We apply the recently developed thermal form factor expansion method to evaluate the real-time longitudinal spin-spin correlation functions of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetically ordered regime at temperature $T=0$. An analytical result containing all types of excitations in the model is obtained, without any approximations. This allows for the accurate calculation of the real-time correlation functions in this strongly interacting quantum system for arbitrary distances and times.
We present exact results on the thermal conductivity of the one-dimensional spin-1/2 XXZ model in the massive antiferromagnetic and ferromagnetic regimes. The thermal Drude weight is calculated by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy as well as analytical results in the low and high temperature limits are presented. At finite temperature, the thermal Drude weight is finite and hence there is non-dissipative thermal transport even in the massive regime. At low temperature, the thermal Drude weight behaves as $D(T)\sim \exp(-δ/T)/\s…
We study the XXZ Heisenberg model in a longitudinal magnetic field using a tensor renormalization method. Built into the tensor representation of the XXZ model is the U(1) symmetry, which is systematically maintained at each renormalization step. This enables rather large tensor representations. We extract ground state properties as well as the low lying spectrum from the fixed point tensors. With rather moderate numerical effort we achieve a very good accuracy as demonstrated by comparison with Bethe Ansatz calculations. The phase structure of the model can be accurately reproduced just from …
We consider multi-point correlation functions in the open XXZ chain with longitudinal boundary fields and in a uniform external magnetic field. We show that, at finite temperature, these correlation functions can be written in the quantum transfer matrix framework as sums over thermal form factors. More precisely, and quite remarkably, each term of the sum is given by a simple product of usual matrix elements of the quantum transfer matrix multiplied by a unique factor containing the whole information about the boundary fields. As an example, we provide a detailed expression for the longitudin…
We derive a novel multiple integral representation for a generating function of the $\s^z$-$\s^z$ correlation functions of the spin-$\2$ XXZ chain at finite temperature and finite, longitudinal magnetic field. Our work combines algebraic Bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics.
Combining a lattice path integral formulation for thermodynamics with the solution of the quantum inverse scattering problem for local spin operators, we derive a multiple integral representation for the time-dependent longitudinal correlation function of the spin-1/2 Heisenberg XXZ chain at finite temperature and in an external magnetic field. Our formula reproduces the previous results in the following three limits: the static, the zero-temperature and the XY limits.
In this paper, we determine the geometric phase for the one-dimensional $XXZ$ Heisenberg chain with spin-$1/2$, the exchange couple $J$ and the spin anisotropy parameter $Δ$ in a longitudinal field(LF) with the reduced field strength $h$. Using the Jordan-Wigner transformation and the mean-field theory based on the Wick's theorem, a semi-analytical theory has been developed in terms of order parameters which satisfy the self-consistent equations. The values of the order parameters are numerically computed using the matrix-product-state(MPS) method. The validity of the mean-filed theory could b…
We derive compact multiple integral formulas for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulas follow from several effective re-summations of the multiple integral representation for the elementary blocks obtained in our previous article (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic fiel…
We present an integral formula for the density matrix of a finite segment of the infinitely long spin-1/2 XXZ chain. This formula is valid for any temperature and any longitudinal magnetic field.
The exact expression for the density matrix of the kink ground state of the ferromagnetic XXZ chain is obtained. Utilizing this, we exactly calculate various correlation functions such as the longitudinal and transverse spin-spin correlation functions, and the ferromagnetic and antiferromagnetic string formation probabilities. The asymptotic behaviors of these correlation functions are also analyzed. As a consequence, we find that the spin-spin correlation functions decay exponentially for large distances, while the string formation probabilities exhibit Gaussian decay for large strings. We al…
The second neighbor correlation functions of the spin-${1/2}$ $XXZ$ chain in the ground state are expressed in the form of three dimensional integrals. We show that these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions for ${Δ>1}$.
Using the algebraic Bethe ansatz method, and the solution of the quantum inverse scattering problem for local spins, we obtain multiple integral representations of the $n$-point correlation functions of the XXZ Heisenberg spin-$1 \over 2$ chain in a constant magnetic field. For zero magnetic field, this result agrees, in both the massless and massive (anti-ferromagnetic) regimes, with the one obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra ${\cal U}_q (\hat{sl}_2)$ together with the corner transfer matrix approach (massive…
We derive exactly the number of complex solutions with two down-spins in the massive regime of the periodic spin-1/2 XXZ spin chain of $N$ sites. Here we remark that every solution of the Bethe ansatz equations is characterized by a set of quantum numbers. We derive them analytically for all the complex solutions in the sector, which we call two-string solutions. We show that in a region of $N$ and $Δ$ the number of two-string solutions is by two larger than the number due to the string hypothesis, i.e., an extra pair of two-strings appears. We determine it exactly and also such regions where …
An explicit construction for Q-operators of the finite XXZ spin-chain with twisted boundary conditions is presented. The massless and the massive regime is considered as well as the root of unity case. It is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous XXZ model on the infinite lattice by Boos, Jimbo, Miwa, Smirnov and Takeyama(hep-th/0412191).
We derive expressions for the form factors of the quantum transfer matrix of the spin-1/2 XXZ chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form-factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. We also explain how to adapt our formulae to the description of ground state correlation functions of the finite cha…
The scaling limit of the Heisenberg XXZ spin chain at zero magnetic field is studied in the gapped antiferromagnetic phase. For a spin-chain ring having $N_x$ sites, the universal Casimir scaling function, which characterises the leading finite-size correction term in the large-$N_x$ expansion of the ground state energy, is calculated by numerical solution of the nonlinear integral equation of the convolution type. It is shown, that the same scaling function describes the temperature dependence of the free energy of the infinite XXZ chain at low enough temperatures in the gapped scaling regime…
The infinite Heisenberg XXZ spin-(1/2) chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field h induces a long-range attractive potential between two adjacent spinons leading to their confinement into 'meson' bound states. Exploiting the integrability of the XXZ model in the deconfined phase h = 0, we perform perturbative calculations of the energy spectra of the two-spinon bound state…
The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of form factors of local operators are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. The expressions are compared with known results upon specializations.Using the spin-reversal property which relates the Hamiltonian with upper and lower triangular boundary conditions, new identities between multiple integrals of infinite products are ex…
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 specula…
We use the form factors of the quantum transfer matrix in the zero-temperature limit in order to study the two-point ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime. We obtain novel form factor series representations of the correlation functions which differ from those derived either from the q-vertex-operator approach or from the algebraic Bethe Ansatz approach to the usual transfer matrix. We advocate that our novel representations are numerically more efficient and allow for a straightforward calculation of the large-distance asymptotic behaviour …
We consider the spectrum of correlation lengths of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetic massive regime. These are given as ratios of eigenvalues of the quantum transfer matrix of the model. The eigenvalues are determined by integrals over certain auxiliary functions and by their zeros. The auxiliary functions satisfy nonlinear integral equations. We analyse these nonlinear integral equations in the low-temperature limit. In this limit we can determine the auxiliary functions and the expressions for the eigenvalues as functions of a finite number of parameters which satisfy…
We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime $Δ>1$, extending our previous results to the full parameter plane of the antiferromagnetic chain ($Δ> -1$ and arbitrary field $h$). The factorized formulae are numerically efficient and allow for taking the isotropic limit ($Δ= 1$) and the Ising limit ($Δ= \infty$). At the cri…
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the …
We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The $n$th summand of the series is a multiple integral accounting for all $n$-particle $n$-hole excitations of the quantum transfer matrix. In previous works the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here we obtain a representation which reduce…
The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed `Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime $-1<q<0$, level one infinite dimensional representation ($q-$vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known result…
The Heisenberg XXZ spin-1/2 chain is considered in the massive antiferromagnetic regime in the presence of a staggered longitudinal magnetic field. The Hamiltonian of the model is characterised by the anisotropy parameter $Δ<-1$, and by the magnetic field strength h. At zero magnetic field, the model is exactly solvable. In the thermodynamic limit, it has two degenerate vacua and the kinks (which are also called spinons) interpolating between these vacua, as elementary excitations. Application of the staggered magnetic field breaks integrability of the model and induces the long-range attracti…
This work constructs a well-defined and operational form factor expansion in a model having a massless spectrum of excitations. More precisely, the dynamic two-point functions in the massless regime of the XXZ spin-1/2 chain are expressed in terms of properly regularised series of multiple integrals. These series are obtained by taking, in an appropriate way, the thermodynamic limit of the finite volume form factor expansions. The series are structured in way allowing one to identify directly the contributions to the correlator stemming from the conformal-type excitations on the Fermi surface …
In the present paper, we consider two species of small spin clusters known as; two spin-1/2 Heisenberg edge-shared tetrahedra and spin-1/2 Heisenberg octahedron with the corresponding Dzyaloshinskii-Moriya terms in a longitudinal magnetic field, then we examine magnetization process and electric polarization of the models as functions of magnetic and electric fields at Low temperature using exact numerical diagonalization. Our exact results are in a good agreement with recent analysis carried out by J. Strečka and K. Kar{\v l}ova \cite{str18b,str17a}. It is demonstrated that the polarization b…
We investigate the large-distance asymptotic behavior of the static density-density and field-field correlation functions in the one-dimensional Bose gas at finite temperature. The asymptotic expansions of the Bose gas correlators are obtained performing a specific continuum limit in the similar low-temperature expansions of the longitudinal and transversal correlation functions of the XXZ spin chain. In the lattice system the correlation lengths are computed as ratios of the largest and next-largest eigenvalues of the XXZ spin chain quantum transfer matrix. In both cases, lattice and continuu…
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.
We compute the momentum- and frequency-dependent longitudinal spin structure factor for the one-dimensional spin-1/2 $XXZ$ Heisenberg spin chain in a magnetic field, using exact determinant representations for form factors on the lattice. Multiparticle contributions are computed numerically throughout the Brillouin zone, yielding saturation of the sum rule to high precision.
We describe a method to derive, from first principles, the long-distance asymptotic behavior of correlation functions of integrable models in the framework of the algebraic Bethe ansatz. We apply this approach to the longitudinal spin- spin correlation function of the XXZ Heisenberg spin-1/2 chain (with magnetic field) in the disordered regime as well as to the density-density correlation func- tion of the interacting one-dimensional Bose gas. At leading order, the results confirm the Luttinger liquid and conformal field theory predictions.
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 demonstrate the universal properties of dissipative Tomonaga-Luttinger (TL) liquids by calculating correlation functions and performing finite-size scaling analysis of a non-Hermitian XXZ spin chain as a prototypical model in one-dimensional open quantum many-body systems. Our analytic calculation is based on effective field theory with bosonization, finite-size scaling approach in conformal field theory, and the Bethe-ansatz solution. Our numerical analysis is based on the density-matrix renormalization group generalized to non-Hermitian systems (NH-DMRG). We uncover that the model in the …
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.
It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole CFT. We reconsider the continuum limit of this spin chain with the exact method of non-linear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the ident…
The false vacuum decay has been a central theme in physics for half a century with applications to cosmology and to the theory of fundamental interactions. This fascinating phenomenon is even more intriguing when combined with the confinement of elementary particles. Due to the astronomical time scales involved, the research has so far focused on theoretical aspects of this decay. The purpose of this Letter is to show that the false vacuum decay is accessible to current optical experiments as quantum analog simulators of spin chains with confinement of the elementary excitations, which mimic t…
We investigate spin-spin correlation functions in the low temperature phase of spin-glasses. Using the replica field theory formalism, we examine in detail their infrared (long distance) behavior. In particular we identify a longitudinal mode that behaves as massive at intermediate length scales (near-infrared region). These issues are then addressed by numerical simulation; the analysis of our data is compatible with the prediction that the longitudinal mode appears as massive, i.e., that it undergoes an exponential decay, for distances smaller than an appropriate correlation length.
