Podcasts about dmrg

Antoine GeorgesPhysique de la matière condenséeAnnée 2022-2023Conférence - Assa Auerbach : Computing Low Energy Effective Hamiltonians of Hubbard and Heisenberg Models in Two Dimensions by Contractor RenormalizationAssa Auerbach, Department of Physics, Technion, Israel Institute of TechnologyThe Higgs Mode and Quantum Criticality in Condensed MatterContractor Renormalization (CORE) invented for computing correlations in lattice gauge theories in 1996, is a very promising approach for deriving low energy effective Hamiltonians of condensed matter lattice models. CORE identifies the low energy degrees of freedom and their interactions up to the range of truncation. Truncation error is controlled by the entanglement lengthscale, similarly to the convergence of DMRG. Examples of CORE results the derivation of Plaquette Boson-Fermion phenomenology for cuprate superconductors, and prediction of p6 symmetry breaking in the gapped spin liquid of the Heisenberg Kagome model.

2017 Arnold Sommerfeld School: Numerical Methods for Correlated Many-Body Systems

2017 Arnold Sommerfeld School: Numerical Methods for Correlated Many-Body Systems

2017 Arnold Sommerfeld School: Numerical Methods for Correlated Many-Body Systems

2017 Arnold Sommerfeld School: Numerical Methods for Correlated Many-Body Systems

2017 Arnold Sommerfeld School: Numerical Methods for Correlated Many-Body Systems

This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin $S = 1/2$ on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully $SU(2)$-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped $Z_2$ quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin $S=3/2$ on the $z=3$ Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.

This thesis is concerned with open quantum systems, and more specifically, quantum impurity models. By this we mean a small local quantum system in contact with a macroscopic non-interacting environment. This can be used to model individual impurities in metals and quantum information systems where the influence from the surrounding environment is not negligible. The numerical renormalization group (NRG) is the traditional method to study quantum impurity models. However its application is limited when dealing with real-time dynamics and bosonic systems. In recent years some of NRG's techniques have been introduced to the density matrix renormalization group method (DMRG), which itself is the most powerful numerical method to study one-dimensional quantum systems. The resulting method shows great potential, and this thesis explores and extends the power of the NRG+DMRG combination in treating open quantum systems. We focus mainly on two types of problem. The first is an open quantum system with a time-dependent Hamiltonian, which for example could be the theoretical description of various problems encountered in qubit manipulation. We combine NRG discretization and adaptive time-dependent DMRG (t-DMRG) to study the dissipative Landau-Zener problem. We also use this method to study the quantum decoherence process and the dynamical properties of the telegraph noise model. The results show that the NRG and t-DMRG combination is a fast, accurate and versatile method for such problems. The second type of problem we study involves the quantum critical properties of one- and two-bath spin-boson models. Unlike fermion and spin models, models with bosons are difficult to treat numerically as each boson basis has an infinite number of dimensions. By introducing the optimal boson basis into the variational matrix product state method (VMPS), which is a variant of DMRG, we can deal with an effective local boson basis as large as $10^{10}$. This is the crucial improvement over NRG which can only deal with at most a few dozen local basis states. With this powerful tool we have settled a controversy about the nature of the quantum phase transition in a sub-Ohmic spin-boson model regarding quantum to classical mapping. There, NRG fails to yield the right result due to the highly truncated local boson basis. We also explore the phase diagram of the two-bath spin-boson model and find a new critical phase. We demonstrate that NRG+VMPS with optimal boson basis represents a powerful setting to study quantum impurity models with a bosonic environment.

The goal of this thesis is to study the transport properties and real-time dynamics of quantum magnets and ultra-cold atomic gases in one spatial dimension using numerical methods. The focus will be on the discussion of diffusive versus ballistic dynamics along with a detailed analysis of characteristic velocities in ballistic regimes. For the simulation of time-dependent density profiles we use the adaptive time-dependent density matrix renormalization group (DMRG). This numerical method allows for the simulation of time-dependent wave functions close to as well as far from equilibrium in a controlled manner. The studies of one-dimensional quantum magnets are partially motivated by the experimental evidence for a highly anisotropic and for insulators comparably high thermal conductivity of certain cuprates. We use linear response theory to study transport coefficients at arbitrary temperatures by diagonalizing small systems exactly and then calculating the current-current correlation functions. As first application we discuss the spin transport in the spin-$1/2$ Heisenberg chain with anisotropic exchange interactions (XXZ-chain). The second application of exact diagonalization, here in combination with time-dependent DMRG, is a discussion of the transverse components of the current-current correlation function. While usually only a Zeeman field is considered in the theory of transport coefficients, we here investigate the dynamic induced by an additional transverse magnetic field. We find that in this scenario the current-current correlation function exhibits coherent oscillations. In addition a second non-trivial frequency, different from the one expected from the usual Larmor precession, emerges and is studied varying temperature and field. Finally we calculate the frequency-dependent spin and heat conductivity of dimerized spin chains in a magnetic field. Motivated by the recent experimental studies of the phase diagram of C$_5$H$_{12}$N$_2$CuBr$_4$ we take the dimerized chain as a minimal model that exhibits features of the low-temperature region of the observed phase diagram. As a main result, the spin and heat conductivity obtained from linear response theory are enhanced in the field-induced gapless phase. The last application in the field of one-dimensional quantum magnets is the simulation of time-dependent energy-density wave-packets close to as well as far from equilibrium using the time-dependent density renormalization group. The main results are ballistic energy dynamics independently of how far out-of-equilibrium the initial state is and a detailed understanding of the average expansion velocity. The applications in the field of ultra-cold atomic gases focus on the sudden expansion of an initially trapped gas into an empty optical lattice. This setup was recently realized in an experiment performed by U. Schneider {it et al.} and discussed in the context of electronic transport in the two-dimensional and the three-dimensional Fermi-Hubbard model. Here we investigate the sudden expansion of three different setups: For the expansion of a spin-balanced cloud of fermions, we identify the ballistic regime, and therein investigate the average expansion velocity of the cloud. As a main result the expansion velocity is determined by a small subset of the initial condition over a wide range of parameters. For instance, the Mott-insulating phase of the Hubbard model is characterized by a constant expansion velocity independently of the strength of the interaction. In the case of spinless bosons, we study the expansion from initial states that have a fixed particle number per lattice site and a certain concentration of defects. We study the expansion velocity as a function of interaction strength and investigate whether the time-dependent momentum distribution functions indicate a dynamical quasi-condensation. The last example is the sudden expansion of a spin-polarized gas of fermions in the presence of attractive interactions. This study is motivated by current effort to experimentally detect the Fulde-Ferrell-Larkin-Ovchinnikov state. Our results for the time-dependent momentum distribution functions and the wave-function of the pair condensate suggest that the signatures of the FFLO state vanish quickly, yet a stationary form of the momentum distribution also emerges fast. The latter is shown to be determined by the initial conditions, which might eventually allow for an indirect detection of the FFLO phase.

This thesis is concerned with two main topics: first, the advancement of the density matrix renormalization group (DMRG) and, second, its applications. In the first project of this thesis we exploit the common mathematical structure of the numerical renormalization group and the DMRG, namely, matrix product states (MPS), to implement an efficient numerical treatment of a two-lead, multi-level Anderson impurity model. By adopting a star-like geometry, where each species (spin and lead) of conduction electrons is described by its own so-called Wilson chain, instead of a single Wilson chain we achieve a very significant reduction in the numerical resources required to obtain reliable results. Moreover, we show that it is possible to find an "optimal" chain basis, in which chain degrees of freedom of different Wilson chains become effectively decoupled from each other further out on the Wilson chains. This basis turns out to also diagonalize the model's chain-to-chain scattering matrix. In the second project we show that Chebychev expansions offer numerically efficient representations for calculating spectral functions of one-dimensional lattice models using MPS methods. The main features of this Chebychev matrix product state (CheMPS) approach are: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it offers a well-controlled broadening scheme; (iii) it is based on using MPS tools to recursively calculate a succession of Chebychev vectors, (iv) whose entanglement entropies were found to remain bounded with increasing recursion order for all cases analyzed here. We present CheMPS results for the structure factor of spin-1/2 antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to benchmark methods, we find that CheMPS yields results comparable in quality to those of correction vector DMRG, at dramatically reduced numerical cost and agrees well with Bethe Ansatz results for an infinite system, within the limitations expected for numerics on finite systems. Following these technologically focused projects we study the so-called Kondo cloud by means of the DMRG in the third project. The Kondo cloud describes the effect of spatially extended spin-spin correlations of a magnetic moment and the conduction electrons which screen the magnetic moment through the Kondo effect at low temperatures. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. We investigate how perturbations which destroy the Kondo effect, like an applied gate potential or a magnetic field, affect the formation of the screening cloud. In a forth project we address the impact of Quantum (anti-)Zeno physics resulting from repeated single-site resolved observations on the many-body dynamics. We use time-dependent DMRG to obtain the time evolution of the full many-body wave function that is then periodically projected in order to simulate realizations of stroboscopic measurements. For the example of a 1-D lattice of spin-polarized fermions with nearest-neighbor interactions, we find regimes for which many-particle configurations are stabilized and destabilized depending on the interaction strength and the time between observations.

This thesis contributes to the field of strongly correlated electron systems with studies in two distinct fields thereof: the specific nature of correlations between electrons in one dimension and quantum quenches in quantum impurity problems. In general, strongly correlated systems are characterized in that their physical behaviour needs to be described in terms of a many-body description, i.e. interactions correlate all particles in a complex way. The challenge is that the Hilbert space in a many-body theory is exponentially large in the number of particles. Thus, when no analytic solution is available - which is typically the case - it is necessary to find a way to somehow circumvent the problem of such huge Hilbert spaces. Therefore, the connection between the two studies comes from our numerical treatment: they are tackled by the density matrix renormalization group (DMRG) [1] and the numerical renormalization group (NRG) [2], respectively, both based on matrix product states. The first project presented in this thesis addresses the problem of numerically finding the dominant correlations in quantum lattice models in an unbiased way, i.e. without using prior knowledge of the model at hand. A useful concept for this task is the correlation density matrix (CDM) [3] which contains all correlations between two clusters of lattice sites. We show how to extract from the CDM, a survey of the relative strengths of the system’s correlations in different symmetry sectors as well as detailed information on the operators carrying long-range correlations and the spatial dependence of their correlation functions. We demonstrate this by a DMRG study of a one-dimensional spinless extended Hubbard model [4], while emphasizing that the proposed analysis of the CDM is not restricted to one dimension. The second project presented in this thesis is motivated by two phenomena under ongoing experimental and theoretical investigation in the context of quantum impurity models: optical absorption involving a Kondo exciton [5, 6, 7] and population switching in quantum dots [8, 9, 10, 11, 12, 13, 14, 15]. It turns out that both phenomena rely on the various manifestations of Anderson orthogonality (AO) [16], which describes the fact that the response of the Fermi sea to a quantum quench (i.e. an abrupt change of some property of the impurity or quantum dot) is a change of the scattering phase shifts of all the single-particle wave functions, therefore drastically changing the system. In this context, we demonstrate that NRG, a highly accurate method for quantum impurity models, allows for the calculation of all static and dynamic quantities related to AO and present an extensive NRG study for population switching in quantum dots.

The main topic of this thesis is the study of many-body effects in strongly correlated one- or quasi one-dimensional condensed matter systems. These systems are characterized by strong quantum and thermal fluctuations, which make mean-field methods fail and demand for a fully numerical approach. Fortunately, a numerical method exist, which allows to treat unusually large one -dimensional system at very high precision. This method is the density-matrix renormalization group method (DMRG), introduced by Steve White in 1992. Originally limited to the study of static problems, time-dependent DMRG has been developed allowing one to investigate non-equilibrium phenomena in quantum mechanics. In this thesis I present the solution of three conceptionally different problems, which have been addressed using mostly the Krylov-subspace version of the time-dependent DMRG. My findings are directly relevant to recent experiments with ultracold atoms, also carried out at LMU in the group of Prof. Bloch. The first project aims the ultimate goal of atoms in optical lattices, namely, the possibility to act as a quantum simulator of more complicated condensed matter system. The underline idea is to simulate a magnetic model using ultracold bosonic atoms of two different hyperfine states in an optical superlattice. The system, which is captured by a two-species Bose-Hubbard model, realizes in a certain parameter range the physics of a spin-1/2 Heisenberg chain, where the spin exchange constant is given by second order processes. Tuning of the superlattice parameters allows one to controlling the effect of fast first order processes versus the slower second order ones. The analysis is motivated by recent experiments, %cite{Folling2007,Trotzky2008} where coherent two-particle dynamics with ultracold bosonic atoms in isolated double wells were detected. My project investigates the coherent many-particle dynamics, which takes place after coupling the double well. I provide the theoretical background for the next step, the observation of coherent many-particle dynamics after coupling the double wells. The tunability between the Bose-Hubbard model and the Heisenberg model in this setup could be used to study experimentally the differences in equilibration processes for non-integrable and Bethe ansatz integrable models. It turns out that the relaxation in the Heisenberg model is connected to a phase averaging effect, which is in contrast to the typical scattering driven thermalization in nonintegrable models In the second project I study a many-body generalization of the original Landau-Zener formula. This formula gives the transition probability between the two states of a quantum mechanical two-level system, where the offset between the two levels is varying linearly in time. In a recent experiment this framework has been extended to a many-body system consisting of pairwise tunnel-coupled one-dimensional Bose liquids. It was found that the tunnel coupling between the tubes and the intertube interactions strongly modify the original Landau-Zener picture. After a introduction to the two-level and the three-level Landau-Zener problem I present my own results for the quantum dynamics of the microscopic model and the comparison to the experimental results. I have calculated both Landau-Zener sweeps as well as the time-evolution after sudden quenches of the energy offset. A major finding is that for sufficiently large initial density quenches can be efficiently used to create quasi-thermal states of arbitrary temperatures. The third project is more mathematical and connects the fields of quantum computation and of quantum information. Here, the main purpose is to analyse systematically the effects of decoherence on maximally entangled multi-partite states, which arise typically during quantum computation processes. The bigger the number of entangled qubits the more fragile is its entanglement under the influence decoherence. As starting point I consider first two entangled qubits, whereby one qubit interacts with an arbitrary environment. For this particular case I have derived a factorization law for the disentanglement. Next, I calculate the decrease of entanglement of two , three and four entangled qubits, general $W$- and general $GHZ$-state, coupled to a global spin-$1/2$ bath or several independent spin-$1/2$ baths , one for each qubit. Although there is no appropriate entanglement measure for three and more qubits, it turns out that this decrease is directly related to the increase of entanglement between the central system and the bath. This implies the formation of a much bigger multipartite entangled network. Thus, using the von Neumann entropy and the Wootters concurrence, I derive a simple upper bound for the bath-induced entanglement breaking power of the initially maximally entangled multi-partite states.