# Publications

###### Theses:

Entanglement & Correlations in exactly solvable models

PhD, University of Pisa (Italy)
March 2012
supervisor: Pasquale Calabrese
[pdf]

Dinamica di non-equilibrio dell’entanglement e delle correlazioni in una catena di spin

Master, University of Pisa (Italy)
March 2009
supervisor: Pasquale Calabrese
[pdf]

La fase di Berry – la fase geometrica in meccanica quantistica

Bachelor, University of Pisa (Italy)
December 2005
supervisor: Mihail Mintchev
[pdf]

Papers

arXiv

Bibliometrics

###### List of papers:

[the articles highlighted in orange are highly cited in the field (according to the database Web of Science on 26/05/2020)]

• L. Zadnik, S. Bocini, K. Bidzhiev, and M. Fagotti
Measurement catastrophe and ballistic spread of charge density with vanishing current
[cited by]

 One of the features of many-body quantum systems with Hilbert-space fragmentation is the existence of stationary states manifesting quantum jamming. It was recently shown that these are “states with memory”, in which, e.g., measuring a localised observable has everlasting macroscopic effects. We study such a measurement catastrophe with an example that stands out for its clarity. We show in particular that at late times the expectation value of a charge density becomes a nontrivial function of the ratio between distance and time notwithstanding the corresponding current approaching zero.
• M. Fagotti
Global Quenches after Localised Perturbations

 We investigate the effect of a single spin flip preceding a global quench between translationally invariant local Hamiltonians in spin-1/2 chains. The effect of the localised perturbation does not fade away however large the distance from the perturbation is. In particular, translational invariance is not restored and the infinite time limit depends on whether the spin was flipped or not. We argue that this phenomenon is more general than the particular example considered and we conjecture that it is triggered by topological properties, specifically, the existence of “semilocal charges”.
• K. Bidzhiev, M. Fagotti, and L. Zadnik
Macroscopic effects of localised measurements in jammed states of quantum spin chains

 A quantum jammed state can be seen as a state where the phase space available to particles shrinks to zero, an interpretation quite accurate in integrable systems, where stable quasiparticles scatter elastically. We consider the integrable dual folded XXZ model, which is equivalent to the XXZ model in the limit of large anisotropy. We perform a jamming-breaking localised measurement in a jammed state. We find that jamming is locally restored, but local observables exhibit nontrivial time evolution on macroscopic, ballistic scales, without ever relaxing back to their initial values.
• V. Alba, B. Bertini, M. Fagotti, L. Piroli, and P. Ruggiero
Generalized-Hydrodynamic approach to Inhomogeneous Quenches: Correlations, Entanglement and Quantum Effects

 We give a pedagogical introduction to the Generalized Hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems: bipartitioning protocols and trap quenches, which represent two prototypical examples of broken translational symmetry in either the system initial state or post-quench Hamiltonian. We report on exact results that have been obtained for generic time-dependent correlation functions and entanglement evolution, and discuss in detail the range of applicability of the theory. Finally, we present some open questions and suggest perspectives on possible future directions.
• L. Zadnik, K. Bidzhiev, and M. Fagotti
The Folded Spin-1/2 XXZ Model:
II. Thermodynamics and Hydrodynamics with a Minimal Set of Charges

 We study the (dual) folded spin-1/2 XXZ model in the thermodynamic limit. We focus, in particular, on a class of “local” macrostates that includes Gibbs ensembles. We develop a thermodynamic Bethe Ansatz description and work out generalised hydrodynamics at the leading order. Remarkably, in the ballistic scaling limit the junction of two local macrostates results in a discontinuity in the profile of essentially any local observable.
• L. Zadnik and M. Fagotti
The Folded Spin-1/2 XXZ Model:
I. Diagonalisation, Jamming, and Ground State Properties

 We study an effective Hamiltonian generating time evolution of states on inter- mediate time scales in the strong-coupling limit of the spin-1/2 XXZ model. To leading order, it describes an integrable model with local interactions. We solve it completely by means of a coordinate Bethe Ansatz that manifestly breaks the translational symmetry. We demonstrate the existence of exponentially many jammed states and estimate their stability under the leading correction to the effective Hamiltonian. Some ground state properties of the model are discussed.
• E. Granet, M. Fagotti, and  F. H. L Essler
Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions

 We consider the problems of calculating the dynamical order parameter two-point function at finite temperatures and the one-point function after a quantum quench in the transverse field Ising chain. Both of these can be expressed in terms of form factor sums in the basis of physical excitations of the model. We develop a general framework for carrying out these sums based on a decomposition of form factors into partial fractions, which leads to a factorization of the multiple sums and permits them to be evaluated asymptotically. This naturally leads to systematic low density expansions. At late times these expansions can be summed to all orders by means of a determinant representation. Our method has a natural generalization to semi-local operators in interacting integrable models.
• Maurizio Fagotti
Locally quasi-stationary states in noninteracting spin chains

 Locally quasi-stationary states (LQSS) were introduced as inhomogeneous generalisations of stationary states in integrable systems. Roughly speaking, LQSSs look like stationary states, but only locally. Despite their key role in hydrodynamic descriptions, an unambiguous definition of LQSSs was not given. By solving the dynamics in inhomogeneous noninteracting spin chains, we identify the set of LQSSs as a subspace that is invariant under time evolution, and we explicitly construct the latter in a generalised XY model. As a by-product, we exhibit an exact generalised hydrodynamic theory (including “quantum corrections”).
• Vincenzo Alba, Bruno Bertini, and Maurizio Fagotti
Entanglement evolution and generalised hydrodynamics: interacting integrable systems

 We investigate the dynamics of bipartite entanglement after the sudden junction of two leads in interacting integrable models. By combining the quasiparticle picture for the entanglement spreading with Generalised Hydrodynamics we derive an analytical prediction for the dynamics of the entanglement entropy between a finite subsystem and the rest. We find that the entanglement rate between the two leads depends only on the physics at the interface and differs from the rate of exchange of thermodynamic entropy. This contrasts with the behaviour in free or homogeneous interacting integrable systems, where the two rates coincide.
• Maurizio Fagotti
On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system

 We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites $$L^d$$. We compute the distribution of the eigenvalues of the time averaged state over a time window $$[t_0,t_0+t]$$ in the limit of large $$L$$. This allows us to infer the size of a subspace that captures time evolution in $$[t_0,t_0+t]$$ with an accuracy $$1-\epsilon$$. We estimate the size to be $$\frac{\sqrt{2\mathfrak{e}_2}}{\pi}\mathrm{erf}^{-1}(1-\epsilon) L^{\frac{d}{2}}t$$, where $$\mathfrak{e}_2$$ is the energy variance per site, and $$\mathrm{erf}^{-1}$$ is the inverse error function.
• Bruno Bertini, Maurizio Fagotti, Lorenzo Piroli, and Pasquale Calabrese
Entanglement evolution and generalised hydrodynamics: noninteracting systems

 The large-scale properties of homogeneous states after quantum quenches in integrable systems have been successfully described by a semiclassical picture of moving quasiparticles. Here we consider the generalisation for the entanglement evolution after an inhomogeneous quench in noninteracting systems in the framework of generalised hydrodynamics. We focus on the protocol where two semi-infinite halves are initially prepared in different states and then joined together, showing that a proper generalisation of the quasiparticle picture leads to exact quantitative predictions. If the system is initially prepared in a quasistationary state, we find that the entanglement entropy is additive and it can be computed by means of generalised hydrodynamics. Conversely, additivity is lost when the initial state is not quasistationary; yet the entanglement entropy in the large-scale limit can be exactly predicted in the quasiparticle picture, provided that the initial state is low entangled.
• Maurizio Fagotti
Higher-order generalized hydrodynamics in one dimension: The noninteracting test
Phys. Rev. B 96, 220302(R) (2017) [cited by]

 We derive a dynamical equation that describes the exact time evolution in generic (inhomogeneous) noninteracting spin-chain models. Assuming quasistationarity, we develop a (generalized) hydrodynamic theory. The question at hand is whether some large-time corrections are captured by higher-order hydrodynamics. We consider in particular the dynamics after two chains, prepared in different conditions, are joined together. In these situations, a light cone, separating regions with macroscopically different properties, emerges from the junction. In free fermionic systems some observables close to the light cone follow a universal behavior, known as Tracy-Widom scaling. Universality means a weak dependence on the system’s details, so this is the perfect setting where hydrodynamics could emerge. For the transverse-field Ising chain and the XX model, we show that hydrodynamics captures the scaling behavior close to the light cone. On the other hand, our numerical analysis suggests that hydrodynamics fails in more general models, whenever a condition is not satisfied.
• Lorenzo Piroli, Jacopo De Nardis, Mario Collura, Bruno Bertini, and Maurizio Fagotti
Transport in out-of-equilibrium XXZ chains: Nonballistic behavior and correlation functions
Phys. Rev. B 96, 115124 (2017) [cited by]

 We consider the nonequilibrium protocol where two semi-infinite gapped XXZ chains, initially prepared in different equilibrium states, are suddenly joined together. At large times, a generalized hydrodynamic description applies, according to which the system can locally be represented by space- and time-dependent stationary states. The magnetization displays an unusual behavior: depending on the initial state, its profile may exhibit abrupt jumps that can not be predicted directly from the standard hydrodynamic equations and which signal nonballistic spin transport. We ascribe this phenomenon to the structure of the local conservation laws and make a prediction for the exact location of the jumps. We find that the jumps propagate at the velocities of the heaviest quasiparticles. By means of time-dependent density matrix renormalization group simulations we show that our theory yields a complete description of the long-time steady profiles of conserved charges, currents, and local correlations.
• Vincenzo Alba and Maurizio Fagotti
Prethermalization at Low Temperature: The Scent of Long-Range Order
Phys. Rev. Lett. 119, 010601 (2017) [cited by]

 Nonequilibrium time evolution in isolated many-body quantum systems generally results in thermalization. However, the relaxation process can be very slow, and quasistationary nonthermal plateaux are often observed at intermediate times. The paradigmatic example is a quantum quench in an integrable model with weak integrability breaking; for a long time, the state cannot escape the constraints imposed by the approximate integrability. We unveil a new mechanism of prethermalization, based on the presence of a symmetry of the prequench Hamiltonian, which is spontaneously broken at zero temperature and is explicitly broken by the postquench Hamiltonian. The typical time scale of the phenomenon is proportional to the thermal correlation length of the initial state, which diverges as the temperature is lowered. We show that the prethermal quasistationary state can be approximated by a mixed state that violates cluster decomposition property. We consider two examples: the transverse-field Ising chain, where the full-time evolution is computed analytically, and the (nonintegrable) anisotropic next-nearest-neighbor Ising model, which is investigated numerically.
• Maurizio Fagotti
Charges and currents in quantum spin chains: late-time dynamics and spontaneous currents
2017 J. Phys. A: Math. Theor. 50 034005 [cited by]

 We review the structure of the conservation laws in noninteracting spin chains and unveil a formal expression for the corresponding currents. We briefly discuss how interactions affect the picture. In the second part, we explore the effects of a localized defect. We show that the emergence of spontaneous currents near the defect undermines any description of the late-time dynamics by means of a stationary state in a finite chain. In particular, the diagonal ensemble does not work. Finally, we provide numerical evidence that simple generic localized defects are not sufficient to induce thermalization.
• Bruno Bertini, Mario Collura, Jacopo De Nardis, and Maurizio Fagotti
Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
Phys. Rev. Lett. 117, 207201 (2016) [cited by]

 We consider the nonequilibrium time evolution of piecewise homogeneous states in the XXZ spin-1/2 chain, a paradigmatic example of an interacting integrable model. The initial state can be thought of as the result of joining chains with different global properties. Through dephasing, at late times, the state becomes locally equivalent to a stationary state which explicitly depends on position and time. We propose a kinetic theory of elementary excitations and derive a continuity equation which fully characterizes the thermodynamics of the model. We restrict ourselves to the gapless phase and consider cases where the chains are prepared: (1) at different temperatures, (2) in the ground state of two different models, and (3) in the “domain wall” state. We find excellent agreement (any discrepancy is within the numerical error) between theoretical predictions and numerical simulations of time evolution based on time-evolving block decimation algorithms. As a corollary, we unveil an exact expression for the expectation values of the charge currents in a generic stationary state.
• Bruno Bertini and Maurizio Fagotti
Determination of the Nonequilibrium Steady State Emerging from a Defect
Phys. Rev. Lett. 117, 130402 (2016) [cited by]

 We consider the nonequilibrium time evolution of a translationally invariant state under a Hamiltonian with a localized defect. We discern the situations where a light cone spreads out from the defect and separates the system into regions with macroscopically different properties. We identify the light cone and propose a procedure to obtain a (quasi)stationary state describing the late time dynamics of local observables. As an explicit example, we study the time evolution generated by the Hamiltonian of the transverse-field Ising chain with a local defect that cuts the interaction between two sites (a quench of the boundary conditions alongside a global quench). We solve the dynamics exactly and show that the late time properties can be obtained with the general method proposed.
• Fabian H L Essler and Maurizio Fagotti

 We review the dynamics after quantum quenches in integrable quantum spin chains. We give a pedagogical introduction to relaxation in isolated quantum systems, and discuss the description of the steady state by (generalized) Gibbs ensembles. We then turn to general features in the time evolution of local observables after the quench, using a simple model of free fermions as an example. In the second part we present an overview of recent progress in describing quench dynamics in two key paradigms for quantum integrable models, the transverse field Ising chain and the anisotropic spin-$$\frac{1}{2}$$ Heisenberg chain.
• Anton S. Buyskikh, Maurizio Fagotti, Johannes Schachenmayer, Fabian Essler, and Andrew J. Daley
Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models
Phys. Rev. A 93, 053620 (2016)

 We investigate the dynamics following a global parameter quench for two one-dimensional models with variable-range power-law interactions: a long-range transverse Ising model, which has recently been realized in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunneling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunneling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions (in terms of separation distance r, decaying slower than $$1/r$$), we find that despite the lack of a light cone, correlations grow slowly as a power law at short times, and that—depending on the structure of the initial state—the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.
• Maurizio Fagotti
Local conservation laws in spin-1/2 XY chains with open boundary conditions
J. Stat. Mech. (2016) 063105

 We revisit the conserved quantities of the spin-$$\frac{1}{2}$$ XY model with open boundary conditions. In the absence of a transverse field, we find new families of local charges and show that half of the seeming conservation laws are conserved only if the number of sites is odd. In even chains the set of noninteracting charges is abelian, like in the periodic case when the number of sites is odd. In odd chains the set is doubled and becomes non-abelian, like in even periodic chains. The dependence of the charges on the parity of the chain’s size undermines the common belief that the thermodynamic limit of diagonal ensembles exists. We consider also the transverse-field Ising chain, where the situation is more ordinary. The generalization to the XY model in a transverse field is not straightforward and we propose a general framework to carry out similar calculations. We conjecture the form of the bulk part of the local charges and discuss the emergence of quasilocal conserved quantities. We provide evidence that in a region of the parameter space there is a reduction of the number of quasilocal conservation laws invariant under chain inversion. As a by-product, we study a class of block-Toeplitz-plus-Hankel operators and identify the conditions that their symbols satisfy in order to commute with a given block-Toeplitz.
• Maurizio Fagotti
Control of global properties in a closed many-body quantum system by means of a local switch

 We consider non-equilibrium time evolution after a quench of a global Hamiltonian parameter in systems described by Hamiltonians with local interactions. Within this background, we propose a protocol that allows to change global properties of the state by flipping a switch that modifies a local term of the Hamiltonian (creating a defect). A light-cone that separates two globally different regions originates from the switch. The expectation values of macroscopic observables, that is to say local observables that are spatially averaged within a subsystem, slowly approach new asymptotic values determined by the defect. The process is almost reversible: flipping again the switch produces a new light-cone with the two regions inverted. Finally, we test the protocol under repeated projective measurements. As explicit example we study the dynamics in a simple exactly solvable model but analogues descriptions apply also to generic models.
• Maurizio Fagotti and Mario Collura
Universal prethermalization dynamics of entanglement entropies after a global quench

 We consider the quantum XY model and study the effects of interacting perturbations on the time evolution of the von Neumann and R\’enyi entropies of spin blocks after global quenches. We show that the entropies are sensitive to perturbations that break hidden symmetries behind the integrability of the model. At times much larger than the characteristic time of the well-known linear increase of the entropies, we identify a time window characterized by a novel linear growth followed by saturation. The typical time of the phenomenon is inversely proportional to the perturbation strength and the behavior is trigger off by the extinction of an infinite number of local conservation laws following a non-abelian algebra. The universality of the crossover is revealed by a semi-classical picture that captures the leading behavior of the entropies. We check our theoretical predictions against iTEBD simulations. The good agreement between theory and numerics substantiates the method developed in [Bertini and Fagotti, J. Stat. Mech. (2015) P07012] for investigating a pre-relaxation limit in weakly interacting models.
• Bruno Bertini and Maurizio Fagotti
Pre-relaxation in weakly interacting models
J. Stat. Mech. (2015) P07012

 We consider time evolution in models close to integrable points with hidden symmetries that generate infinitely many local conservation laws that do not commute with one another. The system is expected to (locally) relax to a thermal ensemble if integrability is broken, or to a so-called generalised Gibbs ensemble if unbroken. In some circumstances expectation values exhibit quasi-stationary behaviour long before their typical relaxation time. For integrability-breaking perturbations, these are also called pre-thermalisation plateaux, and emerge e.g. in the strong coupling limit of the Bose–Hubbard model. As a result of the hidden symmetries, quasi-stationarity appears also in integrable models, for example in the Ising limit of the XXZ model. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-$$\frac{1}{2}$$ chain with additional perturbations that break integrability. One of the most intriguing results of the analysis is the appearance of persistent oscillatory behaviour. To unravel its origin, we study in detail a toy model: the transverse-field Ising chain with an additional nonlocal interaction proportional to the square of the transverse spin per unit length (2013 Phys. Rev. Lett. 111 197203). Despite being nonlocal, this belongs to a class of models that emerge as intermediate steps of the mean-field mapping and shares many dynamical properties with the weakly interacting models under consideration.
• Maurizio Fagotti
Conservation laws for a class of generic Hamiltonians

 Within a strong coupling expansion, we construct local quasi-conserved operators for a class of Hamiltonians that includes both integrable and non-integrable models. We explicitly show that at the lowest orders of perturbation theory the structure of the operators is independent of the system details. Higher order contributions are investigated numerically by means of an ab initio method for computing the time evolution of local operators in the Heisenberg picture. The numerical analysis suggests that the quasi-conserved operators could be approximations of a quasi-local conservation law, even if the model is non-integrable.
• Demetrio Logoteta, Paolo Marconcini, Claudio Bonati, Maurizio Fagotti, Massimo Macucci
High-performance solution of the transport problem in a graphene armchair structure with a generic potential
Phys. Rev. E 89, 063309 (2014)

 We propose an efficient numerical method to study the transport properties of armchair graphene ribbons in the presence of a generic external potential. The method is based on a continuum envelope-function description with physical boundary conditions. The envelope functions are computed in the reciprocal space, and the transmission is then obtained with a recursive scattering matrix approach. This allows a significant reduction of the computational time with respect to finite difference simulations.
• Maurizio Fagotti
On Conservation Laws, Relaxation and Pre-relaxation after a Quantum Quench
J. Stat. Mech. (2014) P03016

 We consider the time evolution following a quantum quench in spin-$$\frac{1}{2}$$ chains. It is well known that local conservation laws constrain the dynamics and, eventually, the stationary behavior of local observables. We show that some widely studied models, such as the quantum XY model, possess extra families of local conservation laws in addition to the translation invariant ones. As a consequence, the additional charges must be included in the generalized Gibbs ensemble that describes the stationary properties. The effects go well beyond a simple redefinition of the stationary state. The time evolution of a non-translation-invariant state under a (translation-invariant) Hamiltonian with a perturbation that weakly breaks the hidden symmetries underlying the extra conservation laws exhibits pre-relaxation. In addition, in the limit of small perturbation, the time evolution following pre-relaxation can be described by means of a time-dependent generalized Gibbs ensemble.
• Maurizio Fagotti, Mario Collura, Fabian H L Essler, and Pasquale Calabrese
Relaxation after quantum quenches in the spin-1/2 Heisenberg XXZ chain
Phys. Rev. B 89, 125101 (2014)

 We consider the time evolution after quantum quenches in the spin-$$\frac{1}{2}$$ Heisenberg XXZ quantum spin chain with Ising-type anisotropy. The time evolution of short-distance spin-spin correlation functions is studied by numerical tensor network techniques for a variety of initial states, including Néel and Majumdar-Ghosh states and the ground state of the XXZ chain at large values of the anisotropy. The various correlators appear to approach stationary values, which are found to be in good agreement with the results of exact calculations of stationary expectation values in appropriate generalized Gibbs ensembles. In particular, our analysis shows how symmetries of the post-quench Hamiltonian that are broken by particular initial states are restored at late times.
• Maurizio Fagotti
Dynamical Phase Transitions as Properties of the Stationary State: Analytic Results after Quantum Quenches in the Spin-1/2 XXZ Chain

 The (Loschmidt) overlap between the state at different times after a quantum quench is attracting increasing interest, as it was recently shown that in the thermodynamic limit its logarithm per unit of length has a non-analytic behavior if a Hamiltonian parameter is quenched across a critical point. This phenomenon was called a “dynamical phase transition” in analogy with the behavior of the canonical partition function at an equilibrium phase transition. We distinguish between local and nonlocal contributions to the aforementioned quantity and derive an analytic expression for the time evolution of the local part after quantum quenches in the XXZ spin-$$\frac{1}{2}$$ chain. The state that describes the stationary properties of (local) observables can be represented by a Gibbs ensemble of a generalized Hamiltonian; we reveal a deep connection between the appearance of singularities and the excitation energies of the generalized Hamiltonian.
• Maurizio Fagotti and Fabian H L Essler
Stationary behaviour of observables after a quantum quench in the spin-1/2 Heisenberg XXZ chain
J. Stat. Mech. (2013) P07012

 We consider a quantum quench in the spin-$$\frac{1}{2}$$ Heisenberg XXZ chain. At late times after the quench it is believed that the expectation values of local operators approach time-independent values that are described by a generalized Gibbs ensemble. Employing a quantum transfer matrix approach we show how to determine short-range correlation functions in such generalized Gibbs ensembles for a class of initial states.
• Maurizio Fagotti and Fabian H L Essler
Reduced density matrix after a quantum quench
Phys. Rev. B 87, 245107 (2013)

 We consider the reduced density matrix (RDM) $$\rho_A(t)$$ for a finite subsystem A after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of $$\rho_A(t)$$ is described by the RDM $$\rho_{{\rm GGE}, A}$$ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how $$\rho_A(t)$$ approaches $$\rho_{{\rm GGE}, A}$$ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law $$t^{-3/2}$$ in time. As the RDM completely determines all local observables within A, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.
• Maurizio Fagotti
Finite-size corrections versus relaxation after a sudden quench
Phys. Rev. B 87, 165106 (2013)

 We consider the time evolution after sudden quenches of global parameters in translational invariant Hamiltonians and study the time average expectation values and entanglement entropies in finite chains. We show that in noninteracting models the time average of spin correlation functions is asymptotically equal to the infinite time limit in the infinite chain, which is known to be described by a generalized Gibbs ensemble. The equivalence breaks down considering nonlocal operators, and we establish that this can be traced back to the existence of conservation laws common to the Hamiltonian before and after the quench. We develop a method to compute the leading finite-size corrections. We find that the finite-size corrections are generally large in observables with large relaxation timescales.
• Fabian H L Essler, Stefano Evangelisti, and Maurizio Fagotti
Dynamical Correlations After a Quantum Quench
Phys. Rev. Lett. 109, 247206 (2012)

 We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.
• Pasquale Calabrese, Fabian H L Essler, and Maurizio Fagotti

 We consider the stationary state properties of the reduced density matrix as well as spin–spin correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. We demonstrate that stationary state properties are described by a generalized Gibbs ensemble. We discuss the approach to the stationary state at late times.
• Pasquale Calabrese, Fabian H L Essler, and Maurizio Fagotti

 We consider the time evolution of order parameter correlation functions after a sudden quantum quench of the magnetic field in the transverse field Ising chain. Using two novel methods based on determinants and form factor sums respectively, we derive analytic expressions for the asymptotic behaviour of one- and two-point correlators. We discuss quenches within the ordered and disordered phases as well as quenches between the phases and to the quantum critical point. We give detailed accounts of both methods.
• Maurizio Fagotti
New insights into the entanglement of disjoint blocks
2012 EPL 97 17007

 We study the entanglement of two disjoint blocks in spin-1/2 chains obtained by merging solvable models, such as XX and quantum Ising models. We focus on the universal quantities that can be extracted from the Rényi entropies $$S_\alpha$$. The most important information is encoded in some functions denoted by $$F_\alpha$$. We compute $$F_2$$ and we show that $$F_\alpha-1$$ and $$F_{\rm v. N.}$$, corresponding to the von Neumann entropy, can be negative, in contrast to what observed in all models examined so far. An exact relation between the entanglement of disjoint subsystems in the XX model and that in a chain embodying two quantum Ising models is a by-product of our investigations.
• Pasquale Calabrese, Fabian H L Essler, and Maurizio Fagotti

 We consider the time evolution of observables in the transverse-field Ising chain after a sudden quench of the magnetic field. We provide exact analytical results for the asymptotic time and distance dependence of one- and two-point correlation functions of the order parameter. We employ two complementary approaches based on asymptotic evaluations of determinants and form-factor sums. We prove that the stationary value of the two-point correlation function is not thermal, but can be described by a generalized Gibbs ensemble (GGE). The approach to the stationary state can also be understood in terms of a GGE. We present a conjecture on how these results generalize to particular quenches in other integrable models.
• Maurizio Fagotti, Claudio Bonati, Demetrio Logoteta, Paolo Marconcini, and Massimo Macucci
Armchair graphene nanoribbons: PT-symmetry breaking and exceptional points without dissipation
Phys. Rev. B 83, 241406(R) (2011)

 We consider a single-layer graphene nanoribbon with armchair edges and with a longitudinally constant external potential, pointing out that it can be described by means of an effective non-Hermitian Hamiltonian. We show that this system has some features typical of dissipative systems, namely, the presence of exceptional points and of PT-symmetry breaking, although it is not dissipative.
• Maurizio Fagotti and Pasquale Calabrese
Universal parity effects in the entanglement entropy of XX chains with open boundary conditions
J. Stat. Mech. (2011) P01017

 We consider the Rényi entanglement entropies in the one-dimensional XX spin-chains with open boundary conditions in the presence of a magnetic field. In the case of a semi-infinite system and a block starting from the boundary, we derive rigorously the asymptotic behavior for large block sizes on the basis of a recent mathematical theorem for the determinant of Toeplitz plus Hankel matrices. We conjecture a generalized Fisher–Hartwig form for the corrections to the asymptotic behavior of this determinant that allows the exact characterization of the corrections to the scaling at order $$o(\ell^{-1})$$ for any n. By combining these results with conformal field theory arguments, we derive exact expressions also in finite chains with open boundary conditions and in the case when the block is detached from the boundary.
•  Maurizio Fagotti, Pasquale Calabrese, and Joel E Moore
Entanglement spectrum of random-singlet quantum critical points
Phys. Rev. B 83, 045110 (2011)

 The entanglement spectrum (i.e., the full distribution of Schmidt eigenvalues of the reduced density matrix) contains more information than the conventional entanglement entropy and has been studied recently in several many-particle systems. We compute the disorder-averaged entanglement spectrum in the form of the disorder-averaged moments $$\mathrm{Tr}[\rho_A^{\alpha}]$$ of the reduced density matrix $$\rho_A$$ for a contiguous block of many spins at the random-singlet quantum critical point in one dimension. The result compares well in the scaling limit with numerical studies on the random XX model and is also expected to describe the (interacting) random Heisenberg model. Our numerical studies on the XX case reveal that the dependence of the entanglement entropy and spectrum on the geometry of the Hilbert space partition is quite different than for conformally invariant critical points.
• Maurizio Fagotti and Pasquale Calabrese
Entanglement entropy of two disjoint blocks in XY chains
J. Stat. Mech. (2010) P04016 [cited by]

 We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.
• Vincenzo Alba, Maurizio Fagotti, and Pasquale Calabrese
Entanglement entropy of excited states
J. Stat. Mech. (2009) P10020

 We study the entanglement entropy of a block of contiguous spins in excited states of spin chains. We consider the XY model in a transverse field and the XXZ Heisenberg spin chain. For the latter, we developed a numerical application of the algebraic Bethe ansatz. We find two main classes of states with logarithmic and extensive behavior in the dimension of the block, characterized by the properties of excitations of the state. This behavior can be related to the locality properties of the Hamiltonian having a given state as the ground state. We also provide several details of the finite size scaling.
• Maurizio Fagotti and Pasquale Calabrese
Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field
Phys. Rev. A 78, 010306(R) (2008)

 The nonequilibrium evolution of the block entanglement entropy is investigated in the XY chain in a transverse magnetic field after the Hamiltonian parameters are suddenly changed from and to arbitrary values. Using Toeplitz matrix representation and multidimensional phase methods, we provide analytic results for large blocks and for all times, showing explicitly the linear growth in time followed by saturation. The consequences of these analytic results are discussed and the effects of a finite block length is taken into account numerically.