm for the bilayer Hubbard model that is sign-problem-free for equal and opposite doping in the two layers and study excitonic order and charge and spin density modulations as a function of chemical potential difference between the two layers, on-site Coulomb repulsion, and interlayer interaction. In the intermediate coupling regime and in proximity to the SU(4)-symmetric point, we find a biexcitonic condensate phase at finite electron-hole doping, as well as a competing (π,π) charge density wave state. We extract the Berezinskii-Kosterlitz-Thouless transition temperature from superfluid density and a finite-size scaling analysis of the correlation functions and explain our results in terms of an effective biexcitonic hard-core boson model.

Slater and Mott insulating states in the SU(6) Hubbard model

We perform large scale projector determinant quantum Monte-Carlo simulations to study the insulating states of the half-filled SU(6) Hubbard model on the square lattice. The transition from the antiferromagnetic state to the valence bond solid state occurs as increasing the Hubbard $U$. In contrast, in the SU(2) and SU(4) cases antiferromagnetism persists throughout the entire interaction range. In the SU(6) case, antiferromagnetism starts to develop in the weak interacting regime based on the Slater mechanism of Fermi surface nesting. As $U$ passes a crossover value $U^*/t\approx 9$, the single-particle gap scales linearly with $U$, marking the onset of Mott physics. In the Mott regime, antiferromagnetism becomes to be suppressed as $U$ increases, and vanishes after $U$ passes the critical value $U_{\rm AF,c}/t=13.3\pm 0.05$. The critical exponents are obtained via critical scalings as $\nu_{\rm AF}=0.60\pm 0.02$ and $\eta_{\rm AF}=0.44\pm 0.03$. As $U$ further increases, the valence bond solid ordering appears exhibiting the anomalous dimension $\eta_{\rm VBS}=0.98\pm 0.01$.

Quantum magnetism of topologically-designed graphene nanoribbons

Based on the Hubbard models, quantum magnetism of topologically-designed graphene nanoribbons (GNRs) is studied using exact numerical simulations. We first study a two-band Hubbard model describing the low-energy topological bands using the density matrix renormalization group (DMRG) and determinant quantum Monte Carlo (DQMC) methods. It is found the spin correlations decay quickly with distance, and the local moment is extrapolated to zero in the presence of symmetry-breaking terms. The results show that the two-band Hubbard chain is nonmagnetic, which is in contrast to the mean-field calculation predicting a critical interaction for the magnetic transition. We then include the Hubbard interaction to the topological-designed GNRs. For large interactions, the spin correlations remain finite for all distances, and the magnetic order develops. The local moment is extrapolated to almost zero for weak interactions, and begins to increase rapidly from a critical interaction. The estimated critical value is much larger than the realistic value in graphene, and we conclude the experimentally relevant GNRs are nonmagnetic, which is consistent with the experimental results.

Accelerating lattice quantum Monte Carlo simulations using artificial neural networks: Application to the Holstein model

Monte Carlo (MC) simulations are essential computational approaches with widespread use throughout all areas of science. We present a method for accelerating lattice MC simulations using fully connected and convolutional artificial neural networks that are trained to perform local and global moves in configuration space, respectively. Both networks take local spacetime MC configurations as input features and can, therefore, be trained using samples generated by conventional MC runs on smaller lattices before being utilized for simulations on larger systems. This new approach is benchmarked for the case of determinant quantum Monte Carlo (DQMC) studies of the two-dimensional Holstein model. We find that both artificial neural networks are capable of learning an unspecified effective model that accurately reproduces the MC configuration weights of the original Hamiltonian and achieve an order of magnitude speedup over the conventional DQMC algorithm. Our approach is broadly applicable to many classical and quantum lattice MC algorithms.

Quantum Monte Carlo study of the dominating pairing symmetry in doped honeycomb lattice**Project supported by the National Natural Science Foundation of China (Grant Nos. 11774019, 11504067, 11574032, and 11734002), the National Key Research and Development Program of China (Grant No. 2016YFA0300304), and the Fundamental Research Funds for the Central Universities, China (Grant No. HIT.NSRIF.2019057).

We perform a systematic determinant quantum Monte Carlo (DQMC) study of the dominating pairing symmetry in a doped honeycomb lattice. The Hubbard model is simulated over a full range of filling levels for both weak and strong interactions. For weak couplings, the d-wave state dominates. The effective susceptibility as a function of filling shows a peak, and its position moves toward half filling as the temperature is increased, from which the optimal filling of the superconducting ground state is estimated. Although the sign problem becomes severe for strong couplings, the simulations access the lowest temperature at which the DQMC method generates reliable results. As the coupling is strengthened, the d-wave state is enhanced in the high-filling region. Our systematic DQMC results provide new insights into the superconducting pairing symmetry in the doped honeycomb lattice.