# Theory Seminar

 Date Seminars 2017   (Go to Seminars 2016, Seminars 2015, Seminars 2014, and earlier therein) 09.01.2017 Bipartite charge fluctuations in Z_2 topological insulators and superconductors Invited speaker: Loïc Herviou (Ecole Polytechnique, ENS) Bipartite charge fluctuations (BCF) have been introduced to provide an experimental indication of many-body entanglement.They are a very efficient and useful tool to characterize phase transitions in a large variety of charge-conserving models in one and two dimensions In this seminar, we study the BCF in generic one- and two-dimensional Z_2 (topological) models such as the Kitaev chain, spin-orbit insulators, the graphene and the Haldane model, where the charge we observe is no longer conserved. In one-dimension, we demonstrate that at phase transitions characterized by a linear dispersion, the BCF probe the change in a winding number that allows to pinpoint the transition and corresponds to the topological invariant for standard models. Additionally, we prove that a sub-dominant logarithmic contribution is still present at the exact critical point. Its quantized coefficient is universal and a characteristic of the critical model. In two dimensions, a similar structure appears. While the area term no longer reveal directly the phase transition, a subdominant logarithmic term is still present. Similarly to the entanglement entropy, it depends on the exact shape of the considered region, with contributions of the corner of the regions only. 25.01.2017 Approaching non-Abelian Lattice Gauge Theories with Tensor Networks Stefan Kühn In recent years the Tensor Network approach to lattice gauge theories has proven itself as promising alternative to the conventional Monte Carlo methods widely used. In contrast to Monte Carlo simulations, numerical methods based on Tensor networks do not suffer from the sign problem, thus allowing to address problems and parameter regimes which are inaccessible with Monte Carlo. However, even for for the simplest non-Abelian gauge models with dynamical fermions, the computational effort typically grows quickly. Hence, current Tensor Network simulations for non-Abelian models are rather limited. In this talk I will address the case of a SU(2) lattice gauge theory. I will show how, starting from a basis of color neutral states, the gauge field for systems on finite lattices with open boundary conditions can be integrated out, thus greatly reducing the degrees of freedom. While this formulation is completely general, it trivially allows to truncate the maximum color-electric flux in the system, thus making it particularly suitable for a Tensor Network approach. As a proof of principle I will present numerical results for the low lying spectrum obtained with Matrix Product States for a family of truncated SU(2) models. 01.02.2017 Decay of correlations in systems of fermions with long-range interactions at non-zero temperature Senaida Hernandez We study correlations in fermionic systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anti-commuting operators based on long-range Lieb-Robinson type bounds. Our result shows that correlations between such operators in fermionic long-range systems of spatial dimension $D$ with at most two-site interactions decaying algebraically with the distance with an exponent $\alpha \geq 2\,D$, decay at least algebraically with an exponent arbitrarily close to $\alpha$. Our bound is asymptotically tight, which we demonstrate by numerically analyzing density-density correlations in a 1D quadratic (free, exactly solvable) model, the Kitaev chain with long-range interactions. Away from the quantum critical point correlations in this model are found to decay asymptotically as slowly as our bound permits. 08.02.2017 An introduction to Variational Monte-Carlo Ivan Glasser Variational Monte-Carlo methods are used to determine the energy of a wave function and to optimize its parameters in order to approximate the ground state of a many-body quantum system. In this talk I will give a general introduction to Variational Monte-Carlo. Starting from the early days of Monte-Carlo integration I will explain how these methods can be used to compute energies of wave functions. I will then give an overview of modern methods to optimize the energy of a wave function with many parameters. 15.02.2017 Simulating non-Equilibrium systems with Matrix Product States Julian Roos Understanding out of equilibrium remains a challenge for classical and quantum systems. There is no general non-equilibrium statistical mechanics framework to resort to, if one is interested in the statistical properties of observables in far from equilibrium situations. The theory of large deviations can fill this gap in some cases and Tensor Networks are one possibility to explore this problem from a numerical side. Matrix Product States can capture the properties of non-equilibrium stationary states of many classical and quantum models. On Wednesday I will discuss how people have used these techniques for the simplest problem of particle hopping on a 1D lattice. 21.02.2017 at 14:00 Dissipation induced topological states: A recipe Invited speaker: Moshe Goldstein (Tel-Aviv University) It has recently been realized that driven-dissipative dynamics, which usually tends to destroy subtle quantum interference and correlation effects, could actually be used as a resource. By proper engineering of the reservoirs and their couplings, one may drive a system towards a desired quantum-correlated steady state, even in the absence of internal Hamiltonian dynamics. An intriguing class of quantum phases is characterized by topology, including the quantum Hall effect and topological insulators and superconductors. Which of these noninteracting topological states can be achieved as the result of purely dissipative Lindblad-type dynamics? Recent studies have only provided partial answers to this question. In this talk I will present a general recipe for the creation, classification, and detection of states of the integer quantum Hall and 2D topological insulator type as the outcomes of coupling a system to reservoirs, and show how the recipe can be realized with ultracold atoms and other quantum simulators. The mixed states so created can be made arbitrarily close to pure states. I will discuss ways to extend this construction to other topological phases, including non-Gaussian ones, such as fractional quantum Hall state. 01.03.2017 at 14:00 Characterizing many-body states at finite temperature via a Klein twist Invited speaker: Hong-Hao Tu (Ludwig-Maximilians-Universität - München ) In this talk, I will describe an ongoing work on how universal data for distinguishing different phases may be extracting from thermal states of quantum many-body systems. This approach relies on a Klein bottle partition function (defined by twisting the usual partition function in imaginary-time axis) and is relevant for situations where non-chiral conformal field theories govern the bulk or edge physics (e.g. 1d critical states and 2d time-reversal invariant topological insulators). Benchmark results will be provided for several 1d critical models. 02.03.2017 at 11:30 Majorana quasi-particles from angular momentum conservation Invited speaker: Fernardo Iemini (International Centre for Theoretical Physics - Trieste, Italy) We show how angular momentum conservation can stabilise a quasi-topological phase of matter supporting Majorana qausi-particles as edge modes. Differently from typical scenarios, where such quasi-particles require the presence of superconductivity, we investigate orbital SU (2) × SU (2) Hubbard models in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional models. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated with numerical simulations. Our results pave the way toward the observation of Majorana edge modes with Alkaline-earth-like fermions in optical lattices, where the basic ingredients for our recipe - spin-orbit coupling and strong inter-orbital interactions - have been observed over the last two years. 08.03.2017 Almost Conserved Local Operators in MBL systems Nicola Pancotti Long time dynamics of non-integrable systems holds the key to fundamental questions (thermalization). Analytical tools can only apply to particular cases (integrable models, perturbative regimes). Numerical simulations, limited in time, have found evidence of different time scales. A new numerical technique for constructing slowly evolving local operators was introduced by Kim et al. in Phys. Rev. E 92, 012128 (2015). Those operators have a small commutator with the Hamiltonian and they might give rise to long time scales. In this work, we apply this technique to the many body localization problem. We show that this method can not only signal the difference between the ergodic and localized phases, but it is also sensitive to the presence of the Griffith region between both. 15.03.2017 Efficient representation of fully many-body localized systems using tensor networks Invited speaker: Thorsten Wahl (Oxford University) Many-body localization (MBL) is currently an intensely studied topic and characterized by the fact that certain strongly disordered systems fail to thermalize. For sufficiently strong disorder in one dimension, all eigenstates of MBL systems fulfill the area law of entanglement. This makes tensor network states ideally suited to represent such fully many-body localized systems. Building on the ansatz proposed in Phys. Rev. B 94, 041116(R) (2016), I will present a tensor network that is able to capture the full set of eigenstates of such MBL systems efficiently: For a given system size, local observables can be approximated with an error that decreases as an inverse polynomial of the computational cost, which is an exponential improvement over the previous ansatz. If the system size is increased, the computational cost needs to grow only linearly with the system size in order to keep the accuracy fixed. The technique turns out to be highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the dynamical phase transition. Finally, the power of the technique is demonstrated on systems of 72 sites, where clear signatures of the phase transition can be seen. 22.03.2017 A generalisation of the injecitvity condition for PEPS Andras Molnar Projected Entangled Pair States (PEPS) is an ansatz believed to be suitable for analytical and numerical investigation of ground states of many-body Hamiltonians. To design a PEPS that admits certain (local) symmetries one has to understand when two different PEPS tensors give rise to the same state. This question in the full generality is however undecidable, it is therefore important to find relevant classes of tensors for which it can be answered. One such class is injcetive PEPS. Two injective PEPS describe the same state if and only if their tensors are related with a gauge transformation on the virtual space. Here we provide a generalisation of this class. This generalisation includes states that fail to be injective for purely geometrical reasons (so called corner problem). We show under which condition can two such states be equal. We also show that symmetries give rise to invertible Matrix Product Operators (MPO) on the boundary degrees of freedom. These MPOs can be used to assign an element of the third cohomology of the symmetry group to the state the same way as in the classification of the Symmetry Protected Topological (SPT) phases. 23.03.2017 at 14:00 DMRG with Subspace Expansion on Symmetry-Protected Tensor Networks Invited speaker: Claudius Hubig (Ludwig-Maximilians-Universität - München) The Density Matrix Renormalisation Group when applied to matrix- product states is the method of choice for ground-state search on one-dimensional systems and still highly competitive even in unfavourable circumstances, such as critical systems and higher dimensions. In this talk, I will discuss two separate methods which can be used to improve the computational efficiency of DMRG and related methods on matrix-product states and beyond. The first component is the implementation of both abelian and non-abelian symmetries in an entirely general way suitable also for higher-rank tensors as encountered in e.g. tree tensor network states. The second ingredient, the subspace expansion, allows for a fully single-site DMRG algorithm with favourable linear scaling in the local dimension of the tensor network. Even for common problems, this results in a considerable speed-up over the traditional two-site DMRG method or the density matrix perturbation approach for ground-state search at reduced algorithmic complexity. Additionally, the subspace expansion can potentially be used in a large set of other algorithms, such as the TDVP or the variational application of a matrix-product operator onto a matrix-product state. 27.03.2017 at 11:30 Decoding Protocols for Classical Communication on Quantum Channels Invited speaker: Matteo Rosati (Scuola Normale Superiore, Pisa, Italy ) I discuss the transmission of classical information via quantum carriers with focus on the decoding stage. While the optimal transmission rate and encoding have been well studied in the past providing viable solutions for free-space or optical-fiber communication, a practical decoder is still difficult to design. This is due to the requirement of performing joint measurements over several transmission modes and their difficult implementation. I approach the problem from several points of view presenting ideas for decoding algorithms and practical devices, especially for communication with coherent states of the electromagnetic field. 29.03.2017 High-Fidelity Hot Gates for Generic Spin-Resonator Systems Invited speaker: Martin Schütz (Harvard University) We propose and analyze a high-fidelity hot gate for generic spin-resonator systems which allows for coherent spin-spin coupling, in the presence of a thermally populated resonator mode. Our scheme is non-perturbative, applies to a broad class of physical systems, including for example spins coupled to circuit-QED and surface acoustic wave resonators as well as nanomechanical oscillators, and can be implemented readily with state-of-the-art experimental setups. We provide and numerically verify simple expressions for the fidelity of creating maximally entangled states under realistic conditions. 30.03.2017 at 14:00 Synthetic gauge fields: from topology to the Unruh effect Invited speaker: Alessio Celi (ICFO) The theoretical and experimental progress in quantum simulation with ultracold atoms of the last decay have pushed the realm of simulable models well deep into condensed matter and has started touching high-energy and gravitational physics. In this talk, I will focus on the possibilities opened by synthetic gauge fields. In the first part, after a brief review of "synthetic dimension" idea, I will discuss in which sense and to what extend a narrow Hofstadter model, as for instance obtained with synthetic lattices, displays the topological properties of a large 2D system pierced by the same flux. In particular, I will show that such narrow systems allow for a surprising accurate measurement of the Chern number through an appropriate Laughlin pump experiment. In the second part of the talk I will move to an apparently unrelated subject, the simulation of Dirac fermions in artificial curved spacetime. In fact, I will show that such simulation can be achieved through laser assisted tunneling and synthetic gauge fields. As an application, I will present our recent proposal for observing the Unruh effect with ultracold atoms. 10.04.2017 at 15:30 Quantum steerability as a nesting problem Invited speaker: Chau Nguyen (Max Planck Institute for the Physics of Complex Systems) We are to discuss quantum steerability, which was recently discovered as the third type of quantum nonlocality besides quantum nonseparability and Bell nonlocality. A bipartite quantum state is said to be steerable (from Alice's side) if the corresponding Einstein-Podolsky-Rosen steering experiment can be convincingly verified (from Alice's side). We show that the problem of determining the steerability of a bipartite quantum state can be stated as a nesting problem of convex objects in a linear space. Nesting criteria are then proposed. As the first application, we prove the conjecture on the steerability of T-states, confirming them as the first class of two-qubit states of lower symmetry than the Werner states of which the steerability can still be fully characterised. As the second application, we discuss our recent progress in understanding the long-standing question on the steerability of the Werner states beyond projective measurements. 11.04.2017 at 14:00 Computational power of symmetry-protected topological phases Invited speaker: David Stephen (University of British Columbia) In many-body physics, many essential properties of a quantum state are determined by the phase of matter in which it resides. Recent years have witnessed tremendous progress in the discovery and classification of quantum phases, and it is thus pertinent to ask: what can a phase of matter be used for? A standout example in quantum information processing is the use of topological phases for error-resilient quantum computation. The exchange statistics of anyonic excitations present in these phases determine the possible logical gates and also label the topological phase itself. In this talk, I will make a similar connection for the symmetry-protected topological (SPT) phases in one dimension. I will show that the computational power of quantum states, defined via their use as resources for measurement-based quantum computation (MBQC), is uniform within certain SPT phases. This uniform computational power is determined using the same algebraic structure that classifies the SPT phases, namely group cohomology. These results give insight into the structure of MBQC resource states, and highlight how the classification of quantum phases can contribute to our understanding of the power of quantum computation. 19.04.2017 A Projector Quantum Monte Carlo Method for non-linear wavefunctions Invited speaker: Lauretta Schwarz (University of Cambridge) The projected imaginary time evolution of Full Configuration Inter-action Quantum Monte Carlo (FCIQMC) can be reformulated in terms of a Lagrangian minimization which naturally admits polynomial complex wavefunction parameterizatons, thereby circumventing the exponential scaling of the FCIQMC approach. While previously these non-linear wavefunctions have traditionally been used in the area of Variational Monte Carlo, we consider recent developments for the identification of deep-learning neural networks to optimize this Lagrangian, which can be written as a modification of the propagator for the wavefunction dynamics. We demonstrate the capability of this approach with a Correlator Product State wavefunction, a form of Tensor Network State, and use it to find solutions to the strongly-correlated Hubbard model, as well as ab-initio systems, including a linear hydrogen chain and a fully periodic, ab-initio graphene sheet. The number of variables which can be simultaneously optimized greatly exceeds alternative formulations of Variational Monte Carlo, allowing for systematic improvability of the wavefunction flexibility towards exactness, whilst combining traditional Variational and Projector Quantum Monte Carlo approaches. 26.04.2017 Effective theory for correlations of states obtained from conformal field theory Benedikt Herwerth We study states of one and two-dimensional spin systems that are constructed as correlators within the conformal field theory of a massless, free boson. In 1d, these states are good descriptions of ground states of XXZ spin chains and in 2d, they are similar to lattice Laughlin states. We show that the zz correlations in these states are determined by an action that is a modification of the original free-boson theory. The truncation of this action to quadratic order provides a solvable, effective theory for the correlations in our states. The mass term in this effective theory explains the behavior of the correlations, which decay polynomially in 1d and at the edge of a 2d system and exponentially in the bulk of a 2d system. We test the validity of our approximation by comparing it to Monte Carlo computations. 03.05.2017 Projected Entangled Pair States wIth Virtual Symmetries Henrik Dreyer Topologically ordered quantum states are the first kind of long-range entangled states that we are beginning to understand. A complete understanding, however, has only been achieved at fine-tuned points (string-nets, quantum doubles, …). Meaningful (i.e., still topological) generalisations of these special points are hard to find. On the other hand, tensor network states have been used to find exact, efficient representations of these fine-tuned points in several cases. The idea of this work is to find a suitable generalisation of quantum double states in the tensor network language. To this end, I will introduce G-injective tensor network states for finite G and then study an example of a SU(2)-invariant class of tensors. In this case, topological order breaks down. I will sketch how we arrive at this conclusion and hint at the mechanism behind the transition. 10.05.2017 PEPS for chiral topological systems Anna Hackenbroich Projected entangled pair states (PEPS) in two spatial dimensions provide an efficient representation for the ground states of many non-chiral systems with instrinsic topological order such as quantum double and string net models. In this framework, a virtual symmetry of the local tensor leads to a non-vanishing topological entanglement entropy and ground state degeneracy on non-trivial surfaces. On the other hand it is much less clear whether PEPS can also describe chiral topological systems. In the case of non-interacting fermions it has been shown that such descriptions do not exist since any gaussian fermionic PEPS with topological features has algebraically decaying correlation functions. After providing some background on chiral topological phases and their possible PEPS representations I will focus on a PEPS describing a critical chiral topological spin liquid for spin 1/2 particles on a square lattice. In particular I will describe the construction of this state in terms of a local tensor with SU(2) symmetry, analyse its virtual symmetries and discuss the properties of the chiral mode appearing in its entanglement spectrum. 17.05.2017 Quantum power methods for ground state problems Yimin Ge Quantum computers are expected to have a deep impact in the simulation of large quantum systems. Of particular interest is the ability to study their ground states, which are classically often intractable. In this talk, I will present a new quantum algorithm for this task inspired by the classical power iteration method. The algorithm prepares a good approximation of the ground state using techniques which were recently developed in the context of QLSP solvers for implementing operators that have suitable Fourier or Chebyshev series representations. Unlike algorithms based on adiabatic evolution or naive phase estimation, this algorithm provides a certification of success. The runtime is quadratically better than the naive phase estimation method and polynomial in the classical iteration length, and moreover has little memory requirements. This makes it an attractive candidate for potential applications of small quantum computers. Joint work with J. Tura and J.I. Cirac. 07.06.2017 Gauge symmetry in MPS Ilya Kull I will describe our ongoing work regarding gauge symmetry in MPS. We are aiming at a classification of all translationally invariant MPS that have a local symmetry property with respect to given groups. Similarly to what has been understood for global symmetries, we investigate how the symmetry property of the state manifests itself in the tensor. I will present our results so far while trying to spare you the details. 19.06.2017 at 14:00 Representations in deep learning and quantum many-body physics Invited speaker: Dr. Peter Wittek (ICFO) Representation is of central importance in both quantum many-body physics and machine learning. Until the advent of deep learning, a key task in machine learning was feature engineering, that is, constructing a space of raw data that would allow a learning algorithm to identify patterns. We see similar 'hand-crafted' representations in physics: for instance, the entanglement spectra often reveals phase transitions. Deep architectures in machine learning automated the extraction of representation, and tensor networks fulfil a similar role in many-body physics. Results proving equivalence between the two paradigms are beginning to emerge. In this talk, we present work-in-progress results on the correspondence between hierarchical tensor networks and deep learning architectures. 20.06.2017 Tensor Network algorithms to study 1D and 2D quantum matter Invited speaker: Augustine Kshetrimayum (Johannes Gutenberg University, Mainz) Tensor Network (TN) algorithms have become increasingly popular in the study of quantum many-body systems recently. In these techniques, the coefficients of the wave function of the quantum many-body states are written as a network of individual tensors based on the amount and structure of entanglement present in them. In this talk, I will summarize some of the works I have been doing in this direction by using these modern techniques in 1D and 2D systems. In particular, I will use Matrix Product States (MPS) to investigate a spin-2 quantum chain and show the emergence of different effective spin-1 ‘Haldane-like’ Symmetry Protected Topological (SPT) phases and their phase transitions. I will then show how I use the iPEPS (infinite Projected Entangled Pair States) algorithm to study frustrated systems such as the kagome lattice for the XXZ model. I will also discuss a new algorithm which we proposed recently based on iPEPS to study dissipative open quantum systems in 2D. In addition, I will mention some possible future work based on these results. 21.06.2017 Anomalies and entanglement renormalization Invited speaker: Jacob Bridgeman (University of Sydney) We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It is numerically demonstrated that this line is a protected gapless phase. Finally, we implement a duality transformation between a pair of critical lines using our subclass of MERA. 22.06.2017 at 14:00 Doped resonating valence bond states: a quantum information study Invited speaker: Sudipto Singha (Harish-Chandra Research Institute, Allahabad, India) Resonating valence bond states have played a crucial role in the description of exotic phases in strongly correlated systems, especially in the realm of Mott insulators and the associated high­-Tc superconducting phase transition. In particular, RVB states are considered to be an important system to study the ground state properties of the doped quantum spin­-1/2 ladder. It is therefore interesting to understand how quantum correlations are distributed among the constituents of these composite systems. In this regard, we formulate an analytical recursive method to generate the wave function of doped short­ range resonating valence bond (RVB) states as a tool to efficiently estimate multi-site entanglement as well as other physical quantities in doped quantum spin ladders. Importantly, our results show that within a specific doping concentration and model parameter regimes, the doped RVB state essentially characterizes the trends of genuine multi-party entanglement in the exact ground states of a Hubbard model with large onsite interactions. Moreover, we consider an isotropic RVB network of spin-­1/2 particles with a finite fraction of defects, where the corresponding wave function of the network is rotationally invariant under the action of local unitaries. By using quantum information­ theoretic concepts like strong sub-additivity of von Neumann entropy and approximate quantum telecloning, we prove analytically that in the presence of defects, caused by loss of a finite fraction of spins, the RVB network sustains genuine multi-site entanglement, and at the same time may exhibit finite moderate­-range bipartite entanglement, in contrast to the case with no defects. 29.08.2017 Solving the pentagon equation with trivalent categories Invited speaker: Ramona Wolf (Leibniz Universität Hannover) Anyons are two-dimensional quasiparticles with exotic statistics that can be used for topological quantum computation. What makes them interesting for this is the concept of fusion and braiding. In the first part of this talk I will give an introduction to the fusion theory of anyons including the so-called pentagon equation, which is crucial for finding consistent anyon models, and point out the difficulties when solving it. The second part of this talk is dedicated to the mathematical construct of trivalent categories, their connection to anyon models and how we can use them to make the process of solving the pentagon equation easier for particular models. 06.09.2017 Non-Standard Bose-Hubbard Model with State-Dependent Tunneling Invited speaker: Daniel Gonzalez Cuadra (ICFO, Barcelona) Non-standard Bose-Hubbard models are relevant for the study of strongly correlated quantum physics thanks to their rich phase diagram and the possibility to simulate them using ultracold atoms in optical lattices. In this talk, I will present a model where the tunnelling coefficient of bosons between neighboring sites depends on the state of 1/2-spin particles sitting on the links. The phase diagram is analyzed using DMRG, finding non-trivial phases, including spin wave insulators and solitonic phases, where the translational invariant symmetry is broken in different manners. Finally, I will show how this model can be implemented using ultracold atoms interacting with fixed impurities, which can be simulated with trapped ions or neutral atoms.