Tensor Product State Simulations of Strongly Correlated Systems

Floquet engineering, the averaging of fast periodic modulations to obtain an effective time-independent system, has in recent years become an ubiquitous toolbox for the creation of novel Hamiltonians for ultracold atoms in optical lattices. We show here that a multicolor modulation of the depth of an optical lattice allows for a flexible independent control of correlated hopping, effective on-site interactions withouth Feshbach resonances, effective inter-site interactions and occupation-dependent gauge fields.

Using the density matrix renormalisation group algorithm, the Bose-Hubbard model is examined on a flat-band lattice by exploring a chain of diamonds. The phase type of the ground state of the model is examined before and after the transition point through the study of the observables and the ground state energy. The diamond chain is considered both for a frustrated case with Aharanov-Bohm cages and a non-frustrated case in order to compare the differences between the two.

For finite groups, the framework of G-injectivity describes a class of states with topological properties that arise as ground states of local Hamiltonians. In this work, we show that some of the properties remain if the finite group is replaced by the compact Lie group SU(2): the resulting PEPS appears as a ground state of a local Hamiltonian and there is a subspace of the ground state manifold that consists of states that are locally indistinguishable but can be transformed into each other by means of inserting string operators. We prove a bound for the entanglement entropy and show that a parameter can be introduced that gives rise to a set of commuting transfer matrices.

When and by which mechanism do closed quantum many-body systems equilibrate? This fundamental question has been in the focus of attention for many years. It lies at the very basis of the connection between thermodynamics, quantum mechanics of many constituents and condensed matter theory. In the setting of free fermionic evolutions, we rigorously capture the time evolution in abstract terms and by basing our proof on intuitive mathematical concepts like Lieb-Robinson bounds, notions of particle transport and an algebraic expansion of operators, we uncover the underlying mechanism how local memory of the initial conditions is forgotten. Specifically, starting from an initially short range correlated fermionic states which can be very far from Gaussian, we show that if the Hamiltonian provides sufficient transport, the system approaches a state that cannot be distinguished from a corresponding Gaussian state by local measurements. For experimentally relevant instances of ultra-cold fermions in optical lattices, our result implies equilibration on realistic physical time scales. Moreover, we characterise the equilibrium state, finding an instance of a rigorous convergence to a fermionic Generalized Gibbs ensemble generated by the non-local constants of motion of the system. Authors: \begin{center} Marek Gluza$^\text{1}$, Christian Krumnow$^\text{1}$, Mathis Friesdorf$^\text{1}$, Christian Gogolin$^\text{2,3}$, Jens\ Eisert$^\text{1}$\\ ${}^\text{1}$Dahlem Center for Complex Quantum Systems, Freie Universit{\"a}t Berlin, Berlin, Germany\\ ${}^\text{2}$ICFO-The Institute of Photonic Sciences, Mediterranean Technology Park, Barcelona, Spain\\ ${}^\text{3}$Max-Planck-Institut f{\"u}r Quantenoptik, Garching, Germany\end{center}

### I can also make a poster for this work, if talk slots are scarce. The phenomenon of many-body localised (MBL) systems has attracted significant interest in recent years, for its intriguing implications from a perspective of both condensed-matter and statistical physics: they are insulators even at non-zero temperature and fail to thermalise, violating expectations from quantum statistical mechanics. What is more, recent seminal experimental developments with ultra-cold atoms in optical lattices constituting analog quantum simulators have pushed many-body localised systems into the realm of physical systems that can be measured with high accuracy. In this work, we introduce experimentally accessible wit- nesses that directly probe distinct features of MBL, distinguishing it from its Anderson counterpart. We insist on building our toolbox from techniques available in the laboratory, including on-site addressing, super-lattices, and time-of-flight measurements, identifying witnesses based on fluctuations, density-density correlators, den- sities, and entanglement. We build upon the theory of out of equilibrium quantum systems, in conjunction with tensor network and exact simulations, showing the effectiveness of the tools for realistic models.

We investigate the ground-state of a $p$-wave Kondo-Heisenberg model introduced by Alexandrov and Coleman with an Ising-type anisotropy in the Kondo interaction and correlated conduction electrons. Our aim is to understand how they affect the stability of the Haldane state obtained in the SU(2) symmetric case without the Hubbard interaction. By applying the density-matrix renormalization group algorithm and calculating the entanglement entropy we show that in the anisotropic case a phase transition occurs and a N\'eel state emerges above a critical value of the Coulomb interaction. These findings are also corroborated by the examination of the entanglement spectrum and the spin profile of the system which clarify the structure of each phase.

In the last few years, experimental and theoretical works have significantly furthered our understanding of the non-equilibrium dynamics of quantum systems characterised by on-site interactions [1]. This on-going effort was recently paralleled by the realisation of longer range interactions in ultracold quantum gases in optical lattices [2]. It is with this exciting development in mind that we investigate the non-equilibrium dynamics of the extended Fermi-Hubbard model [3], with a special interest in understanding if the propagation of correlations is light-cone-like [4,5]. [1] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011). [2] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, F. Sferlaino, arXiv:1507.03500 (2015). [3] T. Giamarchi, Quantum Physics in One Dimension, (Ox- ford University Press, Oxford, UK, 2004). [4] Calabrese, P. & Cardy, J., Phys. Rev. Lett. 96, 136801 (2006). [5] M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Nature 481, 484 (2012).

Quantum state tomography (QST) is the gold standard technique for estimating wave functions of small quantum systems in the laboratory. Applying QST to the larger systems currently being developed in laboratories around the world is impractical due to the large number of measurements and processing time required. In 2010 Cramter et al. proposed a tomography scheme [1] to efficiently reconstruct large quantum states that are well approximated by matrix product states (MPS). On my poster I present the experimental application of this MPS tomography to characterise a trapped ion quantum simulator of spin-1/2 particles. A product state of up to 20 ions is prepared and evolved under an Ising-type interaction, giving rise to many-body entangled states. The MPS reconstruction scheme is then performed at various times during the evolution, and the resulting quantum state investigated. We show that the reconstructed state has a sigificant overlap with the actual state created in the laboratory and reproduces non-classical correlations to a high degree. \\ [1] M. Cramer et al., Nature Communications \textbf{1}, 149 (2010), doi:10.1038$/$ncomms1147

The Bilinear-Biquadratic model for spin-1 chains has attracted great interest in recent years. It has been studied using many different methods including the real space renormalization group, quantum Monte Carlo and the class of MPS algorithms. Using the DMRG algorithm, we study the disordered version of the model with disorder in both the bilinear and biquadratic parts. Motivated by experiments with spinorial condensates in optical lattices, we consider the region of the phase diagram that would correspond, in the clean case, to the ferromagnetic and dimer phase. An important question is whether the dimer phase resists in the disorder case or turns into a random singlet phase as predicted in the literature. Another interesting point is whether the first order transition separating the two phases in the clean case gives way to a continuous transition or an indirect transition through a coexistence phase (large spin phase). To this end we analyse local observables, correlations and entanglement entropy.

Topological matter has been a deep subject of study in the last years, recently recognized with the Nobel Price in Physics. This field describes new phases of matter which the usual Landau approach can not explain and gives rise to an amount of nobel rich phenomena in physics. Of special interest is what happens to topological order when the system is in presence of interactions. We present here the Haldane model with on-site spin interactions and study the topological order for various ranges of the interactions. We probe using a variational Ansatz the robustness of topological order at relatively large values of the interactions compared with the kinetic energy scale.

The spin-$1/2$ Kitaev honeycomb model was originally proposed in the context of topological quantum computation. This analytically solvable model realizes a spin liquid and exhibits rich physical behaviour, such as abelian and non-abelian anyons as excitations. Our aim is to describe the eigenstates of the model using tensor network methods, which offer efficient descriptions of quantum many-body systems. In particular we exploit parity preservation and build a fermionic tensor network to express the eigenstates of the Hamiltonian in the ground state vortex sectors. We implement the network for small lattices with periodic boundary conditions in order to verify the approach for the model in the thermodynamic limit.

I will give a brief overview on how tree tensor product states can be time-evolved with the time-dependent variational principle. I will explain how Krylov subspace methods, the Suzuki-Trotter splitting and the splitting of the node tensors themselves (similar to PEPS) can be used to efficiently apply the TDVP time-evolution operators to higher degree tree nodes. Finally I will give an example of how tree tensor product states can be used to simulate exciton-phonon dynamics in organic molecules.

We perform a full diagonalization study of frustrated spin-1/2 chains (i.e. spin-1/2 chains with nearest and next nearest neighbor interaction) in the presence of an external magnetic eld. The thermal and spin conductivity are computed from Kubo formulae as a function of frustration and eld strenght. We are especially interested in the transport properties in the vector chiral phase that appears in the phase diagram at strong frustration and high eld. We observe an enhanced low-frequency response in the high-eld vector chiral phase which we trace back to a renormalization and enhancement of the characteristic velocity.

Steering a quantum state from one into another finds many applications in quantum computation and simulation. The state is manipulated by controlling the Hamiltonian. Good control of the Hamiltonian is found using optimization techniques from optimal control theory. In this poster the main three methods for calculating such controls is compared for problems relating to ultracold atoms.