# Publications by Fabian Essler

SciPost Physics SciPost (2020)

FHL Essler, AA Ziolkowska

We consider Lindblad equations for one dimensional fermionic models and quantum spin chains. By employing a (graded) super-operator formalism we identify a number of Lindblad equations than can be mapped onto non-Hermitian interacting Yang-Baxter integrable models. Employing Bethe Ansatz techniques we show that the late-time dynamics of some of these models is diffusive.

## How order melts after quantum quenches

PHYSICAL REVIEW B 101 (2020) 41110

M Collura, FHL Essler

© 2020 American Physical Society. Injecting a sufficiently large energy density into an isolated many-particle system prepared in a state with long-range order will lead to the melting of the order over time. Detailed information about this process can be derived from the quantum mechanical probability distribution of the order parameter. We study this process for the paradigmatic case of the spin-1/2 Heisenberg XXZ chain. We determine the full quantum mechanical distribution function of the staggered subsystem magnetization as a function of time after a quantum quench from the classical Néel state. We establish the existence of an interesting regime at intermediate times that is characterized by a very broad probability distribution. Based on our findings we propose a simple general physical picture of how long-range order melts.

## Self-consistent time-dependent harmonic approximation for the sine-Gordon model out of equilibrium

Journal of Statistical Mechanics: Theory and Experiment IOP Publishing 2019 (2019) 084012

Y Van Nieuwkerk, FHL Essler

We derive a self-consistent time-dependent harmonic approximation for the quantum sine-Gordon model out of equilibrium and apply the method to the dynamics of tunnel-coupled one-dimensional Bose gases. We determine the time evolution of experimentally relevant observables and in particular derive results for the probability distribution of subsystem phase fluctuations. We investigate the regime of validity of the approximation by applying it to the simpler case of a nonlinear harmonic oscillator, for which numerically exact results are available. We complement our self-consistent harmonic approximation by exact results at the free fermion point of the sine-Gordon model.

## Almost strong (0, pi) edge modes in clean interacting one-dimensional Floquet systems

PHYSICAL REVIEW B 99 (2019) ARTN 205419

DJ Yates, FHL Essler, A Mitrai

## NMR relaxation in Ising spin chains

Physical Review B: Condensed Matter and Materials Physics American Physical Society 99 (2019) 035156-

J Steinberg, NP Armitage, F Essler, S Sachdev

We examine the low frequency spin susceptibility of the paramagnetic phase of the quantum Ising chain in a transverse field at temperatures well below the energy gap. We find that the imaginary part is dominated by rare quantum processes in which the number of quasiparticles changes by an odd number. We obtain exact results for the NMR relaxation rate in the low temperature limit for the integrable model with nearest-neighbor Ising interactions, and derive exact universal scaling results applicable to generic Ising chains near the quantum critical point. These results resolve certain discrepancies between the energy scales measured with different experimental probes in the quantum disordered paramagnetic phase of the Ising chain system CoNb206

## Full counting statistics in the transverse field Ising chain

SciPost Physics SciPost 4 (2018) 043

S Groha, F Essler, P Calabrese

We consider the full probability distribution for the transverse magnetization of a finite subsystem in the transverse field Ising chain. We derive a determinant representation of the corresponding characteristic function for general Gaussian states. We consider applications to the full counting statistics in the ground state, finite temperature equilibrium states, non-equilibrium steady states and time evolution after global quantum quenches. We derive an analytical expression for the time and subsystem size dependence of the characteristic function at sufficiently late times after a quantum quench. This expression features an interesting multiple light-cone structure.

## Finite-temperature dynamics of the Mott insulating Hubbard chain

PHYSICAL REVIEW B 97 (2018) ARTN 045146

A Nocera, FHL Essler, AE Feiguin

## Critical behavior of the extended Hubbard model with bond dimerization

Physica B: Condensed Matter Elsevier 536 (2017) 474-478

S Ejima, F Lange, FHL Essler, H Fehske

Exploiting the matrix-product-state based density-matrix renormalization group (DMRG) technique we study the one-dimensional extended (U-V) Hubbard model with explicit bond dimerization in the half-filled band sector. In particular we investigate the nature of the quantum phase transition, taking place with growing ratio V/U between the symmetry-protected-topological and charge-density-wave insulating states. The (weak-coupling) critical line of continuous Ising transitions with central charge c=1/2 terminates at a tricritical point belonging to the universality class of the dilute Ising model with c=7/10. We demonstrate that our DMRG data perfectly match with (tricritical) Ising exponents, e.g., for the order parameter β=1/8 (1/24) and correlation length ν=1 (5/9). Beyond the tricritical Ising point, in the strong-coupling regime, the quantum phase transition becomes first order.

## Integrable spin chains with random interactions

PHYSICAL REVIEW B 98 (2018) ARTN 024203

FHL Essler, IR van den Berg, V Gritsev

## Projective phase measurements in one-dimensional Bose gases

SciPost Physics Stichting SciPost 5 (2018) 046

YD van Nieuwkerk, J Schmiedmayer, F Essler

<jats:p>We consider time-of-flight measurements in split one-dimensional Bose gases. It is well known that the low-energy sector of such systems can be described in terms of two compact phase fields <jats:inline-formula><jats:alternatives><jats:tex-math>\hat{\phi}_{a,s}(x)</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mover><mml:mi>ϕ</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mrow><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. Building on existing results in the literature we discuss how a single projective measurement of the particle density after trap release is in a certain limit related to the eigenvalues of the vertex operator <jats:inline-formula><jats:alternatives><jats:tex-math>e^{i\hat{\phi}_a(x)}</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mover><mml:mi>ϕ</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula>. We emphasize the theoretical assumptions underlying the analysis of “single-shot” interference patterns and show that such measurements give direct access to multi-point correlation functions of <jats:inline-formula><jats:alternatives><jats:tex-math>e^{i\hat{\phi}_a(x)}</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:msub><mml:mover><mml:mi>ϕ</mml:mi><mml:mo accent="true">̂</mml:mo></mml:mover><mml:mi>a</mml:mi></mml:msub><mml:mo stretchy="false" form="prefix">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false" form="postfix">)</mml:mo></mml:mrow></mml:msup></mml:math></jats:alternatives></jats:inline-formula> in a substantial parameter regime. For experimentally relevant situations, we derive an expression for the measured particle density after trap release in terms of convolutions of the eigenvalues of vertex operators involving both sectors of the two-component Luttinger liquid that describes the low-energy regime of the split condensate. This opens the door to accessing properties of the symmetric sector via an appropriate analysis of existing experimental data.</jats:p>

## Exotic criticality in the dimerized spin-1 $XXZ$ chain with single-ion anisotropy

SciPost Physics Stichting SciPost 5 (2018) 059

S Ejima, T Yamaguchi, F Essler, F Lange, Y Ohta, H Fehske

<jats:p>We consider the dimerized spin-1 XXZ chain with single-ion anisotropy D. In absence of an explicit dimerization there are three phases: a large-$D$, an antiferromagnetically ordered and a Haldane phase. This phase structure persists up to a critical dimerization, above which the Haldane phase disappears. We show that for weak dimerization the phases are separated by Gaussian and Ising quantum phase transitions. One of the Ising transitions terminates in a critical point in the universality class of the dilute Ising model. We comment on the relevance of our results to experiments on quasi-one-dimensional anisotropic spin-1 quantum magnets.</jats:p>

## Spinon decay in the spin-1/2 Heisenberg chain with weak next nearest neighbour exchange

Journal of Physics A: Mathematical and Theoretical IOP Publishing 50 (2017) 334002

S Groha, F Essler

Integrable models support elementary excitations with infinite lifetimes. In the spin-1/2 Heisenberg chain these are known as spinons. We consider the stability of spinons when a weak integrability breaking perturbation is added to the Heisenberg chain in a magnetic field. We focus on the case where the perturbation is a next nearest neighbour exchange interaction. We calculate the spinon decay rate in leading order in perturbation theory using methods of integrability and identify the dominant decay channels. The decay rate is found to be small, which indicates that spinons remain well-defined excitations even though integrability is broken.

## On truncated generalized Gibbs ensembles in the Ising field theory

Journal of Statistical Mechanics: Theory and Experiment IOP Publishing 2017 (2017) 013103-

F Essler, G Mussardo, M Panfil

<p>We discuss the implementation of two dierent truncated Generalized Gibbs Ensembles (GGE) describing the stationary state after a mass quench process in the Ising Field Theory. One truncated GGE is based on the semi-local charges of the model, the other on regularized versions of its ultra-local charges. We test the efficiency of the two different ensembles by comparing their predictions for the stationary state values of the single-particle Green's function <i>G(x) = ⟨Ψ<sup>†</sup>(x)Ψ(0)⟩</i> of the complex fermion field <i>Ψ(x)</i>. We find that both truncated GGEs are able to recover <i>G(x)</i>, but for a given number of charges the semi-local version performs better.</p>

## Full counting statistics in the spin-1/2 Heisenberg XXZ chain

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 50 (2017) ARTN 414002

M Collura, FHL Essler, S Groha

## Thermalization and light cones in a model with weak integrability breaking

Physical Review B American Physical Society 94 (2017) 245117

B Bertini, F Essler, S Groha, NJ Robinson

We employ equation of motion techniques to study the non-equilibrium dynamics in a lattice model of weakly interacting spinless fermions. Our model provides a simple setting for analyzing the effects of weak integrability breaking perturbations on the time evolution after a quantum quench. We establish the accuracy of the method by comparing results at short and intermediate times to time-dependent density matrix renormalization group computations. For sufficiently weak integrability-breaking interactions we always observe prethermalization plateaux, where local observables relax to non-thermal values at intermediate time scales. At later times a crossover towards thermal behaviour sets in. We determine the associated time scale, which depends on the initial state, the band structure of the non-interacting theory, and the strength of the integrability breaking perturbation. Our method allows us to analyze in some detail the spreading of correlations and in particular the structure of the associated light cones in our model. We find that the interior and exterior of the light cone are separated by an intermediate region, the temporal width of which appears to scale with a universal power-law $t^{1/3}$.

## Spinon confinement in a quasi-one-dimensional anisotropic Heisenberg magnet

PHYSICAL REVIEW B 96 (2017) ARTN 054423

AK Bera, B Lake, FHL Essler, L Vanderstraeten, C Hubig, U Schollwoeck, ATMN Islam, A Schneidewind, DL Quintero-Castro

## Atypical energy eigenstates in the Hubbard chain and quantum disentangled liquids.

Philosophical transactions. Series A, Mathematical, physical, and engineering sciences 375 (2017)

T Veness, FHL Essler, MPA Fisher

We investigate the implications of integrability for the existence of quantum disentangled liquid (QDL) states in the half-filled one-dimensional Hubbard model. We argue that there exist finite energy-density eigenstates that exhibit QDL behaviour in the sense of Grover & Fisher (2014 J. Stat. Mech.2014, P10010. (doi:10.1088/1742-5468/2014/10/P10010)). These states are atypical in the sense that their entropy density is smaller than that of thermal states at the same energy density. Furthermore, we show that thermal states in a particular temperature window exhibit a weaker form of the QDL property, in agreement with recent results obtained by strong-coupling expansion methods in Veness et al. (2016 (http://arxiv.org/abs/1611.02075)).This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

## Quantum disentangled liquid in the half-filled Hubbard model

PHYSICAL REVIEW B 96 (2017) ARTN 195153

T Veness, FHL Essler, MPA Fisher

## Exact Bethe ansatz spectrum of a tight-binding chain with dephasing noise

Physical Review Letters American Physical Society 117 (2016) 137202

MV Medvedyeva, F Essler, T Prosen

We construct an exact map between a tight-binding model on any bipartite lattice in the presence of dephasing noise and a Hubbard model with imaginary interaction strength. In one dimension, the exact many-body Liouvillian spectrum can be obtained by application of the Bethe ansatz method. We find that both the nonequilibrium steady state and the leading decay modes describing the relaxation at late times are related to the η-pairing symmetry of the Hubbard model. We show that there is a remarkable relation between the time evolution of an arbitrary k-point correlation function in the dissipative system and k-particle states of the corresponding Hubbard model.

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

SciPost Physics SciPost (2016)

L Piroli, L, P Calabrese, P, F Essler

We study quantum quenches to the one-dimensional Bose gas with attractive interactions in the case when the initial state is an ideal one-dimensional Bose condensate. We focus on properties of the stationary state reached at late times after the quench. This displays a finite density of multi-particle bound states, whose rapidity distribution is determined exactly by means of the quench action method. We discuss the relevance of the multi-particle bound states for the physical properties of the system, computing in particular the stationary value of the local pair correlation function g2.