Density Matrix Renormalization Group Method Research Articles

Unsupervised Generative Modeling Using Matrix Product States

Generative modeling, which learns joint probability distribution from data and generates samples according to it, is an important task in machine learning and artificial intelligence. Inspired by probabilistic interpretation of quantum physics, we propose a generative model using matrix product states, which is a tensor network originally proposed for describing (particularly one-dimensional) entangled quantum states. Our model enjoys efficient learning analogous to the density matrix renormalization group method, which allows dynamically adjusting dimensions of the tensors and offers an efficient direct sampling approach for generative tasks. We apply our method to generative modeling of several standard datasets including the Bars and Stripes, random binary patterns and the MNIST handwritten digits to illustrate the abilities, features and drawbacks of our model over popular generative models such as Hopfield model, Boltzmann machines and generative adversarial networks. Our work sheds light on many interesting directions of future exploration on the development of quantum-inspired algorithms for unsupervised machine learning, which are promisingly possible to be realized on quantum devices.

Topological Invariants in Terms of Green’s Function for the Interacting Kitaev Chain**Supported by the National Natural Science Foundation of China under Grant No 11274379, and the Research Funds of Renmin University of China under Grant No 14XNLQ07.

A one-dimensional closed interacting Kitaev chain and the dimerized version are studied. The topological invariants in terms of Green’s function are calculated by the density matrix renormalization group method and the exact diagonalization method. For the interacting Kitaev chain, we point out that the calculation of the topological invariant in the charge density wave phase must consider the dimerized configuration of the ground states. The variation of the topological invariant is attributed to the poles of eigenvalues of the zero-frequency Green functions. For the interacting dimerized Kitaev chain, we show that the topological invariant defined by Green’s functions can distinguish more topological nonequivalent phases than the fermion parity.

Communication: An efficient stochastic algorithm for the perturbative density matrix renormalization group in large active spaces

We present an efficient stochastic algorithm for the recently introduced perturbative density matrix renormalization group method for large active spaces. The stochastic implementation bypasses the computational bottleneck involved in solving the first order equation in the earlier deterministic algorithm. We demonstrate the efficiency and accuracy of the algorithm on the C2 and Cr2 molecular benchmark systems.

Phase diagram of the Hubbard-Holstein model on a four-leg tube system at quarter filling

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 $(t/{\ensuremath{\omega}}_{0}\ensuremath{\le}1)$, $t$ 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 $g$ and $U$ are the $e\text{\ensuremath{-}}ph$ coupling constant and Hubbard on-site interaction respectively. For larger $g$, 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 $U$. But for smaller $g$, 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.

Site-occupation embedding theory using Bethe ansatz local density approximations

Site-occupation embedding theory (SOET) is an alternative formulation of density-functional theory (DFT) for model Hamiltonians where the fully-interacting Hubbard problem is mapped, in principle exactly, onto an impurity-interacting (rather than a non-interacting) one. It provides a rigorous framework for combining wavefunction (or Green function) based methods with DFT. In this work, exact expressions for the per-site energy and double occupation of the uniform Hubbard model are derived in the context of SOET. As readily seen from these derivations, the so-called bath contribution to the per-site correlation energy is, in addition to the latter, the key density functional quantity to model in SOET. Various approximations based on Bethe ansatz and perturbative solutions to the Hubbard and single impurity Anderson models are constructed and tested on a one-dimensional ring. The self-consistent calculation of the embedded impurity wavefunction has been performed with the density matrix renormalization group method. It has been shown that promising results are obtained in specific regimes of correlation and density. Possible further developments have been proposed in order to provide reliable embedding functionals and potentials.

Doping evolution of charge and spin excitations in two-leg Hubbard ladders: Comparing DMRG and FLEX results

We study the magnetic and charge dynamical response of a Hubbard model in a two-leg ladder geometry using the density matrix renormalization group (DMRG) method and the random phase approximation within the fluctuation-exchange approximation (RPA+FLEX). Our calculations reveal that RPA+FLEX can capture the main features of the magnetic response from weak up to intermediate Hubbard repulsion for doped ladders, when compared with the numerically exact DMRG results. However, while at weak Hubbard repulsion both the spin and charge spectra can be understood in terms of weakly-interacting electron-hole excitations across the Fermi surface, at intermediate coupling DMRG shows gapped spin excitations at large momentum transfer that remain gapless within the RPA+FLEX approximation. For the charge response, RPA+FLEX can only reproduce the main features of the DMRG spectra at weak coupling and high doping levels, while it shows an incoherent character away from this limit. Overall, our analysis shows that RPA+FLEX works surprisingly well for spin excitations at weak and intermediate Hubbard $U$ values even in the difficult low-dimensional geometry such as a two-leg ladder. Finally, we discuss the implications of our results for neutron scattering and resonant inelastic x-ray scattering experiments on two-leg ladder cuprate compounds.

Thermodynamic properties of the S=1/2 twisted triangular spin tube

Thermodynamic properties of the twisted three-leg spin tube under magnetic field are studied by the finite-$T$ density-matrix renormalization group method. The specific heat, spin, and chiral susceptibilities of the infinite system are calculated for both the original and its low-energy effective models. The obtained results show that the presence of the chirality is observed as a clear peak in the specific heat at low temperature and the contribution of the chirality dominates the low-temperature part of the specific heat as the exchange coupling along the spin tube decreases. The peak structures in the specific heat, spin, and chiral susceptibilities are strongly modified near the quantum phase transition where the critical behaviors of the spin and chirality correlations change. These results confirm that the chirality plays a major role in characteristic low-energy behaviors of the frustrated spin systems.

Spin-1 Bose–Hubbard model with two- and three-body interactions

We investigated the ground state of spin-1 bosons interacting under local two- and three-body interactions in one dimension by means of the density matrix renormalization group method. We found that the even–odd asymmetry will be obtained or not depending on the relative values of the two- and three-body interactions. The Mott insulator lobes are spin isotropic, the first showing a dimerized pattern and the second being composed of singlets. The three-body interactions disfavor a longitudinal polar superfluid and a quantum phase transition to a transverse polar superfluid occurs, which could be continuous or discontinuous.

Ground-state properties of the one-dimensional antiferromagnetic chain with frustrated side spins

The properties of the one-dimensional antiferromagnetic chain with frustrated side spins are studied by means of the spin wave theory, the exact diagonalization method and the density matrix renormalization group method. The results show that the Lieb–Mattis type ferrimagnetic state is always the ground state when α<αc1. The system undergoes a first-order phase transition accompanied by a spontaneous symmetry breaking and its acoustic excitation spectrum softens completely near k=0 at the critical point αc1. In the intermediate parameter region αc1<α<αc2, the ground state is a nonmagnetic state called dimer-tetramer state with a small spin gap. A gapped-gapless Berezinskii-Kosterlitz-Thouless phase transition occurs at another critical point αc2 beyond which the nonmagnetic dimer–monomer state becomes the ground state. The value of the critical point αc2 can be precisely determined not only by the level-spectroscopy method, but also the method based on comparing the magnitude of the two types of correlation functions and the method of a scaling analysis of the gap.

Effect of Interaction on the Majorana Zero Modes in the Kitaev Chain at Half Filling**Supported by the National Natural Science Foundation of China under Grant No 11274379, and the Research Funds of Renmin University of China under Grant No 14XNLQ07.

The one-dimensional interacting Kitaev chain at half filling is studied. The symmetry of the Hamiltonian is examined by dual transformations, and various physical quantities as a function of the fermion-fermion interaction U are calculated systematically using the density matrix renormalization group method. A special value of interaction Up is revealed in the topological region of the phase diagram. We show that at Up the ground states are strictly two-fold degenerate even though the chain length is finite and the zero-energy peak due to the Majorana zero modes is maximally enhanced and exactly localized at the end sites. Here Up may be attractive or repulsive depending on other system parameters. We also give a qualitative understanding of the effect of interaction under the self-consistent mean field framework.

Heisenberg spin- 12 XXZ chain in the presence of electric and magnetic fields

We study the interplay of electric and magnetic order in the one dimensional Heisenberg spin-1/2 XXZ chain with large Ising anisotropy in the presence of the Dzyaloshinskii-Moriya (D-M) interaction and with longitudinal and transverse magnetic fields, interpreting the D-M interaction as a coupling between the local electric polarization and an external electric field. We obtain the ground state phase diagram using the density matrix renormalization group method and compute various ground state quantities like the magnetization, staggered magnetization, electric polarization and spin correlation functions, etc. In the presence of both longitudinal and transverse magnetic fields, there are three different phases corresponding to a gapped N\'{e}el phase with antiferromagnetic (AF) order, gapped saturated phase and a critical incommensurate gapless phase. The external electric field modifies the phase boundaries but does not lead to any new phases. Both external magnetic fields and electric fields can be used to tune between the phases. We also show that the transverse magnetic field induces a vector chiral order in the N\'{e}el phase (even in the absence of an electric field) which can be interpreted as an electric polarization in a direction parallel to the AF order.

Height of a faceted macrostep for sticky steps in a step-faceting zone

The driving force dependence of the surface velocity and the average height of faceted merged steps, the terrace-surface-slope, and the elementary step velocity in the non-equilibrium steady-state are studied using the Monte Carlo method. The Monte Carlo study is based on a lattice model, the restricted solid-on-solid model with point-contact type step--step attraction (p-RSOS model). The temperature is selected to be in the step-faceting zone where the surface is surrounded by the (001) terrace and the (111) faceted step at equilibrium. Long time simulations are performed at this temperature to obtain steady-states for the different driving forces that influence the growth/recession of the surface. A Wulff figure of the p-RSOS model is produced through the anomalous surface tension calculated using the density-matrix renormalization group method. Although the p-RSOS model is a simplified model, the model shows a wide variety of dynamics in the step-faceting zone. There are four characteristic driving forces, $\Delta \mu_y$, $\Delta \mu_f$, $\Delta \mu_{co}$, and $\Delta \mu_R$. For $\Delta \mu_{co}<|\Delta \mu|<\Delta \mu_R$, the surface grows/recedes with the successive attachment-detachment of steps to/from a macrostep. When $|\Delta \mu|$ exceeds $\Delta \mu_R $, the macrostep vanishes and the surface roughens kinetically. Classical 2D heterogeneous multi-nucleation was determined to be valid with slight modifications based on the Monte Carlo results of the step velocity and the change in the surface slope of the "terrace". The finite size effects were also determined to be distinctive near equilibrium.

On compact vector formats in the solution of the chemical master equation with backward differentiation

SummaryA stochastic chemical system with multiple types of molecules interacting through reaction channels can be modeled as a continuous‐time Markov chain with a countably infinite multidimensional state space. Starting from an initial probability distribution, the time evolution of the probability distribution associated with this continuous‐time Markov chain is described by a system of ordinary differential equations, known as the chemical master equation (CME). This paper shows how one can solve the CME using backward differentiation. In doing this, a novel approach to truncate the state space at each time step using a prediction vector is proposed. The infinitesimal generator matrix associated with the truncated state space is represented compactly, and exactly, using a sum of Kronecker products of matrices associated with molecules. This exact representation is already compact and does not require a low‐rank approximation in the hierarchical Tucker decomposition (HTD) format. During transient analysis, compact solution vectors in HTD format are employed with the exact, compact, and truncated generated matrices in Kronecker form, and the linear systems are solved with the Jacobi method using fixed or adaptive rank control strategies on the compact vectors. Results of simulation on benchmark models are compared with those of the proposed solver and another version, which works with compact vectors and highly accurate low‐rank approximations of the truncated generator matrices in quantized tensor train format and solves the linear systems with the density matrix renormalization group method. Results indicate that there is a reason to solve the CME numerically, and adaptive rank control strategies on compact vectors in HTD format improve time and memory requirements significantly.

Ordered states in the Kitaev-Heisenberg model: From 1D chains to 2D honeycomb

We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg (J = cosϕ) and Kitaev (K = sinϕ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-z, ferromagnetic-xy, staggered-xy, Néel-z, and two liquid states: Tomonaga-Luttinger liquid and spiral-xy. The two Kitaev points {boldsymbol{phi }}{boldsymbol{=}}frac{{boldsymbol{pi }}}{{bf{2}}} and {boldsymbol{varphi }}{boldsymbol{=}}frac{{bf{3}}{boldsymbol{pi }}}{{bf{2}}} are singular. The ϕ-dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.

SU(N) fractional quantum Hall effect in topological flat bands

We study $N$-component interacting particles (hardcore bosons and fermions) loaded in topological lattice models with SU$(N)$-invariant interactions based on density matrix renormalization group method. By tuning the interplay of interspecies and intraspecies interactions, we demonstrate that a class of SU$(N)$ fractional quantum Hall states can emerge at fractional filling factors $\nu=N/(N+1)$ for bosons ($\nu=N/(2N+1)$ for fermions) in the lowest Chern band, characterized by the nontrivial fractional Hall responses and the fractional charge pumping. Moreover, we establish a topological characterization based on the $\mathbf{K}$ matrix, and discuss the close relationship to the fractional quantum Hall physics in topological flat bands with Chern number $N$.

Magnetization plateaus in the spin- 12 antiferromagnetic Heisenberg model on a kagome-strip chain

Spin-1/2 Heisenberg model on kagome lattice is a typical frustrated quantum spin system. A basic structure of kagome lattice is also present in kagome-strip lattice in one dimension, where a similar type of frustration is expected. We thus study the magnetization plateaus of the spin-1/2 Heisenberg model on a kagome-strip chain with three-independent antiferromagnetic exchange interactions by the density-matrix renormalization group method. In a certain range of exchange parameters, we find twelve kinds of magnetization plateaus, nine of which have magnetic structures breaking either translational or reflection symmetry spontaneously. The structures are classified by an array of five-site unit cells with specific bond-spin correlations. In a case with nontrivial plateau, 3/10 plateau, we find long-period magnetic structure with a period of four unit cells.

Full Quantum Dynamics Simulation of a Realistic Molecular System Using the Adaptive Time-Dependent Density Matrix Renormalization Group Method.

The accurate theoretical interpretation of ultrafast time-resolved spectroscopy experiments relies on full quantum dynamics simulations for the investigated system, which is nevertheless computationally prohibitive for realistic molecular systems with a large number of electronic and/or vibrational degrees of freedom. In this work, we propose a unitary transformation approach for realistic vibronic Hamiltonians, which can be coped with using the adaptive time-dependent density matrix renormalization group (t-DMRG) method to efficiently evolve the nonadiabatic dynamics of a large molecular system. We demonstrate the accuracy and efficiency of this approach with an example of simulating the exciton dissociation process within an oligothiophene/fullerene heterojunction, indicating that t-DMRG can be a promising method for full quantum dynamics simulation in large chemical systems. Moreover, it is also shown that the proper vibronic features in the ultrafast electronic process can be obtained by simulating the two-dimensional (2D) electronic spectrum by virtue of the high computational efficiency of the t-DMRG method.

Spin-1 two-impurity Kondo problem on a lattice

We present an extensive study of the two-impurity Kondo problem for spin-1 adatoms on square lattice using an exact canonical transformation to map the problem onto an effective one-dimensional system that can be numerically solved using the density matrix renormalization group method. We provide a simple intuitive picture and identify the different regimes, depending on the distance between the two impurities, Kondo coupling $J_K$, longitudinal anisotropy $D$, and transverse anisotropy $E$. In the isotropic case, two impurities on opposite(same) sublattices have a singlet(triplet) ground state. However, the energy difference between the triplet ground state and the singlet excited state is very small and we expect an effectively four-fold degenerate ground state, i.e., two decoupled impurities. For large enough $J_K$ the impurities are practically uncorrelated forming two independent underscreened states with the conduction electrons, a clear non-perturbative effect. When the impurities are entangled in an RKKY-like state, Kondo correlations persists and the two effects coexist: the impurities are underscreened, and the dangling spin-$1/2$ degrees of freedom are responsible for the inter-impurity entanglement. We analyze the effects of magnetic anisotropy in the development of quasi-classical correlations.

Fidelity susceptibility and entanglement entropy in S=1 quantum spin chain with three-site interactions

In the present work, we study the fidelity susceptibility and the entanglement entropy in an antiferromagnetic spin-1 chain with additional next-nearest neighbor interactions and three-site interactions, which are given by H=(J1SiSi+1+ J2SiSi+2)+[J3(SiSi+1)(Si+1Si+2)+ h.c.]. By using the density matrix renormalization group method, the ground-state properties of the system are calculated with very high accuracy. We investigate the effect of the three-site interaction J3 on the fidelity susceptibility numerically, and then analyze its relation with the quantum phase transition (QPT). The fidelity measures the similarity between two states, and the fidelity susceptibility describes the associated changing rate. The QPT is intuitively accompanied by an abrupt change in the structure of the ground-state wave function, so generally a peak of the fidelity susceptibility indicates a QPT and the location of the peak denotes the critical point. For the case of J2=0, a peak of the fidelity susceptibility is found by varying J3, and the height of the peak grows as the system size increases. The location of the peak shifts to a slightly lower J3 up to a particular value as the system size increases. Through a finite size scaling, the critical point J3c=0.111 of the QPT from the Haldane spin liquid to the dimerized phase is identified. We also study the effect of the three-site interaction on the entanglement entropy between the right half part and the rest. It is noted that the peak of the entanglement entropy does not coincide with the critical point. Instead, the critical point is determined by the position at which the first-order derivative of the entanglement entropy takes its minimum, since a second-order QPT is signaled by the first derivative of density matrix element. Moreover, the entanglement entropy disappears when J3=1/6, which corresponds to the size-independent Majumdar-Ghosh point. The positions of quantum critical points extracted from these two quantum information observables agree well with those obtained by the string order parameters, which characterizes the topological order in the Haldane phase. Secondly, we also study the case of J20, and obtain the critical points by both the fidelity susceptibility and the entanglement entropy. Finally we provide a ground-state phase diagram of the system. To sum up, the quantum information observables are effective tools for detecting diverse QPTs in spin-1 models.

Dilute antiferromagnetism in magnetically doped phosphorene

We study the competition between Kondo physics and indirect exchange on monolayer black phos-phorous using a realistic description of the band structure in combination with the density matrixrenormalization group (DMRG) method. The Hamiltonian is reduced to a one-dimensional problemvia an exact canonical transformation that makes it amenable to DMRG calculations, yielding exactresults that fully incorporate the many-body physics. We find that a perturbative description of theproblem is not appropriate and cannot account for the slow decay of the correlations and the completelack of ferromagnetism. In addition, at some particular distances, the impurities decouple formingtheir own independent Kondo states. This can be predicted from the nodes of the Lindhard function.Our results indicate a possible route toward realizing dilute anti-ferromagnetism in phosphorene.&#x0D; Received: 19 September 2017, Accepted: 12 October 2017; Edited by: K. Hallberg; DOI: http://dx.doi.org/10.4279/PIP.090008&#x0D; Cite as: A Allerdt, A E Feiguin, Papers in Physics 9, 090008 (2017)&#x0D; This paper, by A Allerdt, A E Feiguin, is licensed under the Creative Commons Attribution License 3.0.&#x0D;