# Workshop October 2016

The aim of the Research Training Group in organizing this international workshop is to bring together a selected group of junior and senior theorists interested in quantum many-body systems and in developing methods to study these systems, and to foster exchange between the researchers. Each of the six topics "Correlated and Dirac Materials", "Exact Approaches", "Tensor Network Methods", "Approaches to Nonequilibrium", "Entanglement and Topology" and "Ab initio and Many-Body" will be introduced in a keynote presentation, which is then followed by topical talks. Participants are invited to present a poster. Please apply to attend before June 15th, 2016 under rtg1995@physik.rwth-aachen.de (50€ workshop fee).

## Workshop Schedule

## Titles and abstracts

Markus Aichhorn:** Topological invariants from Wannier charge centers: Application to the topological insulator on honeycomb lattices and ribbons without inversion symmetry**

We study the Kane-Mele-Hubbard model with an additional inversion-symmetry breaking term. By a combination of the topological Hamiltonian and Wannier charge center approaches, we calculate the Z_{2}invariant of the system as function of spin-orbit coupling, Hubbard interaction U, and inversion-symmetry breaking on-site potential. The phase diagram calculated in that way shows that on the one hand a large term of the latter kind destroys the topological non-trivial state. On the other hand, however, this inversion-symmetry breaking field can enhance the topological state, since for moderate values the transition from the non-trivial topological to the trivial Mott insulator is pushed to larger values of interaction U. This feature of an enhanced topological state is also found on honeycomb ribbons. With inversion symmetry, the edge of the zigzag ribbon is magnetic for any value of U. This magnetic moment destroys the gapless edge mode. Lifting inversion symmetry allows for a finite region in interaction strength U below which gapless edge modes exist.

Michael Brockmann:** Quantum quenches and equilibration of the spin-1/2 Heisenberg chain**

Jean-Sébastien Caux:** Dynamics and relaxation in integrable quantum systems**

Laura Classen:** Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormalization**

Philippe Corboz: Recent **progress in simulating strongly correlated systems with 2D tensor network methods**

Sebastian Diehl:** Universal phenomena in low dimensional driven open quantum systems**

Quantum optics and many-body physics increasingly merge together in ultracold atomic gases and solid state, such as exciton-polariton systems. This gives rise to new non-equilibrium scenarios in stationary state, where coherent and dissipative dynamics appear on an equal footing. We will report on universal crossovers and phase transitions in one and two dimensions. At long wavelength, the non-equilibrium character reveals itself via a mapping of such systems to a compact variant of the Kardar-Parisi-Zhang equation. This allows for defects without counterpart in the original non-compact equation. This is made manifest by a duality transformation to an open non-linear electrodynamics. We analyze the interplay of non-equilibrium KPZ physics and the proliferation of these defects at weak non-equilibrium drive. They always unbind asymptotically even at low effective temperature. While in two dimensions, the vortex proliferation scale preempts the onset of KPZ scaling, the situation is reversed in one dimension. This suggests one-dimensional exciton-polariton systems as a laboratory to study these universal crossovers. At strong non-equilibrium drive, we furthermore establish a new first order phase transition in one dimension driven by the proliferation of space-time vortex defects.

Katharina Eissing:** Renormalization in periodically driven quantum dots**

A newly developed flexible renormalization-group-based approach for periodic driving is applied to the interacting resonant level model. We aim at the steady state of setups with one or more of the dot or lead parameters varied periodically in time, which is reached after all transients have died out. The interacting resonant level model is characterized by power-law scaling of observables in the relevant energy scales with interaction dependent exponents in equilibrium and if driven by a time constant bias voltage. The functional renormalization group has proven to be a versatile tool to investigate correlated, low-dimensional systems in and out of equilibrium e.g. by capturing these power laws correctly. Hence, we take explicitly advantage of the periodicity by applying the Floquet theorem and setting up the RG procedure in the Floquet basis. This allows to study the role of the driving frequency Ω as an infrared cutoff of the underlying renormalization group flow. I will discuss how the correlation between lead and dot electrons enhance or suppress the amplitude of the driving depending on the sign of the interaction. It is shown analytically that the magnitude of this effect follows a power law in the driving frequency which in turn manifests itself in the pumping power of the resulting single parameter quantum pump.

Maurizio Fagotti:** Transport in out-of-equilibrium spin chains: exact profiles of charges and currents**

Nicole Helbig:** Spin and pure state N-representability constraints in reduced density matrix functional theory**

Reduced Density Matrix Functional Theory is a method that relies on the 1-1 correspondence between the many-body ground-state wave function and the first order reduced density matrix (1RDM) and uses the latter as its fundamental variable. The ground state of a system is determined within this approach by minimizing the energy functional with respect to the 1RDM under the constraint that the 1RDM corresponds to a fermionic ensemble (Coleman's conditions).

Additional constraints can be employed to ensure the existence of a fermionic ensemble with a specific * S _{z} .* However, there remain the two questions if the fermionic system corresponds to a specific total spin and if the system can be represented by a pure state. We show that generally the answer to both questions is negative unless additional constraints are enforced and discuss these constraints exemplary for several small systems.

References:

I. Theophilou et al., J. Chem Phys.

**142**, 154108 (2015)

I. Theophilou , et al., J. Chem. Theory Comput .

**12**, 2668 (2016)

Stephan Heßelmann:** Thermal phase transitions in the vicinity of the quantum critical point of spinless fermions on the honeycomb lattice**

Markus Heyl:** Dynamical Quantum Phase Transitions**

Dynamical quantum phase transitions (DQPTs) have emerged as a nonequilibrium analogue to conventional phase transitions with physical quantities becoming nonanalytic at critical times. I will summarize the recent developments including the first experimental observation of a DQPT in a topological system. While the formal analogies of DQPTs to equilibrium phase transitions are straightforward, a major challenge is to connect to fundamental concepts such as scaling and universality. In this talk I will show that for DQPTs in Ising models exact renormalization group transformations in complex parameter space can be formulated. As a result of this construction, the DQPTs are critical points associated with unstable fixed points of equilibrium Ising models implying scaling and universality in this far-from equilibrium regime.

Hiroke Isobe:** Renormalization group analysis of quantum critical points in Dirac materials**

In spite of their semimetallic nature, Dirac materials at charge neutrality point exhibit the long-range Coulomb interaction. Since they have point nodes and linear energy dispersions, the density of states vanishes, which makes the Coulomb interaction unscreened. This is similar to quantum electrodynamics, and the effect of the long-range Coulomb interaction can be examined by using the renormalization group (RG) analysis.

In this talk, I will briefly review the RG analysis of Dirac materials and report a novel quantum critical point found for j=3/2 Dirac electrons in antiperovskite topological crystalline insulators with cubic crystalline symmetry. The RG analysis reveals three fixed points that have either Lorentz, rotational, or cubic symmetry. The stable fixed point only has the cubic symmetry and exhibits Dirac-like dispersions with finite velocity anisotropy, which gives an interesting counterexample to emergent Lorentz invariance in solids.

Christoph Karrasch:** One-dimensional systems at finite temperature**

We give a pedagogical introduction on how matrix product state techniques can be used to study one-dimensional systems at finite temperatures. As an application, we discuss how very simple (numerical) non-equilibrium setups can be used to efficiently compute linear-response transport properties.

Nicolas Laflorencie:** Quantum entanglement in condensed matter systems**

This talk will review some aspects of quantum entanglement applied to condensed matter physics systems with strong correlations. By tracing out part of the degrees of freedom of correlated quantum systems, useful and non-trivial informations can be obtained through the study of the reduced density matrix, whose eigenvalue spectrum (the entanglement spectrum) and the associated Rényi entropies are now well recognized to contains key features. In particular, the celebrated area law for the entanglement entropy of ground-states will be discussed from the perspective of its subleading corrections which encode universal details of various quantum states of matter, e.g. symmetry breaking states or topological order. Going beyond entropies, the study of the low-lying part of the entanglement spectrum also allows to diagnose topological properties or give a direct access to the excitation spectrum of the edges, and may also raise significant questions about the underlying entanglement Hamiltonian. All these powerful tools can be further applied to shed some light on disordered quantum systems where impurity/disorder can conspire with quantum fluctuations to induce non-trivial effects. Disordered quantum spin systems, the Kondo effect, or the many-body localization problem, which have all been successfully (re)visited through the prism of quantum entanglement, will be discussed.

For further reading, see Physics Report 643, 1-59 (2016)

David Luitz:** Entanglement scaling at the many-body localization transition**

Lisa Markhof:** Spectral function of the Tomonaga Luttinger model revisited**

The momentum- and energy-resolved single-particle spectral function of the Tomonaga-Luttinger model has been examined analytically in [1]. The focus was on the role of the momentum-dependence of the two-particle interaction V(q). Usually this is neglected since it is irrelevant in the renormalization group sense. However, if the momentum k is fixed away from the Fermi momentum k_{F}, with |k-k_{F}| setting a non-vanishing energy scale, the details of V(q) start to matter. It was shown in [1] that the flatness of V(q) at vanishing momentum plays an important role. Using a method to calculate the momentum-resolved spectral function which is based on [2], we could provide strong evidence that any curvature of the two-particle interaction at small transferred momentum q destroys power-law scaling of the momentum-resolved spectral function as a function of energy [3]. Even for |k − k_{F} | much smaller than the momentum-space range of the interaction the spectral line shape depends on the details of V(q). In my talk, I will show those results and shortly discuss the implications of our findings.

[1] V. Meden, Phys. Rev. B 60, 4571 (1999)

[2] K. Schönhammer and V. Meden, Phys. Rev. B 47, 16205 (1993)

[3] L. Markhof and V. Meden, Phys. Rev. B 93, 085108 (2016)

Salvatore Manmana:** Recent progress in dynamical response functions with matrix product states**

Recent developments make it possible to compute finite-temperature spectral functions of low-dimensional strongly correlated quantum systems using flexible matrix product state (MPS) approaches. Using a combination of a Liouvillian formulation and a Chebyshev expansion, the temperature dependence of response functions of the sine-Gordon quantum magnet CuPM and for the S=1/2 dimer chain system BaCu_{2}V_{2}O_{8} are obtained. For CuPM the accuracy of our approach allows to distinguish between nearby lying breather states from boundary bound states. In the dimer chain compound we are able to identify a ferromagnetic inter-dimer coupling by comparing state-of-the-art neutron scattering data with our theoretical results. As an outlook, the development of quantum coherence with temperature in this system is discussed.

Kevin O'Brien:** Classification of Z _{2 }spin liquids in three-dimensional Kitaev models**

Róman Orús:** Progress in tensor network states and methods for quantum lattice systems**

In this talk I will make an overview of some recent developments concerning the application of tensor network states and methods to understand quantum phases of matter for quantum lattice systems. Specifically, I will talk about (i) exact tensor network states for the Kitaev honeycomb model, (ii) entanglement Continuous Unitary Transformations, (iii) simulations of Kagome antiferromagnets in a field, and (iv) exact SU(2) spin liquids with Projected Entangled Pair States.

Olivier Parcollet:** Mott physics and spin fluctuations: A unified framework**

I will present the "TRILEX" formalism for strongly correlated electron systems and its application to the Hubbard model. TRILEX is designed to unify Dynamical Mean Field Theory (DMFT) and spin fluctuation approaches close to the Mott transition in a minimal way. It is based on a local approximation of the dynamical three-leg interaction vertex and solved using a self-consistent local quantum impurity model. It allows to address simultaneously the Mott physics `a la DMFT and the effect of long range antiferromagnetic fluctuations. While its computational cost is comparable to a single site Extended-DMFT computation, the self-energy is momentum-dependent. Moreover TRILEX is the starting point of a systematic and controlled method based on clusters. I will discuss the application of TRILEX to the Hubbard model on a two-dimensional square lattice. As interactions are increased towards the Mott insulating state, the local vertex acquires a strong frequency dependence, driving the system to a Mott transition, while at low enough temperatures the momentum dependence of the self-energy is enhanced due to large spin fluctuations.

Stephan Rachel:** Exotic Landau Levels**

Roman Riwar:** Normal-metal quasiparticle traps for superconducting qubits**

The presence of quasiparticles in superconducting qubits emerges as an intrinsic constraint on their coherence. While it is difficult to prevent the generation of quasiparticles, keeping them away from active elements of the qubit provides a viable way of improving the device performance. Here we develop theoretically and validate experimentally a model for the effect of a single small trap on the dynamics of the excess quasiparticles injected in a transmon-type qubit. The model allows one to evaluate the time it takes to evacuate the injected quasiparticles from the transmon as a function of trap parameters. With the increase of the trap size, this time decreases monotonically, saturating at the level determined by the quasiparticles diffusion constant and the qubit geometry. We determine the characteristic trap size needed for the relaxation time to approach that saturation value. Importantly, the fitted characteristic trap size indicates that the relaxation in the normal metal is not immediate, thus disqualifying a (usually standard) equilibrium assumption for the distribution function in the normal metal.

César Rodríguez-Rosario:** Non-Equilibrium Thermodynamics of Quantum Coherences**

We give a pedagogical introduction to non-equilibrium quantum thermodynamics. We focus on its application to quantum transport, and also, how to treat quantum coherences as a new thermodynamic potential. Quantum coherences challenge the classical notions of a thermodynamic bath. As a result, changes in quantum coherence can lead to a heat flow from baths with no associated temperature, and affect the entropy production rate. From this, we derive a quantum version of the Onsager reciprocal relations that shows that there is a reciprocal relation between thermodynamic forces from coherence and quantum transport. Quantum decoherence can be useful and offers new possibilities of thermodynamic control for quantum transport. We highlight this by demonstrating how a quantum observer can make a thermal flow go against the gradient, and explain how this does not violate any laws of quantum thermodynamics, although it seems paradoxical from a classical point of view.

Pina Romaniello:** Photoemission Spectra beyond GW : the Band Gap in Strongly Correlated Systems**

Photoemission is a powerful tool to obtain insight into the electronic structure and excitations in materials. From the theoretical point of view Many-Body Perturbation Theory, within the so called GW approximation to electron correlation, is the method of choice for calculations of photoemission spectra of many materials. However GW suffers from some fundamental shortcomings, and, in particular, it does not capture strong correlation, unless one

treats the system in a magnetically ordered phase. In this talk we present various efforts to go beyond GW [1-3]. In particular, we focus on a many-body effective-energy theory (MEET) that gives many-body spectral functions in

terms of reduced density matrices (RDMs) [4]. We show that simple approximations, which

require the knowledge of the lowest n-body RDMs only, can provide accurate photoemission spectra in model systems in the weak as well as strong correlation regime. With the example of bulk NiO, we show that our method yields a qualitatively correct picture both in the antiferromagnetic and paramagnetic phases, contrary to

mean-field methods, in which the paramagnet is a metal.

[1] P. Romaniello, S. Guyot, and L. Reining, J. Chem. Phys.131, 154111 (2009)

[2] P. Romaniello, F. Bechstedt, and L. Reining, Phys. Rev. B 85, 155131 (2012)

[3] G. Lani, P. Romaniello, and L. Reining, New J. Phys. 14, 013056 (2012)

[4] S. Di Sabatino, J.A. Berger, L. Reining, and P. Romaniello, arXiv :1607.08410

David Sánchez de la Peña:** Competing electronic instabilities of extended Hubbard models on the honeycomb lattice: A functional Renormalization Group calculation with high wavevector resolution**

We investigate the quantum many-body instabilities for electrons on the honeycomb lattice at half-ﬁlling with extended interactions, motivated by a description of graphene and related materials. We employ a recently developed fermionic functional Renormalization Group scheme which allows for highly resolved calculations of wavevector dependences in the low-energy effective interactions. We encounter the expected anti-ferromagnetic spin density wave for a dominant on-site repulsion between electrons, and charge order with different modulations for dominant pure n-th nearest neighbor repulsive interactions. Novel instabilities towards incommensurate charge density waves take place when non-local density interactions among several bond distances are included simultaneously. Moreover, for more realistic Coulomb potentials in graphene including enough non-local terms there is a suppression of charge order due to competition effects between the different charge ordering tendencies, and if the on-site term fails to dominate, the semi-metallic state is rendered stable. The possibility of a topological Mott insulator being the favored tendency for dominating second nearest neighbor interactions is not realized in our results with high momentum resolution.

Dirk Schuricht:** Time evolution during and after finite-time quantum quenches in one-dimensional systems**

We study the time evolution in the Tomonaga--Luttinger model (TLM) and the transverse-field Ising chain (TFIC) subject to quantum quenches of finite duration, i.e., a continuous change in the interaction strength or transverse magnetic field respectively. We analyze several observables including two-point correlation functions, which show a characteristic bending and delay of the light cone due to the finite quench duration. For example, we extract the universal behavior of the Green functions in the TLM and provide analytic, non-perturbative results for the delay. As another example, for quenches between the phases in the TFIC we show that the Loschmidt echo exhibits characteristic non-analyticities due to a dynamical phase transition, which show clear signatures of the finite quench duration.

Gianluca Stefanucci:** First-principles NonEquilibrium Green’s Function approach to time-resolved photoabsorption**

We will introduce a first-principles nonequilibrium Green’s function approach to calculate the time-resolved photoabsorption spectrum of nanoscale systems. The approach can deal with arbitrary shape, intensity and duration of the laser fields, and it is applicable for overlapping and nonoverlapping pump and probe pulses. If time permits we will present numerical simulations for atomic systems and discuss future challenges and reachable goals.

This talk will outline integrability-based results on the out-of-equilibrium dynamics of low-dimensional systems such as interacting atomic gases and quantum spin chains. A number of recent developments will be explained, including a new method for explicitly calculating the relaxation of observables after a quantum quench. Exact solutions to the interaction turn-on quench in the Lieb-Liniger model and to the Néel-to-XXZ quench in spin chains will be presented. Particular emphasis will be given to interesting open issues, including the failure of the (local) Generalized Gibbs Ensemble to properly describe post-quench steady-state properties and the necessity to include quasilocal conserved charges to obtain correct answers.

Ilya Tokatly:** Quantum Electrodynamical Time Dependent Density Functional Theory**

In this talk I present a recently proposed Quantum Electrodynamical Time Dependent Density Functional Theory (QED-TDDFT). I will start from a general physical picture and the formal mathematical statement of the many-body problem for non-relativistic electrons strongly coupled to photon modes of a microcavity. Then I will sketch a proof of the basic QED-TDDFT mapping theorem and introduce an approximate xc functional that generalizes the concept of optimized effective potential (OEP) and makes a connection to the many-body perturbation theory. Finally I will discuss implication of the QED-TDDFT to the theory of quantum dissipative systems.