Continuous-time Quantum Monte Carlo Method Research Articles

Spontaneously Symmetry-Breaking States in the Attractive SU(N) Hubbard Model

We investigate spontaneously symmetry breaking states in the attractive SU($N$) Hubbard model at half filling. Combining dynamical mean-field theory with the continuous-time quantum Monte Carlo method, we obtain the finite temperature phase diagrams for the superfluid state. When $N>2$, the second-order phase transition occurs in the weak coupling region, while the first-order phase transition with the hysteresis appears in the strong coupling region. We also discuss the stability of the density wave state and clarify the component dependence of the maximum critical temperature.

Fractional Spin Fluctuations as a Precursor of Quantum Spin Liquids: Majorana Dynamical Mean-Field Study for the Kitaev Model.

Experimental identification of quantum spin liquids remains a challenge, as the pristine nature is to be seen in asymptotically low temperatures. We here theoretically show that the precursor of quantum spin liquids appears in the spin dynamics in the paramagnetic state over a wide temperature range. Using the cluster dynamical mean-field theory and the continuous-time quantum MonteCarlo method, which are newly developed in the Majorana fermion representation, we calculate the dynamical spin structure factor, relaxation rate in nuclear magnetic resonance, and magnetic susceptibility for the honeycomb Kitaev model whose ground state is a canonical example of the quantum spin liquid. We find that dynamical spin correlations show peculiar temperature and frequency dependence even below the temperature where static correlations saturate. The results provide the experimentally accessible symptoms of the fluctuating fractionalized spins evincing the quantum spin liquids.

Worm-improved estimators in continuous-time quantum Monte Carlo

We derive the improved estimators for general interactions and employ these for the continuous-time quantum Monte Carlo method. Using a worm algorithm we show how measuring higher-ordered correlators leads to an improved high-frequency behavior in irreducible quantities such as the one-particle self-energy or the irreducible two-particle vertex for non-density-density interactions. A good knowledge of the asymptotics of the two-particle vertex is essential for calculating non-local electronic correlations using diagrammatic extensions to the dynamical mean field theory as well as for calculating susceptibilities. We test our algorithm against analytic results for the multi-orbital atomic-limit and the Falicov-Kimball model.

Finite-temperature phase transitions in theSU(N)Hubbard model

We investigate the $\mathrm{SU}(N)$ Hubbard model for the multicomponent fermionic optical lattice system, combining dynamical mean-field theory with the continuous-time quantum Monte Carlo method. We obtain the finite-temperature phase diagrams with $N\ensuremath{\le}6$ and find that low-temperature properties depend on the parity of the components. The magnetically ordered state competes with the correlated metallic state in the system with an even number of components $(N\ensuremath{\ge}4)$, yielding the first-order phase transition. It is also clarified that in the odd-component system, the ordered state is realized at relatively lower temperatures and the critical temperature is constant in the strong coupling limit.

Phase transitions in the Hubbard model for the bismuth nickelate

We study low temperature properties of the Hubbard model for the bismuth nickelate, where degenerate orbitals in the nickel ions and a single orbital in the bismuth ions are taken into account, combining dynamical mean-field theory with the continuous-time quantum Monte Carlo method. We discuss the effect of the attractive interactions to mimic the valence skipping phenomenon in the bismuth ions. We demonstrate how the charge and magnetically ordered states are stable against thermal fluctuations. It is furthermore clarified that the ferromagnetically ordered and orbital ordered states are stabilized due to the presence of the orbital degeneracy at low temperatures. The crossover between metallic and insulating states is also discussed.

Quantum phase transition, universality, and scaling behaviors in the spin-1/2 Heisenberg model with ferromagnetic and antiferromagnetic competing interactions on a honeycomb lattice.

The quantum phase transition, scaling behaviors, and thermodynamics in the spin-1/2 quantum Heisenberg model with antiferromagnetic coupling J>0 in the armchair direction and ferromagnetic interaction J^{'}<0 in the zigzag direction on a honeycomb lattice are systematically studied using the continuous-time quantum Monte Carlo method. By calculating the Binder ratio Q_{2} and spin stiffness ρ in two directions for various coupling ratios α=J^{'}/J under different lattice sizes, we found that a quantum phase transition from the dimerized phase to the stripe phase occurs at the quantum critical point α_{c}=-0.93. Through the finite-size scaling analysis on Q_{2},ρ_{x}, and ρ_{y}, we determined the critical exponent related to the correlation length ν to be 0.7212(8), implying that this transition falls into a classical Heisenberg O(3) universality. A zero magnetization plateau is observed in the dimerized phase, whose width decreases with increasing α. A phase diagram in the coupling ratio α-magnetic field h plane is obtained, where four phases, including dimerized, stripe, canted stripe, and polarized, are identified. It is also unveiled that the temperature dependence of the specific heat C(T) for different α's intersects precisely at one point, similar to that of liquid ^{3}He under different pressures and several magnetic compounds under various magnetic fields. The scaling behaviors of Q_{2},ρ, and C(T) are carefully analyzed. The susceptibility is compared with the experimental data to give the magnetic parameters of both compounds.

Real-space renormalized dynamical mean field theory

We propose real-space renormalized dynamical mean field theory (rr-DMFT) to deal with large clusters in the framework of a cluster extension of the DMFT. In the rr-DMFT, large clusters are decomposed into multiple smaller clusters through a real-space renormalization. In this work, the renormalization effect is taken into account only at the lowest order with respect to the intercluster coupling, which nonetheless reproduces exactly both the noninteracting and atomic limits. Our method allows us large cluster-size calculations which are intractable with the conventional cluster extensions of the DMFT with impurity solvers, such as the continuous-time quantum Monte Carlo and exact diagonalization methods. We benchmark the rr-DMFT for the two-dimensional Hubbard model on a square lattice at and away from half filling, where the spatial correlations play important roles. Our results on the spin structure factor indicate that the growth of the antiferromagnetic spin correlation is taken into account beyond the decomposed cluster size. We also show that the self-energy obtained from the large-cluster solver is reproduced by our method better than the solution obtained directly for the smaller cluster. When applied to the Mott metal-insulator transition, the rr-DMFT is able to reproduce the reduced critical value for the Coulomb interaction comparable to the large cluster result.

DMFT Study for Valence Fluctuations in the Extended Periodic Anderson Model

We study valence fluctuations at finite temperatures in the extended periodic Anderson model, where the Coulomb interaction between conduction and localized f-electrons is taken into account, using dynamical mean-field theory combined with the continuous-time quantum Monte Carlo (CT-QMC) method. The valence transition with the hysteresis is clearly found, indicating the first-order phase transition between the Kondo and mixed-valence states. We demonstrate that spin correlation rapidly develops when the system approaches the valence transition point. The comparison of the impurity solvers, the CT-QMC, non-crossing approximation, and one-crossing approximation, is also addressed.

Split Orthogonal Group: A Guiding Principle for Sign-Problem-Free Fermionic Simulations

We present a guiding principle for designing fermionic Hamiltonians and quantum MonteCarlo (QMC) methods that are free from the infamous sign problem by exploiting the Lie groups and Lie algebras that appear naturally in the MonteCarlo weight of fermionic QMC simulations. Specifically, rigorous mathematical constraints on the determinants involving matrices that lie in the split orthogonal group provide a guideline for sign-free simulations of fermionic models on bipartite lattices. This guiding principle not only unifies the recent solutions of the sign problem based on the continuous-time quantum MonteCarlo methods and the Majorana representation, but also suggests new efficient algorithms to simulate physical systems that were previously prohibitive because of the sign problem.

Quantum Monte Carlo study of mass-imbalanced Hubbard models

Building on recent solutions of the fermion sign problem for specific models we present two continuous-time quantum Monte Carlo methods for efficient simulation of mass-imbalanced Hubbard models on bipartite lattices at half-filling. For both methods we present the solutions to the fermion sign problem and the algorithms to achieve efficient simulations. As applications, we calculate the dependence of the spin correlation on the mass imbalance in a one-dimensional lattice and study the thermal and quantum phase transitions to an antiferromagnetic Ising long-range ordered state in two dimensions. These results offer unbiased predictions for experiments on ultracold atoms and bridge known exact solutions of Falicov-Kimball model and previous studies of the SU(2)-symmetric Hubbard model.

Exact Dynamics of Charge Fluctuations in the Multichannel Interacting Resonant Level Model

A modified version of the spinless Anderson model is studied by means of the continuous-time quantum Monte Carlo method. This study is motivated by the peculiar heavy-fermion behavior observed in certain Samarium compounds, which is insensitive to magnetic field. The model involves M channels for conduction electrons, all of which interact with local f electron via the Coulomb repulsion Ufc, while only one channel has hybridization with the local state. The effective hybridization is reduced by the Anderson orthogonality effect, and a quantum critical point occurs with increasing M and/or increasing Ufc. The numerical results at finite temperature of the local charge susceptibility are well fitted by a simple scaling theory for all M. However, the single-particle spectrum is described by a double Lorentzian for M>1, in contrast with the single Lorentzian with M=1. A quasi-particle perturbation theory is presented that reproduces the quantum critical point for large M. The quasi-particle theory gives not only the renormalized energy scale, but its extrapolation toward higher energies being consistent with the double Lorentzian spectrum.

Slope-reversed Mott transition in multiorbital systems

We examine finite-temperature phase transitions in the two-orbital Hubbard model with different bandwidths by means of the dynamical mean-field theory combined with the continuous-time quantum Monte Carlo method. It is found that there emerges a peculiar slope-reversed first-order Mott transition between the orbital-selective Mott phase and the Mott insulator phase in the presence of Ising-type Hund's coupling. The origin of the slope-reversed phase transition is clarified by the analysis of the temperature dependence of the energy density. It turns out that the increase of Hund's coupling lowers the critical temperature of the slope-reversed Mott transition. Beyond a certain critical value of Hund's coupling the first-order transition turns into a finite-temperature crossover. We also reveal that the orbital-selective Mott phase exhibits frozen local moments in the wide orbital, which is demonstrated by the spin-spin correlation functions.

Numerical calculation of spectral functions of the Bose-Hubbard model using bosonic dynamical mean-field theory

We calculate the momentum dependent spectral function of the Bose-Hubbard model on a simple cubic lattice in three dimensions within the bosonic dynamical mean-field theory (B-DMFT). The continuous-time quantum Monte Carlo method is used to solve the self-consistent B-DMFT equations together with the maximum entropy method for the analytic continuation to real frequencies. Results for weak, intermediate, and strong interactions are presented. In the limit of weak and strong interactions very good agreement with results obtained by perturbation theory is found. By contrast, at intermediate interactions the results differ significantly, indicating that in this regime perturbative methods fail do describe the dynamics of interacting bosons.

Efficient continuous-time quantum Monte Carlo method for the ground state of correlated fermions

We present the ground state extension of the efficient quantum Monte Carlo algorithm for lattice fermions of arXiv:1411.0683. Based on continuous-time expansion of imaginary-time projection operator, the algorithm is free of systematic error and scales \emph{linearly} with projection time and interaction strength. Compared to the conventional quantum Monte Carlo methods for lattice fermions, this approach has greater flexibility and is easier to combine with powerful machinery such as histogram reweighting and extended ensemble simulation techniques. We discuss the implementation of the continuous-time projection in detail using the spinless $t-V$ model as an example and compare the numerical results with exact diagonalization, density-matrix-renormalization-group and infinite projected entangled-pair states calculations. Finally we use the method to study the fermionic quantum critical point of spinless fermions on a honeycomb lattice and confirm previous results concerning its critical exponents.

Excitation spectra and correlation functions of quantum Su-Schrieffer-Heeger models

We study one-dimensional Su-Schrieffer-Heeger (SSH) models with quantum phonons using a continuous-time quantum Monte Carlo method. Within statistical errors, we obtain identical results for the SSH model with acoustic phonons, and a related model with a coupling to an optical bond phonon mode. Based on this agreement, we first study the Peierls metal-insulator transition of the spinless SSH model, and relate it to the Kosterlitz-Thouless transition of a spinless Luttinger liquid. In the Peierls phase, the spectral functions reveal the single-particle and charge gap, and a central peak related to long-range order. For the spinful SSH model, which has a dimerized ground state for any nonzero coupling, we reveal a symmetry-related degeneracy of spin and charge excitations, and the expected spin and charge gaps as well as a central peak. Finally, we study the SSH-$UV$ model with electron-phonon and electron-electron interaction. We observe a Mott phase with critical spin and bond correlations at weak electron-phonon coupling, and a Peierls phase with gapped spin excitations at strong coupling. We relate our findings to the extended Hubbard model, and discuss the physical origin of the agreement between optical and acoustic phonons.

Formula omitted]: An open source continuous-time quantum Monte Carlo impurity solver toolkit

Quantum impurity solvers have a broad range of applications in theoretical studies of strongly correlated electron systems. Especially, they play a key role in dynamical mean-field theory calculations of correlated lattice models and realistic materials. Therefore, the development and implementation of efficient quantum impurity solvers is an important task. In this paper, we present an open source interacting quantum impurity solver toolkit (dubbed iQIST). This package contains several highly optimized quantum impurity solvers which are based on the hybridization expansion continuous-time quantum Monte Carlo algorithm, as well as some essential pre- and post-processing tools. We first introduce the basic principle of continuous-time quantum Monte Carlo algorithm and then discuss the implementation details and optimization strategies. The software framework, major features, and installation procedure for iQIST are also explained. Finally, several simple tutorials are presented in order to demonstrate the usage and power of iQIST. Program summaryProgram title:iQISTCatalogue identifier: AEWQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEWQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 226270No. of bytes in distributed program, including test data, etc.: 5263144Distribution format: tar.gzProgramming language: Fortran 2008 and Python.Computer: Desktop PC, laptop, high performance computing cluster.Operating system: Unix, Linux, Mac OS X, Windows.Has the code been vectorized or parallelized?: Yes, it is parallelized by MPI and OpenMPRAM: Depends on the complexity of the problemClassification: 7.3.External routines: BLAS, LAPACK, Latex is required to build the user manual.Nature of problem:Quantum impurity models were originally proposed to describe magnetic impurities in metallic hosts. In these models, the Coulomb interaction acts between electrons occupying the orbitals of the impurity atom. Electrons can hop between the impurity and the host, and in an action formulation, this hopping is described by a time-dependent hybridization function. Nowadays quantum impurity models have a broad range of applications, from the description of heavy fermion systems, and Kondo insulators, to quantum dots in nano-science. They also play an important role as auxiliary problems in dynamical mean-field theory and its diagrammatic extensions [1–3], where an interacting lattice model is mapped onto a quantum impurity model in a self-consistent manner. Thus, the accurate and efficient solution of quantum impurity models becomes an essential task.Solution method:The quantum impurity model can be solved by the numerically exact continuous-time quantum Monte Carlo method, which is the most efficient and powerful impurity solver for finite temperature simulations. In the iQIST software package, we implemented the hybridization expansion version of continuous-time quantum Monte Carlo algorithm. Both the segment representation and general matrix formalism are supported. The key idea of this algorithm is to expand the partition function diagrammatically in powers of the impurity-bath hybridization, and to stochastically sample these diagrams to all relevant orders using the Metropolis Monte Carlo algorithm. For a detailed review of the continuous-time quantum Monte Carlo algorithms, please refer to [4].Running time:Depends on the complexity of the problem. The sample run supplied in the distribution takes about 1.5 minutes.

Spin-orbit-induced exotic insulators in a three-orbital Hubbard model with(t2g)5electrons

On the basis of the multi-orbital dynamical mean field theory, a three-orbital Hubbard model with a relativistic spin-orbit coupling (SOC) is studied at five electrons per site. The numerical calculations are performed by employing the continuous-time quantum Monte Carlo (CTQMC) method based on the strong coupling expansion. We find that appropriately choosing bases, i.e., the maximally spin-orbit-entangled bases, drastically improve the sign problem in the CTQMC calculations, which enables us to treat exactly the full Hund's coupling and pair hopping terms. This improvement is also essential to reach at low temperatures for a large SOC region where the SOC most significantly affects the electronic structure. We show that a metal-insulator transition is induced by the SOC for fixed Coulomb interactions. The insulating state for smaller Coulomb interactions is antiferromagnetically ordered with the local effective total angular momentum $j=1/2$, in which the $j=1/2$ based band is essentially half-filled while the $j=3/2$ based bands are completely occupied. More interestingly, for larger Coulomb interactions, we find that an excitonic insulating state emerges, where the condensation of an electron-hole pair in the $j=1/2$ and $j=3/2$ based bands occurs. The origin of the excitonic insulator as well as the experimental implication is discussed.

Preserving spin rotational symmetry in quantum Monte Carlo for description of magnetic properties of strongly correlated compounds

We present a modification of the continuous-time hybridization-expansion quantum Monte Carlo method, which allows one to take into account spin rotational symmetry. At present, studies of real compounds by means of dynamical mean-field theory employ an approximate form of the Coulomb interaction which breaks the rotational symmetry, leading, in particular, to considerable overestimation of the Curie-Weiss temperature. The obtained temperature dependence of magnetic susceptibility and Curie-Weiss temperature for the three-band model are in good agreement with the exact solution. Owing to its computational efficiency, the presented method is suitable for investigation of magnetic properties of real compounds with strong electronic correlations.

Superconductivity in the two-band Hubbard model

We study the two-band Hubbard model in infinite dimensions by solving the dynamical mean-field equations with a strong coupling continuous-time quantum Monte Carlo method and show that an $s$-wave superconducting state can be stabilized in the repulsively interacting case. We discuss how this superconducting state competes with the metallic and paired Mott states. The effects of the Hund coupling and crystalline electric field are also addressed.

Dynamical Characteristics of the Mott Transition: Examination of Doublon Dynamics in a Triangular-lattice Hubbard Model

Dynamics in a triangular-lattice Hubbard model is studied at half filling on the basis of the cluster dynamical mean field theory. Numerical calculations use a solver of the continuous-time quantum Monte Carlo method based on the strong-coupling expansion. We examine the change in the nearest-neighbor dynamical correlations of doublon and holon in the time domain with varying the Coulomb repulsion near the Mott transition. We demonstrate that the nearest-neighbor doublon-holon pair shows a strong attractive correlation, particularly in the insulating phase, while the nearest-neighbor doublon-doublon pair shows a repulsive correlation. Useful information is provided by the trajectories in the complex plane of the dynamical correlation functions. The short-time dynamics of the nearest-neighbor doublon and holon in the metallic phase indicates larger fluctuations than in the insulating phase. Their trajectories in the complex plane show that the short-time dynamics of the doublon-holon pair has an opposite behavior to that of the doublon-doublon pair. We find that their time scale of dynamics can be characterized by the period in which the phase rotates π around the long-time limit in both phases. In the long-time region, fluctuations persist up to a very long time in the metallic phase, while they quickly vanish in the insulating phase.