Error estimates for extrapolations with matrix-product states

We introduce a new error measure for matrix-product states without requiring the relatively costly two-site density matrix renormalization group (2DMRG). This error measure is based on an approximation of the full variance $\langle \psi | ( \hat H - E )^2 |\psi \rangle$. When applied to a series of matrix-product states at different bond dimensions obtained from a single-site density matrix renormalization group (1DMRG) calculation, it allows for the extrapolation of observables towards the zero-error case representing the exact ground state of the system. The calculation of the error measure is split into a sequential part of cost equivalent to two calculations of $\langle \psi | \hat H | \psi \rangle$ and a trivially parallelized part scaling like a single operator application in 2DMRG. The reliability of the new error measure is demonstrated at four examples: the $L=30, S=\frac{1}{2}$ Heisenberg chain, the $L=50$ Hubbard chain, an electronic model with long-range Coulomb-like interactions and the Hubbard model on a cylinder of size $10 \times 4$. Extrapolation in the new error measure is shown to be on-par with extrapolation in the 2DMRG truncation error or the full variance $\langle \psi | ( \hat H - E )^2 |\psi \rangle$ at a fraction of the computational effort.Comment: 10 pages, 11 figure

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

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, 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.Comment: 8.5 pages, 11 figure

Generic Construction of Efficient Matrix Product Operators

Matrix Product Operators (MPOs) are at the heart of the second-generation Density Matrix Renormalisation Group (DMRG) algorithm formulated in Matrix Product State language. We first summarise the widely known facts on MPO arithmetic and representations of single-site operators. Second, we introduce three compression methods (Rescaled SVD, Deparallelisation and Delinearisation) for MPOs and show that it is possible to construct efficient representations of arbitrary operators using MPO arithmetic and compression. As examples, we construct powers of a short-ranged spin-chain Hamiltonian, a complicated Hamiltonian of a two-dimensional system and, as proof of principle, the long-range four-body Hamiltonian from quantum chemistry.Comment: 13 pages, 10 figure

Dynamical topological quantum phase transitions in nonintegrable models

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.Comment: 5+1 pages, 4+1 figure

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

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, whose groundstates would be visited in an appropriately defined charge pump. First, we investigate the grandcanonical 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 conceptual explanation and argue that these properties could be observed in quantum-gas experiments

Evaluation of time-dependent correlators after a local quench in iPEPS: hole motion in the t-J model

Infinite projected entangled pair states (iPEPS) provide a convenient variational description of infinite, translationally-invariant two-dimensional quantum states. However, the simulation of local excitations is not directly possible due to the translationally-invariant ansatz. Furthermore, as iPEPS are either identical or orthogonal, expectation values between different states as required during the evaluation of non-equal-time correlators are ill-defined. Here, we show that by introducing auxiliary states on each site, it becomes possible to simulate both local excitations and evaluate non-equal-time correlators in an iPEPS setting under real-time evolution. We showcase the method by simulating the t-J model after a single hole has been placed in the half-filled antiferromagnetic background and evaluating both return probabilities and spin correlation functions, as accessible in quantum gas microscopes.Comment: 12 pages, 5 figures, minor revision requested by SciPost Physic

Spinon confinement in a quasi one dimensional anisotropic Heisenberg magnet

Confinement is a process by which particles with fractional quantum numbers bind together to form quasiparticles with integer quantum numbers. The constituent particles are confined by an attractive interaction whose strength increases with increasing particle separation and as a consequence, individual particles are not found in isolation. This phenomenon is well known in particle physics where quarks are confined in baryons and mesons. An analogous phenomenon occurs in certain magnetic insulators; weakly coupled chains of spins S=1/2. The collective excitations in these systems is spinons (S=1/2). At low temperatures weak coupling between chains can induce an attractive interaction between pairs of spinons that increases with their separation and thus leads to confinement. In this paper, we employ inelastic neutron scattering to investigate the spinon confinement in the quasi-1D S=1/2 XXZ antiferromagnet SrCo2V2O8. Spinon excitations are observed above TN in quantitative agreement with established theory. Below TN the pairs of spinons are confined and two sequences of meson-like bound states with longitudinal and transverse polarizations are observed. Several theoretical approaches are used to explain the data. A new theoretical technique based on Tangent-space Matrix Product States gives a very complete description of the data and provides good agreement not only with the energies of the bound modes but also with their intensities. We also successfully explained the effect of temperature on the excitations including the experimentally observed thermally induced resonance between longitudinal modes below TN ,and the transitions between thermally excited spinon states above TN. In summary, our work establishes SrCo2V2O8 as a beautiful paradigm for spinon confinement in a quasi-1D quantum magnet and provides a comprehensive picture of this process.Comment: 17 pages, 18 figures, submitted to PR

Idiographische und nomothetische Forschung in wissenschaftstheoretischer Sicht

Die Biographik als Forschungsrichtung rekonstruiert also sowohl die Regeln, die in Biographien exemplifiziert sind, als auch die spezifische Art, wie diese Regeln im Leben des Biographierten modifiziert werden. Allerdings wird diese Blickrichtung nicht durch einen direkten Bezug zum "Leben" ermöglicht, denn dieses liegt ja nicht als Gegenstand, sondern nur in Zeichen vor. Auch dieser Blick ist also auf das Zeichenmaterial der Biographie angewiesen, sieht es jedoch unter dem Gesichtspunkt, der gerade nicht auf die Einstimmigkeit des Regelvollzug abhebt, sondern auf die Vielschichtigkeit und Widersprüchlichkeit der Äußerungen, in denen diese Regeln modifiziert werden

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

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

Gaussian time-dependent variational principle for the Bose-Hubbard model

We systematically extend Bogoliubov theory beyond the mean field approximation of the Bose-Hubbard model in the superfluid phase. Our approach is based on the time dependent variational principle applied to the family of all Gaussian states (i.e. Gaussian TDVP). First, we find the best ground state approximation within our variational class using imaginary time evolution in 1d, 2d and 3d. We benchmark our results by comparing to Bogoliubov theory and DMRG in 1d. Second, we compute the approximate 1- and 2-particle excitation spectrum as eigenvalues of the linearized projected equations of motion (linearized TDVP). We find the gapless Goldstone mode, a continuum of 2-particle excitations and a doublon mode. We discuss the relation of the gap between Goldstone mode and 2-particle continuum to the excitation energy of the Higgs mode. Third, we compute linear response functions for perturbations describing density variation and lattice modulation and discuss their relations to experiment. Our methods can be applied to any perturbations that are linear or quadratic in creation/annihilation operators. Finally, we provide a comprehensive overview how our results are related to well-known methods, such as traditional Bogoliubov theory and random phase approximation