Nonlinear dynamics of the classical isotropic Heisenberg antiferromagnetic chain: The sigma model sector and the kink sector

We identify two distinct low-energy sectors in the classical isotropic antiferromagnetic Heisenberg spin-S chain, in the continuum limit. We show that two types of rotation generators arise for the field in each sector. Using these, the Lagrangian for sector I is shown to be that of the nonlinear sigma model. Sector II has a null Lagrangian. Its Hamiltonian density is just the Pontryagin term. Exact solutions are found in the form of magnons and precessing pulses in I and moving kinks in II. The kink has ‘spin’ S. Sector I has a higher minimum energy than II.     

On the dynamics of the classical Heisenberg chain

We investigate the space- and time-dependent spin-correlation-function for a linear chain of classical magnetic moments coupled by a purely isotropic exchange interaction. The correlation functions are given in terms of relative variables which at low temperatures turn out to be the normal coordinates appropriate to a system without long-range-order. Analytic expressions for the correlation functions, including correctly the fluctuations responsible for the absence of long-range-order, are given at low temperatures. Well-defined spin wave oscillations exist for wavelengths smaller than the correlation length. In this region our results are consistent with experimental data for the isotropic Heisenberg antiferromagnet TMMC.     

On the dynamics of impure classical Heisenberg chains

We present theoretical results on dynamic correlation functions of impure, classical, one-dimensional Heisenberg magnets with both ferromagnetic and antiferromagnetic coupling. The results are obtained in the framework of Mori's theory and in an approximation which includes correctly all moments of the relaxation function to lowest order in impurity concentration and temperature and which preserves the rotational symmetry of the system even at zero temperature. Explicit results are given for lineshifts and linewidths of quasi-spinwaves and the behaviour of correlation functions in the hydrodynamic limit is discussed.     

LMG model: Markovian evolution of classical and quantum correlations under decoherence

We have investigated the quantum phase transition in the ground state of collective Lipkin-Meshkov-Glick model (LMG model) subjected to decoherence due to its interaction, represented by a quantum channel, with an environment. We discuss the behavior of quantum and classical pair wise correlations in the system, with the quantumness of correlations measured by quantum discord (QD), entanglement of formation (EOF), measurement-induced disturbance (MID) and the Clauser-Horne-Shimony-Holt-Bell function (CHSH-Bell function). The time evolution established by system-environment interactions is assumed to be Markovian in nature and the quantum channels studied include the amplitude damping (AD), phase damping (PD), bit-flip (BF), phase-flip (PF), and bit-phase-flip (BPF) channels. One can identify appropriate quantities associated with the dynamics of quantum correlations signifying quantum phase transition in the model. Surprisingly, the CHSH-Bell function is found to detect all the phase transitions, even when quantum and classical correlations are zero for the relevant ground state.     

General optimality of the Heisenberg limit for quantum metrology

Quantum metrology promises improved sensitivity in parameter estimation over classical procedures. However, there is a debate over the question of how the sensitivity scales with the resources and the number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a completely general optimality proof of the Heisenberg limit for quantum metrology. We give an example of how our proof resolves paradoxes that suggest sensitivities beyond the Heisenberg limit, and we show that the Heisenberg limit is an information-theoretic interpretation of the Margolus-Levitin bound, rather than Heisenberg's uncertainty relation.     

The dynamics of a central spin in quantum Heisenberg XY model

In the thermodynamic limit, we present an exact calculation of the time dynamics of a central spin coupling with its environment at finite temperatures. The interactions belong to the Heisenberg XY type. The case of an environment with finite number of spins is also discussed. To get the reduced density matrix, we use a novel operator technique which is mathematically simple and physically clear, and allows us to treat systems and environments that could all be strongly coupled mutually and internally. The expectation value of the central spin and the von Neumann entropy are obtained.     

Transverse Ising model: Markovian evolution of classical and quantum correlations under decoherence

The transverse Ising Model (TIM) in one dimension is the simplest model which exhibits a quantum phase transition (QPT). Quantities related to quantum information theoretic measures like entanglement, quantum discord (QD) and fidelity are known to provide signatures of QPTs. The issue is less well explored when the quantum system is subjected to decoherence due to its interaction, represented by a quantum channel, with an environment. In this paper we study the dynamics of the mutual information I(ρ _AB ), the classical correlations C(ρ _AB ) and the quantum correlations Q(ρ _AB ), as measured by the QD, in a two-qubit state the density matrix of which is the reduced density matrix obtained from the ground state of the TIM in 1d. The time evolution brought about by system-environment interactions is assumed to be Markovian in nature and the quantum channels considered are amplitude damping, bit-flip, phase-flip and bit-phase-flip. Each quantum channel is shown to be distinguished by a specific type of dynamics. In the case of the phase-flip channel, there is a finite time interval in which the quantum correlations are larger in magnitude than the classical correlations. For this channel as well as the bit-phase-flip channel, appropriate quantities associated with the dynamics of the correlations can be derived which signal the occurrence of a QPT.     

Renarks on the classical limit of quantum field theories

Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called constructive approach to quantum field theory.After a section on generalities, we discuss in Section 2 the situation where no phase transitions occur in the limith0 and in Section 3 we reformulate one result in the case where such a transition occurs (Glimmet al. [7]). We discuss the validity of the loop expansion. It seems however that the tools to show the rigorous validity of soliton calculations are not yet prepared.     

Renormalization group solution of the one-dimensional classical Heisenberg model

It is shown that the method of Van Leeuwen^1,2) and Nauenberg and Nienhuis³) in the application of the renormalization theory to Ising-like spin systems, can easily be extended to include all one-dimensional classical spin systems with nearest neighbor interactions. The series for the free energy converges very rapidly towards the known exact value (for Heisenberg interaction), provided that the temperature is not too close to the critical temperature T = 0.     

Flow equations and extended Bogoliubov transformation for the Heisenberg antiferromagnet near the classical limit

The Heisenberg spin-S quantum antiferromagnet is studied near the large-spin limit, applying a new continuous unitary transformation which extends the usual Bogoliubov transformation to higher order in the 1/S-expansion of the Hamiltonian. This allows to diagonalize the bosonic Hamiltonian resulting from the Holstein-Primakoff representation beyond the conventional spin-wave approximation. The zero-temperature flow equations derived from the extension of the Bogoliubov transformation to order for the ground-state energy, the spin-wave velocity, and the staggered magnetization are solved exactly and yield results which are in agreement with those obtained by a perturbative treatment of the magnon interactions. Received: 19 March 1998 / Revised: 2 June 1998 / Accepted: 8 June 1998     

Differential equations of spin dynamics and asymptotic expansion of quantum expectation values near the classical limit. II

The method developed in Part I is used to obtain asymptotic expansions of the functions that represent spin operators and of the quantum expectation values near the classical limit J→∞. We calculate with accuracy to the first quantum correction the correlation functions responsible for the statistical properties of cooperative spontaneous emission (super-radiance) of a gas of twolevel molecules. It is shown that the light emitted by a finite number of molecules does not have coherence of higher orders. The statistics of the field of quantum fluctuations is discussed.     

Differential equations of spin dynamics and asymptotic expansion of quantum mean values near the classical limit. III

A study is made of the asymptotic behavior of the functional analogs of the coefficients of vector addition (Clebsch-Gordan coefficients) introduced by means of atomic coherent states for large values of the angular momenta. The cases J₁ J₂, J₃ 1 and J₁, J₃ J₂, where J₁ and J₂ are the original angular momenta and J₃ is the resultant, are investigated. It is shown that the investigated functions are transformed for large J into narrow distributions, which makes it possible to expand integrals containing such functions in asymptotic series in powers of J^–1. The quantum rule for adding angular momenta, formulated in functional language, goes over into the classical one in the limit J The possibility of using these relations to describe molecules with rotational degrees of freedom is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 142–148, April, 1977.     

Differential equations of spin dynamics and asymptotic expansion of quantum expectation values near the classical limit. I

A large class of functional representations for spin operators in the spaces of the phase functions of a two-dimensional oscillator and functions on the unit sphere is constructed by means of generating functionals. The group-theoretical meaning of the formalism and its connection with the method of coherent states are discussed. Rules are found for rewriting the operator equations of motion in the form of differential equations for the functions that represent the operators.     

The massive thirring model as continuum limit of the Heisenberg model

The continuum limit of the lattice fermion version of the anisotropic spin-1/2 Heisenberg chain is reconsidered. It is shown that certain matrix elements of the Heisenberg Hamiltonian converge towards corresponding matrix elements of a massive Thirring model as previously suggested by Luther and Peschel. However, the result is only to first order consistent with the exactly known spectral and critical properties of the two models. Going beyond previous results in addition to the coupling constant of the massive Thirring model the kinetic energy coefficient comes out correctly to the first order of the lattice fermion interaction, too. Emphasizing the role of the underlying Hilbert spaces the discrepancy in higher orders is explained.Work performed within the research program of the Sonderforschungsbereich 125 Aachen, Jülich, Köln     

Hysteresis behavior of the anisotropic quantum Heisenberg model

The effect of the anisotropy in the exchange interaction on the hysteresis loops within the anisotropic quantum Heisenberg model has been investigated with the effective field theory for two spin cluster. Particular attention has been devoted on the behavior of the hysteresis loop area, coercive field and remanent magnetization with the anisotropy in the exchange interaction for both ferromagnetic and paramagnetic phases.