We consider an interaction quench in the critical spin-1/2 Heisenberg XXZ chain. We numerically compute the time evolution of the two-point correlation functions of spin operators in the thermodynamic limit and compare the results to predictions obtained in the framework of the Luttinger liquid approximation. We find that the transverse correlation function $\langle S^x_jS^x_{j+\ell}\rangle$ agrees with the Luttinger model prediction to a surprising level of accuracy. The agreement for the longitudinal two-point function $\langle S^z_jS^z_{j+\ell}\rangle$ is found to be much poorer. We speculate that this difference between transverse and longitudinal correlations has its origin in the locality properties of the respective spin operator with respect to the underlying fermionic modes.
The ground-state phase diagram of a spin-1/2 XXZ chain with competing ferromagnetic nearest-neighbor (J_1<0) and antiferromagnetic second-neighbor (J_2>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range -4<J_1/J_2<0 has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid st…
We investigate the edge modes at T=0 in the intermediate-D (ID) phase and the large-D (LD) phase of the S=2 quantum spin chain with the XXZ anisotropy and the generalized on-site anisotropies by use of the DMRG. There exists a gapless edge mode in the ID phase, while no gapless edge mode in the LD phase. These results are consistent with the physical pictures of these phases. We also show the ground-state phase diagrams obtained by use of the exact diagonalization and the level spectroscopy analysis.
We consider a quantum quench in the spin-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.
The steady state after a quantum quench from the Néel state to the anisotropic Heisenberg model for spin chains is investigated. Two methods that aim to describe the postquench non-thermal equilibrium, the generalized Gibbs ensemble and the quench action approach, are discussed and contrasted. Using the recent implementation of the quench action approach for this Néel-to-XXZ quench, we obtain an exact description of the steady state in terms of Bethe root densities, for which we give explicit analytical expressions. Furthermore, by developing a systematic small-quench expansion around the anti…
Following our previous work [PRL 113 (2014) 09020] we present here a detailed comparison of the quench action approach and the predictions of the generalized Gibbs ensemble, with the result that while the quench action formalism correctly captures the steady state, the GGE does not give a correct description of local short-distance correlation functions. We extend our studies to include another initial state, the so-called q-dimer state. We present important details of our construction, including new results concerning exact overlaps for the dimer and q-dimer states, and we also give an exact …
We investigate the dynamics of the XXZ spin chain after a geometric quench, which is realized by connecting two half-chains prepared in their ground states with zero and maximum magnetizations, respectively. The profiles of magnetization after the subsequent time evolution are studied numerically by density-matrix renormalization group methods, and a comparison to the predictions of generalized hydrodynamics yields a very good agreement. We also calculate the profiles of entanglement entropy and propose an ansatz for the noninteracting XX case, based on arguments from conformal field theory. I…
We consider the time evolution after quantum quenches in the spin-1/2 Heisenberg XXZ quantum spin chain with Ising-like 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 G…
We discuss entanglement and critical properties of the spin-3/2 XXZ chain in its entire gapless region. Employing density-matrix renormalization group calculations combined with different methods based on level spectroscopy, correlation functions and entanglement entropies, we determine the sound velocity and the Luttinger parameter of the model as a function of the anisotropy parameter. Then, we focus on entanglement properties by systematically studying the behavior of Rényi entropies under both open and periodic boundary conditions, providing further evidence of recent findings about entang…
Quench dynamics of the quantum XXZ chain with staggered interactions: Exact results and simulations on digital quantum computers
National Chengchi University（国立政治大学）；Wigner Research Centre for Physics, Institute for Solid State Physics and Optics（维格纳物理研究中心，固体物理与光学研究所）；University of Szeged（塞格德大学）
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 evolut…
We investigate the ground-state properties of the XXZ model with $1/r^α$ interactions, describing spins interacting with long-range (LR) transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hardcore bosons with LR repulsion and hopping. The LR nature of the couplings allows us to quantitatively study the spectral, correlation and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three dis…
We study the return probability and its imaginary ($τ$) time continuation after a quench from a domain wall initial state in the XXZ spin chain, focusing mainly on the region with anisotropy $|Δ|< 1$. We establish exact Fredholm determinant formulas for those, by exploiting a connection to the six vertex model with domain wall boundary conditions. In imaginary time, we find the expected scaling for a partition function of a statistical mechanical model of area proportional to $τ^2$, which reflects the fact that the model exhibits the limit shape phenomenon. In real time, we observe that in the…
We study the nonequilibrium time evolution of the spin-1/2 anisotropic Heisenberg (XXZ) spin chain, with a choice of dimer product and Neel states as initial states. We investigate numerically various short-ranged spin correlators in the long-time limit and find that they deviate significantly from predictions based on the generalized Gibbs ensemble (GGE) hypotheses. By computing the asymptotic spin correlators within the recently proposed quench-action formalism [Phys. Rev. Lett. 110, 257203 (2013)], however, we find excellent agreement with the numerical data. We, therefore, conclude that th…
The ground-state phase diagram of frustrated S=1 XXZ spin chains with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied using the infinite-system density-matrix renormalization-group method. We find six different phases, namely, the Haldane, gapped chiral, gapless chiral, double Haldane, Néel, and double Néel (uudd) phases. The gapped and gapless chiral phases are characterized by the spontaneous breaking of parity, in which the long-range order parameter is a chirality, κ_l = S_l^xS_{l+1}^y-S_l^yS_{l+1}^x, whereas the spin correlation decays either expone…
We study quantum quenches in the XXZ spin-$1/2$ Heisenberg chain from families of ferromagnetic and antiferromagnetic initial states. Using Bethe ansatz techniques, we compute short-range correlators in the complete generalized Gibbs ensemble (GGE), which takes into account all local and quasi-local conservation laws. We compare our results to exact diagonalization and numerical linked cluster expansion calculations for the diagonal ensemble finding excellent agreement and thus providing a very accurate test for the validity of the complete GGE. Furthermore, we compute the diagonal entropy in …
We consider the computation of the Loschmidt echo after quantum quenches in the interacting $XXZ$ Heisenberg spin chain both for real and imaginary times. We study two-site product initial states, focusing in particular on the Néel and tilted Néel states. We apply the Quantum Transfer Matrix (QTM) approach to derive generalized TBA equations, which follow from the fusion hierarchy of the appropriate QTM's. Our formulas are valid for arbitrary imaginary time and for real times at least up to a time $t_0$, after which the integral equations have to be modified. In some regimes, $t_0$ is seen to …
We compute the dynamics of localized excitations produced by a quantum quench in the spin 1/2 XXZ chain. Using numerics combining the density matrix renormalization group and exact time evolution, as well as analytical arguments, we show that fractionalization due to interactions in the pre-quench state gives rise to "ultrarelativistic" density waves that travel at the maximum band velocity. The system is initially prepared in the ground state of the chain within the gapless XY phase, which admits a Luttinger liquid (LL) description at low energies and long wavelengths. The Hamiltonian is then…
In a two or three dimensional ferromagnetic XXZ model, a low energy excitation mode above a magnetic domain wall is gapless, whereas all of the usual spin wave excitations moving around the whole crystal are gapful. Although this surprising fact was already proved in a mathematically rigorous manner, the gapless excitations have not yet been detected experimentally. For this issue, we show theoretically that the gapless excitations appear as the dynamical fluctuations of the experimental observable, magnetoresistance, in a ferromagnetic wire. We also discuss other methods (e.g., ferromagnetic …
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalization group theory. The nearest-neighbor spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lyi…
A general relation between quantum phase transitions and the second derivative of the fidelity (or the "fidelity susceptibility") is proposed. The validity and the limitation of the fidelity susceptibility in characterizing quantum phase transitions is thus established. Moreover, based on the bosonization method, general formulas of the fidelity and the fidelity susceptibility are obtained for a class of one-dimensional gapless systems known as the Tomonaga-Luttinger liquid. Applying these formulas to the one-dimensional spin-1/2 $XXZ$ model, we find that quantum phase transitions, even of the…
We consider the Renyi alpha-entropies for Luttinger liquids (LL). For large block lengths l these are known to grow like ln l. We show that there are subleading terms that oscillate with frequency 2k_F (the Fermi wave number of the LL) and exhibit a universal power-law decay with l. The new critical exponent is equal to K/(2 alpha), where K is the LL parameter. We present numerical results for the anisotropic XXZ model and the full analytic solution for the free fermion (XX) point.
The dynamical quantum phase transitions (DQPTs) in quantum spin chains with gapless phases after a sudden quench are studied. We mainly consider the general systems with asymmetrical quasiparticle excitation spectra and obtain the general expression of the Loschmidt echo as well as the general conditions for the occurrence of DQPTs. As two examples, we study the DQPTs in the \emph{XY} chains with Dzyaloshinskii-Moriya interaction and \emph{XZY-YZX} type of three-site interaction. It's found that the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether th…
The ground-state phase diagram of the frustrated spin-S XXZ chain with the competing nearest- and next-nearest-neighbor antiferromagnetic couplings is studied numerically by using the density-matrix renormalization-group method for the cases of S = 1/2, 3/2, and 2. We are paricularly interested in the possible gapless and gapped chiral phases, in which the chirality κ_l = S_l^x S_{l+1}^y - S_l^y S_{l+1}^x exhibits a finite long-range order whereas the spin correlation decays either algebraically or exponentially. We show that the gapless chiral phase appears in a broad region of the phase diag…
The density-matrix renormalization-group technique is used to calculate the spin correlation functions <S^x_jS^x_k> and <S^z_jS^z_k> of the one-dimensional S=1/2 XXZ model in the gapless regime. The numerical results for open chains of 200 spins are analyzed by comparing them with correlation functions calculated from a low-energy field theory. This gives precise estimates of the amplitudes of the correlation functions in the thermodynamic limit. The exact amplitude recently conjectured by Lukyanov and Zamolodchikov and the logarithmic correction in the Heisenberg model are confirmed numerical…
In this review, we study the quenching dynamics of a one-dimensional XY Hamiltonian in a transverse field under linear variation of different parameters of the Hamiltonian so that the system is driven through various critical points and gapless lines. It is observed that the density of defects falls off as a power law with the quenching rate $1/τ$ where the power law is determined by the dimensionality of the underlying lattice and the critical exponents associated with the quantum critical point as predicted by Kibble-Zurek scaling. However, we show that when the system is driven through a mu…
We consider a class of quantum quenches in the spin-1/2 XXZ chain, where the initial state is of a simple product form. Specific examples are the Néel state, the dimer state and the q-deformed dimer state. We compute determinant formulas for finite volume overlaps between the initial state and arbitrary eigenstates of the spin chain Hamiltonian. These results could serve as a basis for calculating the time dependence of correlation functions following the quantum quench.
We calculate the entropy and cooling rate of the antiferromagnetic spin-1/2 XXZ chain under an adiabatic demagnetization process using the quantum transfer-matrix technique and non-linear integral equations. The limiting case of the Ising chain (corresponding to infinitely large anisotropy) is presented for comparison. Our exact results for the Heisenberg chain are used as a crosscheck for the numerical exact diagonalization as well as Quantum Monte Carlo simulations and allow us to benchmark the numerical methods. Close to field-induced quantum phase transitions we observe a large magnetocalo…
We investigate the critical scaling of the spin-1/2 antiferromagnet on the square lattice in the easy-plane (XXZ) regime, via numerical measurements of the entanglement entropy constructed from the zeroes of a polynomial ring. We relate these results to conformal field theory predictions for the area law scaling of entanglement entropyin the vicinity of quantum critical points in gapless conformally invariant 2d quantum systems, and in gapless systems with finite Fermi surfaces. Our measurements are focused in three low-temperature regions at fixed XXZ anisotropies of $Δ=1.01$, $Δ=1.78$ and $Δ…
We consider a class of global quantum quenches in the Heisenberg XXZ spin chain, where the initial states are given by products of local two-site states. The two main examples are the Néel state and the dimer state. We derive an exact analytic result for the "Loschmidt echo per site" at imaginary times and also consider the analytic continuation back to real times. As a by-product we obtain an exact result for the "overlap per site" between the Néel state and the ground state of the XXZ Hamiltonian in the massive regime.
We compute all dynamical spin-spin correlation functions for the spin-1/2 $XXZ$ anisotropic Heisenberg model in the gapless antiferromagnetic regime, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier t…
We study the dynamics of a quench-prepared domain wall state released into a system whose unitary time evolution is dictated by the Hamiltonian of the Heisenberg spin-1/2 gapped antiferromagnetic chain. Using exact wavefunctions and their overlaps with the domain wall state allows us to describe the release dynamics to high accuracy, up to the long-time limit, for finite as well as infinite systems. The results for the infinite system allow us to rigorously prove that the system in the gapped regime ($Δ>1$) cannot thermalize in the strict sense.
We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise …
We compute, by means of exact diagonalization of systems of N=16 and 18 spins, the correlation function <σ^z_0σ^z_n> at nonzero temperature for the XXZ model with anisotropy Δ. In the gapless ferromagnetic region -1<Δ<0 for fixed separation the temperature can always be made sufficiently low so that the correlation is always negative for n \neq 0. However we find that for sufficiently large temperatures and fixed separation or for fixed temperature greater than some T0(\DElta) and sufficiently large separations the correlations are always positive. This sign changing effect has not been previo…
We present a conjectured exact formula for overlaps between the Bethe states of the spin-1/2 XXZ chain and generic two-site states. The result takes the same form as in the previously known cases: it involves the same ratio of two Gaudin-like determinants, and a product of single-particle overlap functions, which can be fixed using a combination of the Quench Action and Quantum Transfer Matrix methods. Our conjecture is confirmed by numerical data from exact diagonalization. For one-site states the formula is found to be correct even in chains with odd length, where existing methods can not be…
Under a perfect periodic potential, the electric current density induced by a constant electric field may exhibit nontrivial oscillations, so-called Bloch oscillations, with an amplitude that remains nonzero in the large system size limit. Such oscillations have been well studied for nearly noninteracting particles and observed in experiments. In this work, we revisit Bloch oscillations in strongly interacting systems. By analyzing the spin-1/2 XXZ chain, which can be mapped to a model of spinless electrons, we demonstrate that the current density at special values of the anisotropy parameter …
Low energy physics of quasi-one-dimensional ultracold atomic gases is often described by a gapless Luttinger liquid (LL). It is nowadays routine to manipulate these systems by changing their parameters in time but, no matter how slow the manipulation is, it must excite a gapless system. We study a smooth change of parameters of the LL (a smooth "quench") with a variable quench time and find that the excitation energy decays with an inverse power of the quench time. This universal exponent is -2 at zero temperature, and -1 for slow enough quenches at finite temperature. The smooth quench does n…
We employ the mean-field approach in the fermionic picture of the spin-1/2 XXZ chain to investigate the dynamics of bipartite quantum discord and concurrence under sudden quenching. In the case, when quenching is performed in the anisotropy from an initial value to the critical point, the quantum correlations show periodic cusps in time. Moreover, the first suppression (cusp) in quantum correlations can be explained in terms of the semi-classical picture of quasiparticle propagation. On the other hand, quenching to, as well as away from the criticality point shows that the longtime pairwise qu…
The XXZ quantum spin chain has a triple point in its ground state $h$-$1/Δ$ phase diagram. This first order critical point is located at the joint end point of the two second order phase transition lines marking the transition from the gapless phase to the fully polarized phase and to the Néel ordered phase, respectively. We explore the magnetization and the short-range correlation functions in its vicinity using the exact solution of the model. In the critical regime above the triple point we observe a strong variation of all physical quantities on a low energy scale of order $1/Δ$ induced by…
It is shown that the sl(2)q-invariant open antiferromagnetic XXZ spin chain with a boundary field has a gapless sector in the thermodynamic limit when its length is odd. Owing to a Temperley-Lieb equivalence of the spectra, the same conclusion is drawn for the purely biquadratic spin 1 chain with open boundaries and odd length.
In the context of slow quenching dynamics of a $p$-wave superconducting chain, it has been shown that a Majorana edge state can not be adiabatically transported from one topological phase to the other separated by a quantum critical line. On the other hand, the inclusion of a phase factor in the hopping term, that breaks the extended time reversal invariance, results in an extended gapless region between two topological phases. We show that for a finite chain with an open boundary condition there exists a non-zero probability that an edge Majorana can be adiabatically transported from one topo…
We consider the non-equilibrium 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 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 cons…
We study the spreading of entanglement produced by the time evolution of a local fermionic excitation created above the ground state of the XXZ chain. The resulting entropy profiles are investigated via density-matrix renormalization group calculations, and compared to a quasiparticle ansatz. In particular, we assume that the entanglement is dominantly carried by spinon excitations traveling at different velocities, and the entropy profile is reproduced by a probabilistic expression involving the density fraction of the spinons reaching the subsystem. The ansatz works well in the gapless phase…
We study slow variation (both spatial as well as temporal) of a parameter of a system in the vicinity of discontinuous quantum phase transitions, in particular, a discontinuity critical point (DCP) (or a first-order critical point). We obtain the universal scaling relations of the density of defects and the residual energy after a temporal quench, while we also unravel the scaling of the characteristic length scale associated with a spatial quench of a symmetry breaking field. Considering a spin-1/2 XXZ chain we establish how these scaling relations get modified when the DCP is located at the …
For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume $2s$-strings for the ground-state solution of the Bethe ansatz equations for the spin-$s$ XXZ chain, and solve the integral equations for the spin-$s$ Gaudin matrix. In terms of the XXZ coupling $Δ$ we define $ζ$ by $Δ=\cos…
We discuss a general interaction quench in a Luttinger liquid described by a paired bosonic Hamiltonian. By employing $\mathsf{su}(1,1)$ Lie algebra, the post-quench time-evolved wavefunctions are obtained analytically, from which the time evolution of the entanglement in momentum space can be investigated. We note that depending on the choice of Bogoliubov quasiparticles, the expressions of wavefunctions, which describe time-evolved paired states, can take different forms. The correspondence between the largest entanglement eigenvalue in momentum space and the wavefunction overlap in quench d…
An analytic method is proposed to compute the surface energy and elementary excitations of the XXZ spin chain with generic non-diagonal boundary fields. For the gapped case, in some boundary parameter regimes the contributions of the two boundary fields to the surface energy are non-additive. Such a correlation effect between the two boundaries also depends on the parity of the site number $N$ even in the thermodynamic limit $N\to\infty$. For the gapless case, contributions of the two boundary fields to the surface energy are additive due to the absence of long-range correlation in the bulk. A…
We analyze universal terms that appear in the large system size scaling of the overlap between the Néel state and the ground state of the spin-1/2 XXZ chain in the antiferromagnetic regime. In a critical theory, the order one term of the asymptotics of such an overlap may be expressed in terms of $g$-factors, known in the context of boundary conformal field theory. In particular, for the XXZ model in its gapless phase, this term provides access to the Luttinger parameter. In its gapped phase, on the other hand, the order one term simply reflects the symmetry broken nature of the phase. In orde…
It is well known that the dynamics of a quantum system is always non-adiabatic in passage through a quantum critical point and the defect density in the final state following a quench shows a power-law scaling with the rate of quenching. However, we propose here a possible situation where the dynamics of a quantum system in passage across quantum critical regions is adiabatic and the defect density decays exponentially. This is achieved by incorporating additional interactions which lead to quantum critical behavior and gapless phases but do not participate in the time evolution of the system.…
We study the low-temperature transport properties of out-of-equilibrium XXZ spin-$1/2$ chains. We consider the protocol where two semi-infinite chains are prepared in two thermal states at small but different temperatures and suddenly joined together. We focus on the qualitative and quantitative features of the profiles of local observables, which at large times $t$ and distances $x$ from the junction become functions of the ratio $ζ=x/t$. By means of the generalized hydrodynamic equations, we analyse the rich phenomenology arising by considering different regimes of the phase diagram. In the …
We investigate the entanglement spreading in the anisotropic spin-1/2 Heisenberg (XXZ) chain after a geometric quench. This corresponds to a sudden change of the geometry of the chain or, in the equivalent language of interacting fermions confined in a box trap, to a sudden increase of the trap size. The entanglement dynamics after the quench is associated with the ballistic propagation of a magnetization wavefront. At the free fermion point (XX chain), the von-Neumann entropy S_A exhibits several intriguing dynamical regimes. Specifically, at short times a logarithmic increase is observed, si…
Information scrambling, characterized by the out-of-time-ordered correlator (OTOC), has attracted much attention, as it sheds new light on chaotic dynamics in quantum many-body systems. The scale invariance, which appears near the quantum critical region in condensed matter physics, is considered to be important for the fast decay of the OTOC. In this paper, we focus on the one-dimensional spin-1/2 XXZ model, which exhibits quantum criticality in a certain parameter region, and investigate the relationship between scrambling and the scale invariance. We quantify scrambling by the averaged OTOC…
In this letter we consider expectation values of local correlators in highly excited states of the spin-1/2 XXZ chain. Assuming that the string hypothesis holds we formulate the following conjecture: The correlation functions can be computed using the known factorized formulas of the finite temperature situation, if the building blocks are computed via certain linear integral equations using the string densities only. We prove this statement for the nearest neighbour z-z correlator for states with arbitrary string densities. Also, we check the conjecture numerically for other correlators in th…
We study the Rényi entropies in the spin-$1/2$ anisotropic Heisenberg chain after a quantum quench starting from the Néel state. The quench action method allows us to obtain the stationary Rényi entropies for arbitrary values of the index $α$ as generalised free energies evaluated over a calculable thermodynamic macrostate depending on $α$. We work out this macrostate for several values of $α$ and of the anisotropy $Δ$ by solving the thermodynamic Bethe ansatz equations. By varying $α$ different regions of the Hamiltonian spectrum are accessed. The two extremes are $α\to\infty$ for which the t…
