Ulrich Schollwöck

Theoretical Nanophysics

Ludwig-Maximilians-Universität München

Theresienstr. 37

80333 Munich

schollwoeck[at]lmu.de

Group Webpage

Description

Research focus: computational methods, condensed-matter and quantum many-body physics

The research interests of the group of Uli Schollwöck focus on the computational study of many-body systems that show strong interaction effects such that conventional many-body perturbation techniques mostly do not work. Such systems range from quantum spin chains, frustrated quantum magnets through low-dimensional superconductors and conjugate polymers to ultracold atom gases in optical lattices, which are particularly important for us: The merger between traditional condensed matter physics and quantum optics is a special focus of our work.

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While we have focused previously on the equilibrium and linear response regime, more advanced techniques now allow us to go very far from equilibrium, where physics is very poorly understood: we are looking at quantum quenches, expansion experiments, transport properties and relaxation physics.

The dominant role of quantum and interaction effects usually implies that no simple analytical approximations are available. More advanced techniques, such as bosonization in one dimension or field theoretical approaches in arbitrary dimensions, have to be used. In the absence of a large number of exact solutions the ultimate test is in numerical approaches, which are a central part of our efforts. In low dimensional quantum systems, methods of choice are the density-matrix renormalization group (DMRG), which is a variational method within the ansatz class of matrix product states (MPS), and arguably the most powerful method for one-dimensional quantum systems, and tensor network states (TNS). Here our interest is in both applying and developing these methods with a special emphasis on the guidance provided by insights from quantum information theory.

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In the context of out-of-equilibrium physics, recent highlights of our research include the first dynamical quantum simulator, which resulted from a collaboration with the Bloch group, the development of expansion dynamics in the presence of optical lattices as a tool to study interactions and probe integrability vs. non-integrability in quantum systems, and the dynamics of (spin) impurities in magnets simulated by ultracold atoms.

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We are also pushing applications of conventional DMRG to two-dimensional quantum systems: while this is an exponentially hard task in the thermodynamic limit, the very high numerical stability of the method allows us to obtain very precise results on such quantum systems nevertheless. A recent highlight was the identification of the topological nature of the spin liquid ground state of a key model in frustrated magnetism, the kagome Heisenberg model, where we determined the topological entanglement entropy.

Starting from the density-matrix renormalization group (DMRG), we are also interested in merging this method with the dynamical mean-field theory (DMFT) and in developing new algorithms at finite temperature and for higher dimensions.



Featured

Publications

BaOsO3: A Hund's metal in the presence of strong spin-orbit coupling

M. Bramberger, J. Mravlje, M. Grundner, U. Schollwöck, M. Zingl

Physical Review B 103, 165133 (2021).

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We investigate the 5d transition metal oxide BaOsO3 within a combination of density functional theory and dynamical mean-field theory, using a matrix-product-state impurity solver. BaOsO3 has four electrons in the t2g shell akin to ruthenates but stronger spin-orbit coupling (SOC) and is thus expected to reveal an interplay of Hund's metal behavior with SOC. We explore the paramagnetic phase diagram as a function of SOC and Hubbard interaction strengths, identifying metallic, band (van Vleck) insulating, and Mott insulating regions. At the physical values of the two couplings, we find that BaOsO3 is still situated inside the metallic region and has a moderate quasiparticle renormalization m∗/m≈2, consistent with specific heat measurements. SOC leads to a splitting of a van Hove singularity close to the Fermi energy and a subsequent reduction of electronic correlations (found in the vanishing SOC case), but the SOC strength is insufficient to push the material into an insulating van Vleck regime. In spite of the strong effect of SOC, BaOsO3 can be best pictured as a moderately correlated Hund's metal.

DOI: 10.1103/PhysRevB.103.165133

Bosonic Pfaffian State in the Hofstadter-Bose-Hubbard Model

F. A. Palm, M. Buser, J. Léonard, M. Aidelsburger, U. Schollwöck, F. Grusdt

Physical Review B 103, L161101 (2021).

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Topological states of matter, such as fractional quantum Hall states, are an active field of research due to their exotic excitations. In particular, ultracold atoms in optical lattices provide a highly controllable and adaptable platform to study such new types of quantum matter. However, finding a clear route to realize non-Abelian quantum Hall states in these systems remains challenging. Here we use the density-matrix renormalization-group (DMRG) method to study the Hofstadter-Bose-Hubbard model at filling factor ν=1 and find strong indications that at α=1/6 magnetic flux quanta per plaquette the ground state is a lattice analog of the continuum non-Abelian Pfaffian. We study the on-site correlations of the ground state, which indicate its paired nature at ν=1, and find an incompressible state characterized by a charge gap in the bulk. We argue that the emergence of a charge density wave on thin cylinders and the behavior of the two- and three-particle correlation functions at short distances provide evidence for the state being closely related to the continuum Pfaffian. The signatures discussed in this letter are accessible in current cold atom experiments and we show that the Pfaffian-like state is readily realizable in few-body systems using adiabatic preparation schemes.

DOI: 10.1103/PhysRevB.103.L161101

Probing the Hall Voltage in Synthetic Quantum Systems

M. Buser, S. Greschner, U. Schollwoeck, T. Giamarchi

Physical Review Letters 126 (3), 030501 (2021).

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YIn the context of experimental advances in the realization of artificial magnetic fields in quantum gases, we discuss feasible schemes to extend measurements of the Hall polarization to a study of the Hall voltage, allowing for direct comparison with solid state systems. Specifically, for the paradigmatic example of interacting flux ladders, we report on characteristic zero crossings and a remarkable robustness of the Hall voltage with respect to interaction strengths, particle fillings, and ladder geometries, which is unobservable in the Hall polarization. Moreover, we investigate the site-resolved Hall response in spatially inhomogeneous quantum phases.

DOI: 10.1103/PhysRevLett.126.030501

Concept of Orbital Entanglement and Correlation in Quantum Chemistry

L.X. Ding, S. Mardazad, S. Das, S. Szalay, U. Schollwoeck, Z. Zimboras, C. Schilling

Journal of Chemical Theory and Computation 17 (1), 79-95 (2021).

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A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.

DOI: 10.1021/acs.jctc.0c00559

Interacting bosonic flux ladders with a synthetic dimension: Ground-state phases and quantum quench dynamics

M. Buser, D. Hubig, U. Schollwoeck, L. Tarruell, F. Heidrich-Meisner

Physical Review A 102 (5), 053314 (2020).

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Flux ladders constitute the minimal setup enabling a systematic understanding of the rich physics of interacting particles subjected simultaneously to strong magnetic fields and a lattice potential. In this paper, the ground-state phase diagram of a flux-ladder model is mapped out using extensive density-matrix renormalization-group simulations. The emphasis is put on parameters which can be experimentally realized exploiting the internal states of potassium atoms as a synthetic dimension. The focus is on accessible observables such as the chiral current and the leg-population imbalance. Considering a particle filling of one boson per rung, we report the existence of a Mott-insulating Meissner phase as well as biased-ladder phases on top of superfluids and Mott insulators. Furthermore, we demonstrate that quantum quenches from suitably chosen initial states can be used to probe the equilibrium properties in the transient dynamics. Concretely, we consider the instantaneous turning on of hopping matrix elements along the rungs or legs in the synthetic flux-ladder model, with different initial particle distributions. We show that clear signatures of the biased-ladder phase can be observed in the transient dynamics. Moreover, the behavior of the chiral current in the transient dynamics is discussed. The results presented in this paper provide guidelines for future implementations of flux ladders in experimental setups exploiting a synthetic dimension.

DOI: 10.1103/PhysRevA.102.053314

Sr2MoO4 and Sr2RuO4: Disentangling the Roles of Hund's and van Hove Physics

J. Karp, M. Bramberger, M. Grundner, U. Schollwoeck, A.J. Millis, M. Zingl

Physical Review Letters 125 (16), 166401 (2020).

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Sr2MoO4 is isostructural to the unconventional superconductor Sr2RuO4 but with two electrons instead of two holes in the Mo/Ru-t(2g) orbitals. Both materials are Hund's metals, but while Sr2RuO4 has a van Hove singularity in close proximity to the Fermi surface, the van Hove singularity of Sr2MoO4 is far from the Fermi surface. By using density functional plus dynamical mean-field theory, we determine the relative influence of van Hove and Hund's metal physics on the correlation properties. We show that theoretically predicted signatures of Hund's metal physics occur on the occupied side of the electronic spectrum of Sr2MoO4, identifying Sr2MoO4 as an ideal candidate system for a direct experimental confirmation of the theoretical concept of Hund's metals via photoemission spectroscopy.

DOI: 10.1103/PhysRevLett.125.166401

Topological phases in the Fermi-Hofstadter-Hubbard model on hybrid-space ladders

L. Stenzel, A.L.C. Hayward, U. Schollwoeck, F. Heidrich-Meisner

Physical Review A 102 (2), 023315 (2020).

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In recent experiments with ultracold atoms, both two-dimensional (2D) Chern insulators and one-dimensional topological charge pumps have been realized. Without interactions, both systems can be described by the same Hamiltonian, when some variables are being reinterpreted. In this paper, we study the relation of both models when Hubbard interactions are added, using the density-matrix renormalization-group algorithm. To this end, we express the fermionic Hofstadter model in a hybrid-space representation, and define a family of interactions, which connects 1D Hubbard charge pumps to 2D Hubbard Chern insulators. We study a three-band model at particle density rho = 2/3, where the topological quantization of the 1D charge pump changes from Chern number C = 2 to C = -1 as the interaction strength increases. We find that the C = -1 phase is robust when varying the interaction terms on narrow-width cylinders. However, this phase does not extend to the limit of the 2D Hofstadter-Hubbard model, which remains in the C = 2 phase. We discuss the existence of both topological phases for the largest cylinder circumferences that we can access numerically. We note the appearance of a ferromagnetic ground state between the strongly interacting 1D and 2D models. For this ferromagnetic state, one can understand the C = -1 phase from a band structure argument. Our method for measuring the Hall conductivity could similarly be realized in experiments: We compute the current response to a weak, linear potential, which is applied adiabatically. The Hall conductivity converges to integer-quantized values for large system sizes, corresponding to the system's Chern number.

DOI: 10.1103/PhysRevA.102.023315

Absence of Superconductivity in the Pure Two-Dimensional Hubbard Model

M.P. Qin, C.M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwoeck, S.R. White, S.W. Zhang

Physical Review X 10 (3), 031016 (2020).

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We study the superconducting pairing correlations in the ground state of the doped Hubbard model-in its original form without hopping beyond nearest neighbor or other perturbing parameters-in two dimensions at intermediate to strong coupling and near optimal doping. The nature of such correlations has been a central question ever since the discovery of cuprate high-temperature superconductors. Despite unprecedented effort and tremendous progress in understanding the properties of this fundamental model, a definitive answer to whether the ground state is superconducting in the parameter regime most relevant to cuprates has proved exceedingly difficult to establish. In this work, we employ two complementary, state-of-the-art, many-body computational methods-constrained-path (CP) auxiliary-field quantum Monte Carlo (AFQMC) and density matrix renormalization group (DMRG) methods-deploying the most recent algorithmic advances in each. Systematic and detailed comparisons between the two methods are performed. The DMRG is extremely reliable on small width cylinders, where we use it to validate the AFQMC. The AFQMC is then used to study wide systems as well as fully periodic systems, to establish that we have reached the thermodynamic limit. The ground state is found to be nonsuperconducting in the moderate to strong coupling regime in the vicinity of optimal hole doping.

DOI: 10.1103/PhysRevX.10.031016

Plaquette versus ordinary d-wave pairing in the t '-Hubbard model on a width-4 cylinder

C.M. Chung, M.P. Qin, S.W. Zhang, U. Schollwoeck, S.R. White

Physical Review B 102 (4), 041106 (2020).

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The Hubbard model and its extensions are important microscopic models for understanding high-Tc superconductivity in cuprates. In the model with next-nearest-neighbor hopping t' (the t'-Hubbard model), pairing is strongly influenced by t'. In particular, a recent study on a width-4 cylinder observed quasi-long-range superconducting order, associated with a negative t', which was taken to imply superconductivity in the two-dimensional (2D) limit. In this work we study more carefully pairing in the width-4 t'-Hubbard model. We show that in this specific system, the pairing symmetry with t' < 0 is not the ordinary d-wave one would expect in the 2D limit. Instead we observe a so-called plaquette d-wave pairing. We show that the plaquette d-wave or its extension is difficult to generalize in other geometries, for example a 4-leg ladder with open boundaries or a width-6 cylinder. We find that a negative t' suppresses the conventional d-wave, leading to plaquette pairing. In contrast, a different t '' coupling acting diagonally on the plaquettes suppresses plaquette pairing, leading to conventional d-wave pairing.

DOI: 10.1103/PhysRevB.102.041106

Thermal Control of Spin Excitations in the Coupled Ising-Chain Material RbCoCl3

M. Mena, N. Hänni, S. Ward, E. Hirtenlechner, R. Bewley, C. Hubig, U. Schollwöck, B. Normand, K.W. Krämer, D.F. McMorrow, C. Rüegg

Physical Review Letters 124, 257201 (2020).

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We have used neutron spectroscopy to investigate the spin dynamics of the quantum (S=1/2) antiferromagnetic Ising chains in RbCoCl3. The structure and magnetic interactions in this material conspire to produce two magnetic phase transitions at low temperatures, presenting an ideal opportunity for thermal control of the chain environment. The high-resolution spectra we measure of two-domain-wall excitations therefore characterize precisely both the continuum response of isolated chains and the “Zeeman-ladder” bound states of chains in three different effective staggered fields in one and the same material. We apply an extended Matsubara formalism to obtain a quantitative description of the entire dataset, Monte Carlo simulations to interpret the magnetic order, and finite-temperature density-matrix renormalization-group calculations to fit the spectral features of all three phases.

DOI: 10.1103/PhysRevLett.124.257201

Imaginary-time matrix product state impurity solver in a real material calculation: Spin-orbit coupling in Sr2RuO4

N.O. Linden, M. Zingl, C. Hubig, O. Parcollet, U. Schollwoeck

Physical Review B 101 (4), 041101 (2020).

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Using an imaginary-time matrix-product state (MPS) based quantum impurity solver we perform a realistic dynamical mean-field theory (DMFT) calculation combined with density functional theory (DFT) for Sr2RuO4. We take the full Hubbard-Kanamori interactions and spin-orbit coupling (SOC) into account. The MPS impurity solver works at essentially zero temperature in the presence of SOC, a regime of parameters currently inaccessible to continuous-time quantum Monte Carlo methods, due to a severe sign problem. We show that earlier results obtained at high temperature, namely, that the diagonal self-energies are nearly unaffected by SOC and that interactions lead to an effective enhancement of the SOC, hold even at low temperature. We observe that realism makes the numerical solution of the impurity model with MPS much more demanding in comparison to earlier works on Bethe lattice models, requiring several algorithmic improvements.

DOI: 10.1103/PhysRevB.101.041101

Time-dependent density matrix renormalization group quantum dynamics for realistic chemical systems

X. Xie, Y. Liu, Y. Yao, U. Schollwöck, C. Liu, H. Ma

Journal of Chemical Physics 151 (22), 224101 (2019).

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Electronic and/or vibronic coherence has been found by recent ultrafast spectroscopy experiments in many chemical, biological, and material systems. This indicates that there are strong and complicated interactions between electronic states and vibration modes in realistic chemical systems. Therefore, simulations of quantum dynamics with a large number of electronic and vibrational degrees of freedom are highly desirable. Due to the efficient compression and localized representation of quantum states in the matrix-product state (MPS) formulation, time-evolution methods based on the MPS framework, which we summarily refer to as tDMRG (time-dependent density-matrix renormalization group) methods, are considered to be promising candidates to study the quantum dynamics of realistic chemical systems. In this work, we benchmark the performances of four different tDMRG methods, including global Taylor, global Krylov, and local one-site and two-site time-dependent variational principles (1TDVP and 2TDVP), with a comparison to multiconfiguration time-dependent Hartree and experimental results. Two typical chemical systems of internal conversion and singlet fission are investigated: one containing strong and high-order local and nonlocal electron-vibration couplings and the other exhibiting a continuous phonon bath. The comparison shows that the tDMRG methods (particularly, the 2TDVP method) can describe the full quantum dynamics in large chemical systems accurately and efficiently. Several key parameters in the tDMRG calculation including the truncation error threshold, time interval, and ordering of local sites were also investigated to strike the balance between efficiency and accuracy of results.

DOI: 10.1063/1.5125945

Time-evolution methods for matrix-product states

S. Packel, T. Kohler, A. Swoboda, S. Manmana, U. Schollwock, C. Hubig.

Annals of Physics 411, 167998 (2019).

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Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO W-I,W-II method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics.

DOI: 10.1016/j.aop.2019.167998

Time-evolution methods for matrix-product states

S. Paeckel, T. Köhler, A. Swoboda, S.R. Manmana, U. Schollwöck, C. Hubig

Annals of Physics 411, 167998 (2019).

Show Abstract

Matrix-product states have become the de facto standard for the representation of one-dimensional quantum many body states. During the last few years, numerous new methods have been introduced to evaluate the time evolution of a matrix-product state. Here, we will review and summarize the recent work on this topic as applied to finite quantum systems. We will explain and compare the different methods available to construct a time-evolved matrix-product state, namely the time-evolving block decimation, the MPO method, the global Krylov method, the local Krylov method and the one- and two-site time-dependent variational principle. We will also apply these methods to four different representative examples of current problem settings in condensed matter physics.

DOI: 10.1016/j.aop.2019.167998

Dynamical Topological Quantum Phase Transitions in Nonintegrable Models

I. Hagymasi, C. Hubig, O. Legeza, U. Schollwoeck

Physical Review Letters 122 (25), 250601 (2019).

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We consider sudden quenches across quantum phase transitions in the S = 1 XXZ model starting from the Haldane phase. We demonstrate that dynamical phase transitions may occur during these quenches that are identified by nonanalyticities in the rate function for the return probability. In addition, we show that the temporal behavior of the string order parameter is intimately related to the subsequent dynamical phase transitions. We furthermore find that the dynamical quantum phase transitions can be accompanied by enhanced two-site entanglement.

DOI: 10.1103/PhysRevLett.122.250601

Quantum phases and topological properties of interacting fermions in one-dimensional superlattices

L. Stenzel, A.L.C. Hayward, C. Hubig, U. Schollwöck, F. Heidrich-Meisner.

Physical Review A 99, 053614 (2019).

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The realization of artificial gauge fields in ultracold atomic gases has opened up a path towards experimental studies of topological insulators and, as an ultimate goal, topological quantum matter in many-body systems. As an alternative to the direct implementation of two-dimensional lattice Hamiltonians that host the quantum Hall effect and its variants, topological charge-pumping experiments provide an additional avenue towards studying many-body systems. Here, we consider an interacting two-component gas of fermions realizing a family of one-dimensional superlattice Hamiltonians with onsite interactions and a unit cell of three sites, the ground states of which would be visited in an appropriately defined charge pump. First, we investigate the grand canonical quantum phase diagram of individual Hamiltonians, focusing on insulating phases. For a certain commensurate filling, there is a sequence of phase transitions from a band insulator to other insulating phases (related to the physics of ionic Hubbard models) for some members of the manifold of Hamiltonians. Second, we compute the Chern numbers for the whole manifold in a many-body formulation and show that, related to the aforementioned quantum phase transitions, a topological transition results in a change of the value and sign of the Chern number. We provide both an intuitive and a conceptual explanation and argue that these properties could be observed in quantum-gas experiments.

DOI: 10.1103/PhysRevA.99.053614

Finite-temperature properties of interacting bosons on a two-leg flux ladder

M. Buser, F. Heidrich-Meisner, U. Schollwöck

Physical Review A 99, 053601 (2019).

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Quasi-one-dimensional lattice systems such as flux ladders with artificial gauge fields host rich quantum-phase diagrams that have attracted great interest. However, so far, most of the work on these systems has concentrated on zero-temperature phases while the corresponding finite-temperature regime remains largely unexplored. The question if and up to which temperature characteristic features of the zero-temperature phases persist is relevant in experimental realizations. We investigate a two-leg ladder lattice in a uniform magnetic field and concentrate our study on chiral edge currents and momentum-distribution functions, which are key observables in ultracold quantum-gas experiments. These quantities are computed for hard-core bosons as well as noninteracting bosons and spinless fermions at zero and finite temperatures. We employ a matrix-product-state based purification approach for the simulation of strongly interacting bosons at finite temperatures and analyze finite-size effects. Our main results concern the vortex-fluid-to-Meissner crossover of strongly interacting bosons. We demonstrate that signatures of the vortex-fluid phase can still be detected at elevated temperatures from characteristic finite-momentum maxima in the momentum-distribution functions, while the vortex-fluid phase leaves weaker fingerprints in the local rung currents and the chiral edge current. In order to determine the range of temperatures over which these signatures can be observed, we introduce a suitable measure for the contrast of these maxima. The results are condensed into a finite-temperature crossover diagram for hard-core bosons.

DOI: 10.1103/PhysRevA.99.053601

Density-matrix embedding theory study of the one-dimensional Hubbard-Holstein model

T.E. Reinhard, U. Mordovina, C. Hubig, J.S. Kretchmer, U. Schollwöck, H. Appel, M.Journal of Chemical Theory and ComputationJournal of Chemical Theory and ComputationA. Sentef, A. Rubio,

Journal of Chemical Theory and Computation 15 (4), 2221-2232 (2019).

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We present a density-matrix embedding theory (DMET) study of the one-dimensional Hubbard–Holstein model, which is paradigmatic for the interplay of electron–electron and electron–phonon interactions. Analyzing the single-particle excitation gap, we find a direct Peierls insulator to Mott insulator phase transition in the adiabatic regime of slow phonons in contrast to a rather large intervening metallic phase in the anti-adiabatic regime of fast phonons. We benchmark the DMET results for both on-site energies and excitation gaps against density-matrix renormalization group (DMRG) results and find good agreement of the resulting phase boundaries. We also compare the full quantum treatment of phonons against the standard Born–Oppenheimer (BO) approximation. The BO approximation gives qualitatively similar results to DMET in the adiabatic regime but fails entirely in the anti-adiabatic regime, where BO predicts a sharp direct transition from Mott to Peierls insulator, whereas DMET correctly shows a large intervening metallic phase. This highlights the importance of quantum fluctuations in the phononic degrees of freedom for metallicity in the one-dimensional Hubbard–Holstein model.

DOI: 10.1021/acs.jctc.8b01116

Interaction quench and thermalization in a one-dimensional topological Kondo insulator

I. Hagymási, C. Hubig, and U. Schollwöck

Physical Review B 99, 075145 (2019).

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We study the nonequilibrium dynamics of a one-dimensional topological Kondo insulator, modelled by a p-wave Anderson lattice model, following a quantum quench of the on-site interaction strength. Our goal is to examine how the quench influences the topological properties of the system, and therefore our main focus is the time evolution of the string order parameter, entanglement spectrum, and the topologically protected edge states. We point out that postquench local observables can be well captured by a thermal ensemble up to a certain interaction strength. Our results demonstrate that the topological properties after the interaction quench are preserved. Though the absolute value of the string order parameter decays in time, the analysis of the entanglement spectrum, Loschmidt echo and the edge states indicates the robustness of the topological properties in the time-evolved state. These predictions could be directly tested in state-of-the-art cold-atom experiments.

DOI: 10.1103/PhysRevB.99.075145

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