Tensor Network based approaches to Quantum Many-Body Systems

We study the properties of a two dimensional complex scalar field with a U(1) gauge field with a non-zero $\theta$ term. This is accomplished using a tensor network formulation evaluated via the tensor renormalisation group.

Thermalisation in classical systems is driven by chaos. In quantum systems the story looks very different. The time dependent variational principle over matrix product states yields a classical hamiltonian dynamics, which can becomes chaotic if we constrain ourselves to matrix product states with lower bond dimension. What can we learn about thermalisation from the chaos properties of this Hamiltonian dynamics?

In contrast to inorganic semiconductors, conjugated polymer systems display a much greater level of disorder in their structural and electronic properties. Energetic disorder, arising from the fluctuations of polarisation energies, and structural disorder, manifesting in molecular bending and variation of torsional angles, dictate the activation barrier to charge migration across the system, and therefore overall charge mobility. Charge mobility is calculated in our novel model representing a bulk poly(para-phenylene vinylene) (PPV) polymer system. Localised states of the system are determined using the Holstein Hamiltonian, followed by the application of Marcus theory to find hopping rates between them, which are then employed in kinetic Monte Carlo simulations. The effects of varying disorder and temperature are studied across a range of electric field strengths, and interpreted in light of the experimentally observed Poole-Frenkel behaviour, which states that the relation $\ln{\mu}\propto E^{1/2}$ holds for a wide range of field strengths.

In this work we study the high temperature spin transport through the anisotropic Heisenberg spin chain where the integrability has been broken by a single impurity or defect at the centre of the chain. For finite impurity strength the level spacing statistics of this model are known to be strongly Wigner-Dyson like. The aim of the paper is to understand if a signature of this integrability breaking is manifested in the hight temperature transport of spin current. We focus first on the non equilibrium steady state (NESS) where the chain is connected to spin baths which act as sources and sinks for spin excitations at the boundaries. Using a combination of open quantum system theory and matrix product operators techniques we extract the transport properties by means of a finite size scaling of the expectation value of the current in the NESS. Our results indicate that despite the formation of a domain wall in the steady state magnetisation the transport remains ballistic. We contrast this behaviour with global integrability breaking by means of a staggered magnetic field where it is known that the transport is fully diffusive. We demonstrate that our findings are fully consistent with Kubo theory by performing a numerical computation of the the real part of the spin conductivity. In addition, we also analyse subtleties associated with the apparent vanishing of the Drude weight for our model.

This paper addresses properties of interacting bosons on a two-leg ladder subjected to a uniform gauge field. Especially, it concentrates on the Vortex-to-Meissner crossover of hard-core bosons at finite temperatures. For this purpose, a canonical purification approach is employed. Focusing on momentum distributions, temperatures with clearly detectable characteristics of the Vortex phase are determined.

Joint work with Daniel Nagaj, arXiv:1805.07168 Investigating translationally invariant qudit spin chains with a low local dimension, we ask what is the best possible tradeoff between the scaling of the entanglement entropy of a large block and the inverse-polynomial scaling of the spectral gap. Restricting ourselves to Hamiltonians with a "rewriting" interaction, we find the pair-flip model, a family of spin chains with nearest neighbor, translationally invariant, frustration-free interactions, with a very entangled ground state and an inverse-polynomial spectral gap. For a ground state in a particular invariant subspace, the entanglement entropy across a middle cut scales as $\log n$ for qubits (it is equivalent to the XXX model), while for qutrits and higher, it scales as $\sqrt{n}$. Moreover, we conjecture that this particular ground state can be made unique by adding a small translationally-invariant perturbation that favors neighboring letter pairs, adding a small amount of frustration, while retaining the entropy scaling.

We provide a comprehensive framework to address quantum metrological problems via tensor networks, namely, matrix product operators (MPO). The MPO formalism allows for the consideration of complex problems involving spatial and temporal noise correlations, in contrast to presently available methods which are limited to models with either uncorrelated or fully correlated noise. The MPO method allows the determination of the maximal achievable estimation precision in such models, as well as the optimal probe states in the particle-number regime inaccessible for other state-of-the-art methods. Moreover, the application of infinite MPO (iMPO) techniques allows for a direct and efficient determination of the asymptotic behavior of the precision of optimal protocols in the limit of infinite particle number. We illustrate the potential of the framework via the example of atomic clock stabilization (temporal noise correlation) as well as magnetic field sensing in the presence of locally correlated magnetic field fluctuations (spatial noise correlations). As a byproduct, the developed methods for calculating quantum fisher information via MPO may be used as a method to calculate the fidelity susceptibility, a parameter widely used in many-body physics to study phase transitions.

In recent years, the scientific activity in the area of thermal engines has been boosted by the importance society is placing on sustainable energy. Central to this activity are quantum autonomous thermal machines, which convert heat to work through non equilibrium steady-state flows of microscopic particles without macroscopic moving parts. Quantum thermoelectrics, for example, would allow us to convert to electricity heat that is unaccessible and wasted by conventional semiconductor devices. Their efficiency, however, is still far below the threshold for technological applications. Since there is not any fundamental physical reason preventing quantum engines from reaching higher efficiencies, we explore the capability of one dimensional wires on a quasi-periodic geometry as working mediums. In particular, we focus on the highly non trivial spectral features of the class of extended Aubre-André-Harper models. We exploit the presence of an energy mobility edge in the spectrum to function as an energy filter and increase the efficiency at maximum power.

The two topological phases of a linear chain of ions differ in their harmonic yields by several orders of magnitude due to the difference in the destructive interference of all valence band and edge state electrons. A program to solve the time-dependent Kohn-Sham equations has been developed. It allows for the simulation of a linear chain in an intense laser field in an all-electron, self-consistent way. The robustness of the differing harmonic yield was investigated with respect to finite-size and disorder effects. A remarkable robustness was observed, which might allow applications that steer topological electronics by all-optical means or control strong-field-based light sources electronically.

Quantum entanglement and its main quantitative measure, the entanglement entropy, is playing a central role in many body systems. Among other uses, the amount of entanglement determines the applicability of tensor network algorithms. An interesting twist arises when the system considered has symmetries leading to conserved quantities: A recent study \cite{GS} introduced a way to define, represent in field theory, calculate for $1+1D$ conformal systems, and measure, the contribution of individual charge sectors to the entanglement measures between different parts of a system in its ground state. In my research, I apply these methods for studying the time evolution of the charge-resolved contributions to the entanglement entropy after a local quantum quench. I calculate them both numerically, using TEBD simulations on various $1D$ lattice models, as well as analytically for $1+1D$ conformal field theory description, and find very good agreement. * Here is the LaTeX version of my reference: \begin{thebibliography} \bibitem ={GS} M. Goldstein and E. Sela, \textit{Phys. Rev. Lett.} vol 120, p. 200602, May 2018. \end{thebibliography}

Generic closed driven interacting many-body systems absorb energy indefinitely from the drive. As a consequence, these systems tend to an infinite temperature state, exceeding any energy scale for the observation of quantum effects. A key challenge thus, is preventing systems from heating. On the contrary, we present a one dimensional topological pump model [Lindner, Berg, Rudner, PRX 2017] for which heating is desired to reach a pre-thermal quasi-steady and consequently enabling universal non-trivial charge pump for a long time-window. Furthermore, we study the effect of disorder on the thermalization and the quasi steady state and possibilities of extending our studies with TN schemes.

It has been shown that matchgate circuits [1] can be performed as a compressed quantum computation, i.e., the computation can be performed on an quantum computer using exponentially less qubits and only polynomial overhead in runtime [2]. Here, we consider whether this property translates over to adiabatic quantum simulation and computation. Moreover, we elaborate on recent results on classical simulability of matchgate circuits [3]. [1] L. Valiant, SIAM J. Comput. 31, 1229 (2002) [2] R. Jozsa, B. Kraus, A. Miyake, J. Watrous, Proc. R. Soc. A 466, 809 (2010) [3] D. J. Brod, PRA 93, 062332 (2016)

We review different tensor network approaches to study the spreading of operators in chaotic quantum systems. As a common ground to all methods, we quantify this spreading by means of the norm of the commutator of a spreading operator with a local operator, which is often referred to as the out of time order correlation function. We compare two approaches based on matrix product states in the Schrödinger picture: the time dependent block decimation (TEBD) and the time dependent variational principle (TDVP), as well as TEBD based on matrix product operators directly in the Heisenberg picture. We compare the results of all methods to numerically exact results using exact Krylov space time evolution in the Heisenberg picture. We also compare our results to the distance of reduced density matrices of two states which differ only by a local perturbation, which has recently been proposed as a measure of quantum chaos. We find that the short time dynamics for fixed bond dimension is accurately described by our TDVP approach , which yields a ballistic front and is accurate for longer times compared to TEBD. We also present results for larger systems for longer times, and assess what physical behaviours can be extracted from these different methods.

We combine the strengths of adiabatic local TDDFT and the NEGF formalism to describe real-time dynamics in strongly correlared systems. The adiabatic local density approximation (ALDA) with nonperturbative XC-potentials is successful to describe adiabatic dynamics in strongly correlated systems. In fast regimes there is an importance of non-adiabatic and non-local effects, there is neccesity to go beyond ALDA. To include these missing effects we propose a general systematic approach. Nonperturbative DFT exchange-correlation potentials in ALDA are combined with self-energies of many-body pertubation theory without double countings. Performance of the proposed approach is tested against exact solutions on Hubbard-type Hamiltonians.

We have studied a mobile impurity in a bath of two leg bosonic ladder. We have computed Green's function of impurity as a function of time for different momentum. We find a power law decay of Green's function at zero momentum unlike exponential decay, and quasi-particle description of impurity breaks down. For small interaction between impurity and bath, we find that the impurity effectively moves in a one dimensional bath, Green's function of impurity is very similar to that of impurity in a one dimensional bath, for infinite interaction Green's function decays as a power law for small momentum.

We study the applicability of the time-dependent variational principle in matrix product state manifolds for the long time description of quantum interacting systems. By studying integrable and nonintegrable systems for which the long time dynamics are known we demonstrate that convergence of long time observables is subtle and needs to be examined carefully. Remarkably, for the disordered nonintegrable system we consider the long time dynamics are in good agreement with the rigorously obtained short time behavior and with previous obtained numerically exact results, suggesting that at least in this case the apparent convergence of this approach is reliable. Our study indicates that while great care must be exercised in establishing the convergence of the method, it may still be asymptotically accurate for a class of disordered nonintegrable quantum systems.

Area laws are a far-reaching consequence of the locality of physical interactions, and they are relevant in a range of systems, from black holes to quantum many-body systems. Typically, these laws concern the entanglement entropy or the quantum mutual information of a subsystem at a single time. However, when considering information propagating in spacetime, while carried by a physical system with local interactions, it is intuitive to expect area laws to hold for spacetime regions; in this work, we prove such a law in the case of quantum lattice systems. We consider a sequence of local quantum operations performed at discrete times on a spin-lattice, such that each operation is associated to a point in spacetime. In the time between operations, the time evolution of the spins is governed by a finite range Hamiltonian. By considering a purification of the quantum instruments and analyzing the quantum mutual information between the ancillas used in the purification, we obtain a spacetime area law bound for the correlations between the measurement outcomes inside a spacetime region, and those outside of it.

We present some improvements for variational optimization of finite projected entangled-pair states (PEPS). A gauge-fixing scheme is proposed to achieve numerically stable ground state search. In addition, we demonstrate an algebraic approach to extend the PEPS ansatz to simulate fermionic systems. Using this approach, the minus sign upon interchange of two fermions is accounted for within tensor index operations. The improved PEPS algorithm is benchmarked by studying the ground states of the two-dimensional Hubbard model on finite-square lattices.

Canonical form in Matrix Product State makes numerical methods well-conditioned, bringing general eigenvalue problems to standard eigenvalue problems and avoiding solving linear systems of equations in iterative variational compression, and leads to the success of MPS. We show that it is possible to define (quasi) canonical form in finite PEPS, with which contraction of PEPS in imaginary TEBD algorithm is avoided. Different from previous works, e.g. arxiv:1405.3259, there is no approximation made for PEPS contraction and better conditioning than gauge fixing methods comes automatically. In order to impose and shift the canonical form of PEPS, we introduce and compare different splitter(/disentangler) for realizing the canonical form in terms of complexity and accuracy.

The accurate description of the non-equilibrium steady state properties of quantum impurities (such as an interacting quantum dot in the Kondo regime under the application of a bias voltage) is a central open problem in condensed matter physics. In a recent work [arXiv:1708.06315 [cond-mat.str-el]] an NRG-DMRG based quench scheme was developed. First, equilibrium NRG is used to find the relevant Hilbert subspace of the impurity and its vicinity. This part of the system is then coupled to leads at different chemical potential and time evolved using tDMRG, to find an intermediate time finite system approximation of the infinite system steady state. However, this approach suffers from a growth in entanglement entropy, which limits the evolution time, often before meaningful observables can be extracted. To allow for a true steady state we introduce open quantum system dynamics, modeling the effect of infinite reservoirs by finite ones with Lindblad relaxation dynamics. This dynamics is studied using a time-dependent MPDO (Matrix Product Density Operator) algorithm. Our results show that the Lindblad dissipation, if appropriately tuned, can cut off the entanglement entropy growth, while at the same time giving rise to the correct steady state observables.

As quantum-computation devices are currently under development, but have only a small number of qubits, it is interesting to ask whether one can emulate gate-based quantum computations for certain cases using a classical computer. Often, approaches based on exact diagonalizations and a complete representation of the states are used, which severely restrict the number of emulated qubits. Here, we use matrix-product states, which allow for the treatment of significantly more qubits. We implement the time evolution of a many-body Hamiltonian in this setup by formulating the Trotter decomposition via universal gates. The matrix-product operators of these gates are represented as finite-states machines. This allows us to test the reliability of the implemented emulator by comparing time evolutions to exact results.

Gapless free fermion states are notoriously challenging to represent with tensor network state methods. In a recent breakthrough, Fishman and White [PRB 92, 075132 (2015)] described an algorithm for efficiently representing the ground states of fermionic quadratic Hamiltonians in one spatial dimension as matrix product states (MPSs). We investigate generalizations of this method to construct efficient MPS representations of Gutzwiller-projected model variational wavefunctions for various quantum spin liquid states in 1D and quasi-1D. We benchmark our approach on a single half-filled band of spin-$1/2$ fermionic spinons---Gutzwiller projection of this state is known to be a quantitatively accurate description of the ground state of the 1D nearest-neighbor Heisenberg antiferromagnet. We then march toward 2D by considering quasi-1D incarnations of U(1) spin liquids on both the triangular and kagome lattices. We will compare the numerical effort of these calculations to that required for traditional variational Monte Carlo techniques, as well as comment on the feasibility of our approach for constructing good initial states for ground-state DMRG simulations of model Hamiltonians.

Over the last two decades, Bose-Einstein condensation (BEC) has enabled the study of non-linear waves in atomic systems, the so-called matter-wave solitons. These structures emerge in mean field descriptions of BECs, as provided by the Gross-Pitaevskii equation (GP), which incorporate a non-linear term proportional to the interatomic interaction strength $g$. An interesting scenario within mean field descriptions is the study of multicomponent BECs systems, which can be described by a set of coupled GP equations. This fact inspired the study of matter-wave solitons in this setting, and a new solitonic structure has been found: dark-dark solitons [1, 2, 3]. Particular attention has been given to the so-called Manakov limit in 1D settings, where the intra- and inter-condensate particle interactions match [2]. We study the 1D dynamics of dark-dark solitons in binary mixtures of Bose Einstein condensates away from the Manakov limit ($g\neq g_{12}$) with repulsive interparticle interactions within each condensate ($g > 0$) [4]. By using an adiabatic perturbation theory in the parameter $g_{12}/g$, we show that, contrary to the case of two solitons in scalar condensates, the interactions between solitons are attractive when the interparticle interactions between condensates are repulsive $g_{12} > 0$. As a result, the relative motion of dark solitons with equal chemical potential is well approximated by harmonic oscillations of angular frequency $w_r=\frac{\mu}{\hbar}\sqrt{\frac{8g_{12}}{15g}}. We also show that in finite systems, the resonance of this anomalous excitation mode with the spin density mode of lowest energy gives rise to alternating dynamical instability and stability fringes as a function of the perturbative parameter $g_{12}/g$. In the presence of harmonic trapping (with angular frequency $\Omega$) the solitons are driven by the superposition of two harmonic motions. When $g_{12} < 0$, these two oscillators compete to give rise to an overall effective potential that can be either single well or double well through a pitchfork bifurcation. All our theoretical results are compared with numerical solutions of the Gross-Pitaevskii equation for the dynamics and the Bogoliubov equations for the linear stability. A good agreement is found between them. [1] P. Ohberg and L. Santos, Phys. Rev. Lett. 86, 2918 (2001). [2] D. Yan, J. J. Chang, C. Hamner, M. Hoefer, P. G. Kevrekidis, P. Engels, V. Achilleos, D. J. Frantzeskakis, and J. Cuevas, J. Phys. B: At. Mol. Opt. Phys. 45, 115301 (2012). [3] M. A. Hoefer, J. J. Chang, C. Hamner, and P. Engels, Phys. Rev. A 84, 041605(R) (2011). [4] I. Morera, A. M. Mateo, A. Polls, and B. Juliá-Díaz, Phys. Rev. A 97, 043621 (2018).

The entanglement structure of cMERA states for free quantum field theories was studied. To this end, the (R\'enyi) entanglement entropies and the entanglement Hamiltonian were determined for cMERA states of both bosonic Klein-Gordon and fermionic Dirac theories in (1+1) dimensions. Where the entanglement entropies could be calculated directly, the determination of the entanglement Hamiltonian was more challenging as it diverges for cMERA states. However, by implementing an additional regularization, a non-divergent version of the entanglement Hamiltonian was derived. Both for this regularized entanglement Hamiltonian and the entanglement entropy a satisfactory agreement with theoretical predictions was obtained.

Simulating open quantum systems interacting strongly with their environments is known to be a challenging task. However, there have been developed powerful techniques to simulate systems that are linearly coupled to fermionic as well as bosonic environments in the past decades. In this poster I give a general overview of the "Time Evolving Density Matrix with Orthogonal Polynomial Algorithm" (TEDOPA) which is widely accepted in fields like quantum biology. In particular, I show how an arbitrary system interacting with a continuous, bosonic or fermionic, environment can be mapped unitarily onto a semi-infinite, one-dimensional chain. Due to the nature of the orthogonal polynomials involved in this mapping, the system is then only coupled to the first bath degree of freedom and the bath Hamiltonian itself contains only nearest-neighbor interactions. Hence, after truncating the infinite chain, we obtain a setting that is well-suited for all kinds of Matrix Product State (MPS) algorithms, such as "Time Evolving Block Decimation" (TEBD) or variational DMRG.

The question whether the mixed phase of a gapless superconductor can support a Landau level is a celebrated problem in the context of $d$-wave superconductivity, with a negative answer: The scattering of the subgap excitations (massless Dirac fermions) by the vortex lattice obscures the Landau level quantization. Here we show that the same question has a positive answer for a Weyl superconductor: The chirality of the Weyl fermions protects the zeroth Landau level by means of a topological index theorem. As a result, the heat conductance parallel to the magnetic field has the universal value $G=1/2 g_0 \Phi/\Phi_0$, with $\Phi$ the magnetic flux through the system, $\Phi_0$ the superconducting flux quantum, and $g_0$ the thermal conductance quantum.

In order to simulate quantum systems strongly coupled to an environment, we use an exact numerical approach that describes the non-Markovian time-evolution of the system. The method we use is called the time-evolving matrix product operator, or TEMPO. TEMPO exploits the augmented density tensor (ADT) in order to represent the time-evolution of the system in terms of tensors. The aim of this PhD thesis is to apply the TEMPO method in order to understand how the laws of thermodynamics generalize to arbitrary quantum systems both at and away from equilibrium, and in both Markovian and non-Markovian settings.

Fracton phases are characterized by excitations that exhibit restricted mobility. These mobility constraints are thought to be related to the conservation of the dipole moment associated to a given charge quantum number. Motivated by recent results on random unitary circuits [arXiv:1807.09776 [cond-mat.stat-mech]], we study one dimensional spin-1/2 and spin-1 Hamiltonian systems that conserve a U(1)-charge and its associated dipole moment. Our goal is: to understand the implications of these conservation laws and their effect on Hamiltonian dynamics, identify the 'fractonic' phenomena, and compare our results to those previously obtained for random unitary circuits.

A chain of Hydrogen atoms is studied using the Sliced Basis Density Matrix Renormalization Group (SBDMRG) in order to assess the fidelity of standard model Hamiltonians. Given the hydrogen chain’s Mott like behavior as a function of atomic separation, a natural choice for modeling the physics within the charge sector is the Hubbard model, leaving the spin sector to be modeled by a Heisenberg model. Calculations of various observables and correlation functions are then carried out within both sectors and compared. In addition to the direct comparison between observables, Hubbard and Heisenberg model parameters are extracted from the simulation of the hydrgoen chain at several atomic spacings. These parameters are shown to converge in the large U limit as is expected from the traditional mapping of a Hubbard model onto a Heisenberg model.

The formalism of tensor networks can be applied quantum field theories through use of continuous matrix product states. On this poster, the time-dependent variational principle is applied to these continuous matrix product states in an attempt to simulate modern ultracold atom gas experiments.

Tensor network methods have been very successful to accurately simulate one- and two-dimensional quantum models, with White's density-matrix renormalization group (DMRG) as a famous example [PRL 69, 2863 (1992)]. A way to generalize tensor network methods to three dimensions would be highly desirable, especially for simulating fermionic and frustrated models which in many cases are hard or impossible to solve using other methods. Here a novel way of contracting three-dimensional tensor networks using clusters is introduced. These clusters consist of a certain number of tensors which are contracted exactly or with a higher accuracy as compared to the environment of the cluster, which is approximated effectively. The method is implemented on the square and cubic lattices and applied to the quantum Ising model with a transverse field and the Heisenberg model. The results are compared to corner transfer matrix (CTM) results and other reference values. Also, as a non-trivial example, a simulation is made for the SU(3) Heisenberg model. A three-sublattice striped state is found to be lower in energy than a two-sublattice striped state.

In this work, we develop a new method for studying excitations in quantum spin chains. The method is based on the framework of tensor network states, and, in particular, on the quasiparticle ansatz for capturing elementary excitations on top of strongly-correlated ground states. The method forms a logic extension of the quasiparticle ansatz and is especially suited for the description of low-energy bound states. We successfully apply the method on paradigmatic models such as the AKLT model and the Z2 -symmetry broken Ising model. We also use the developed techniques to simulate the excitation spectrum around a first-order phase transition in a frustrated Heisenberg spin-1 chain, a situation in which we expect a stack of soliton/anti-soliton bound states to emerge.