Publications

Order by: publication date number of citations

• The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

  Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Jünemann, Daniel Jaschke, Matteo Rizzi, Simone Montangero

  SciPost Phys. Lect. Notes 8 (2019) · published 18 March 2019 | Toggle abstract · pdf

  We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

  65 citations

• Emergence of curved light-cones in a class of inhomogeneous Luttinger liquids

  Jérôme Dubail, Jean-Marie Stéphan, Pasquale Calabrese

  SciPost Phys. 3, 019 (2017) · published 6 September 2017 | Toggle abstract · pdf

  The light-cone spreading of entanglement and correlation is a fundamental and ubiquitous feature of homogeneous extended quantum systems. Here we point out that a class of inhomogenous Luttinger liquids (those with a uniform Luttinger parameter $K$) at low energy display the universal phenomenon of curved light cones: gapless excitations propagate along the geodesics of the metric $ds^2=dx^2+v(x)^2 d\tau^2$, with $v(x)$ being the calculable spatial dependent velocity induced by the inhomogeneity. We confirm our findings with explicit analytic and numerical calculations both in- and out-of-equilibrium for a Tonks-Girardeau gas in a harmonic potential and in lattice systems with artificially tuned hamiltonian density.

  64 citations

• Search for non-Abelian Majorana braiding statistics in superconductors

  C. W. J. Beenakker

  SciPost Phys. Lect. Notes 15 (2020) · published 4 August 2020 | Toggle abstract · pdf

  This is a tutorial review of methods to braid the world lines of non-Abelian anyons (Majorana zero-modes) in topological superconductors. That "Holy Grail" of topological quantum information processing has not yet been reached in the laboratory, but there now exists a variety of platforms in which one can search for the Majorana braiding statistics. After an introduction to the basic concepts of braiding we discuss how one might be able to braid immobile Majorana zero-modes, bound to the end points of a nanowire, by performing the exchange in parameter space, rather than in real space. We explain how Coulomb interaction can be used to both control and read out the braiding operation, even though Majorana zero-modes are charge neutral. We ask whether the fusion rule might provide for an easier pathway towards the demonstration of non-Abelian statistics. In the final part we discuss an approach to braiding in real space, rather than parameter space, using vortices injected into a chiral Majorana edge mode as "flying qubits".

  63 citations

• Entanglement evolution and generalised hydrodynamics: interacting integrable systems

  Vincenzo Alba, Bruno Bertini, Maurizio Fagotti

  SciPost Phys. 7, 005 (2019) · published 8 July 2019 | Toggle abstract · pdf

  We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable systems, where the two rates coincide.

  60 citations

• Quantum quenches to the attractive one-dimensional Bose gas: exact results

  Lorenzo Piroli, Pasquale Calabrese, Fabian H. L. Essler

  SciPost Phys. 1, 001 (2016) · published 14 September 2016 | Toggle abstract · pdf

  We study quantum quenches to the one-dimensional Bose gas with attractive interactions in the case when the initial state is an ideal one-dimensional Bose condensate. We focus on properties of the stationary state reached at late times after the quench. This displays a finite density of multi-particle bound states, whose rapidity distribution is determined exactly by means of the quench action method. We discuss the relevance of the multi-particle bound states for the physical properties of the system, computing in particular the stationary value of the local pair correlation function $g_2$.

  59 citations

• Krotov: A Python implementation of Krotov's method for quantum optimal control

  Michael H. Goerz, Daniel Basilewitsch, Fernando Gago-Encinas, Matthias G. Krauss, Karl P. Horn, Daniel M. Reich, Christiane P. Koch

  SciPost Phys. 7, 080 (2019) · published 12 December 2019 | Toggle abstract · pdf

  We present a new open-source Python package, krotov, implementing the quantum optimal control method of that name. It allows to determine time-dependent external fields for a wide range of quantum control problems, including state-to-state transfer, quantum gate implementation and optimization towards an arbitrary perfect entangler. Krotov's method compares to other gradient-based optimization methods such as gradient-ascent and guarantees monotonic convergence for approximately time-continuous control fields. The user-friendly interface allows for combination with other Python packages, and thus high-level customization.

  58 citations

• Inhomogeneous quenches in the transverse field Ising chain: scaling and front dynamics

  Márton Kormos

  SciPost Phys. 3, 020 (2017) · published 8 September 2017 | Toggle abstract · pdf

  We investigate the non-equilibrium dynamics of the transverse field quantum Ising chain evolving from an inhomogeneous initial state given by joining two macroscopically different semi-infinite chains. We obtain integral expressions for all two-point correlation functions of the Jordan-Wigner Majorana fermions at any time and for any value of the transverse field. Using this result, we compute analytically the profiles of various physical observables in the space-time scaling limit and show that they can be obtained from a hydrodynamic picture based on ballistically propagating quasiparticles. Going beyond the hydrodynamic limit, we analyze the approach to the non-equilibrium steady state and find that the leading late time corrections display a lattice effect. We also study the fine structure of the propagating fronts which are found to be described by the Airy kernel and its derivatives. Near the front we observe the phenomenon of energy back-flow where the energy locally flows from the colder to the hotter region.

  58 citations

• Geometry of variational methods: dynamics of closed quantum systems

  Lucas Hackl, Tommaso Guaita, Tao Shi, Jutho Haegeman, Eugene Demler, J. Ignacio Cirac

  SciPost Phys. 9, 048 (2020) · published 8 October 2020 | Toggle abstract · pdf

  We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: K\"ahler and non-K\"ahler. Traditional variational methods typically require the variational family to be a K\"ahler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-K\"ahler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

  52 citations

• Survival probability in Generalized Rosenzweig-Porter random matrix ensemble

  Giuseppe De Tomasi, Mohsen Amini, Soumya Bera, Ivan M. Khaymovich, Vladimir E. Kravtsov

  SciPost Phys. 6, 014 (2019) · published 30 January 2019 | Toggle abstract · pdf

  We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability $R(t)$, the probability of finding the initial state after time $t$. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that $R(t)$ can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified in different phases. The ergodic phase is characterized by the standard power-law decay of $R(t)$ with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of $R(t\to\infty)=k$, finite in the thermodynamic limit $N\rightarrow\infty$, which approaches $k=R(t\to 0)$ in this limit.

  52 citations

• Multifractality and its role in anomalous transport in the disordered XXZ spin-chain

  David J. Luitz, Ivan M. Khaymovich, Yevgeny Bar Lev

  SciPost Phys. Core 2, 006 (2020) · published 30 April 2020 | Toggle abstract · pdf

  The disordered XXZ model is a prototype model of the many-body localization transition (MBL). Despite numerous studies of this model, the available numerical evidence of multifractality of its eigenstates is not very conclusive due severe finite size effects. Moreover it is not clear if similarly to the case of single-particle physics, multifractal properties of the many-body eigenstates are related to anomalous transport, which is observed in this model. In this work, using a state-of-the-art, massively parallel, numerically exact method, we study systems of up to 24 spins and show that a large fraction of the delocalized phase flows towards ergodicity in the thermodynamic limit, while a region immediately preceding the MBL transition appears to be multifractal in this limit. We discuss the implication of our finding on the mechanism of subdiffusive transport.

  51 citations