Temperley-Lieb-Jones Algebraic Structures in Higher Spin Chain Models

We investigate the Temperley-Lieb-Jones algebraic structures in isomorphic higher spin chain modgls with nearest neighbour interactions and SU(2) symmetry. The Ternperley-Lieb-Jones algebraic constructions for such chains of spins from 1 to 7/2 are presented. For the case of spin-1 we also give the Temperley-Lieb-Jones algebraic representations related to the anisomorphic spin chain with q-deformed SU_q(2) symmetry. Their related spin chain Harniltonians with SU(2) and SU_q(2) symmetries as well as the corresponding solutions of Yang-Baxter equation are also discussed.     

Spectral Theory of Massless Pauli-Fierz Models

We study the spectral theory of massless Pauli-Fierz models using an extension of the Mourre method. We prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant. In addition we show that the expectation value of the number operator is finite on all eigenvectors.Supported by Carlsbergfondet     

Lagrangian Curves on Spectral Curves of Monopoles

We study Lagrangian points on smooth holomorphic curves in ${\rm T}{\mathbb P}^1$ equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of ${\rm T}{\mathbb P}^1$ with the space ${\mathbb L}({\mathbb E}^3)$ of oriented affine lines in Euclidean 3-space ${\mathbb E}^3$ , these Lagrangian curves give rise to ruled surfaces in ${\mathbb E}^3$ , which we prove have zero Gauss curvature. Each ruled surface is shown to be the tangent lines to a curve in ${\mathbb E}^3$ , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in ${\mathbb E}^3$ where the number of oriented lines in the complex curve Σ that pass through the point is less than the degree of Σ. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.     

Quantum Communication in Spin Chain with Multiple Spin Exchange Interaction

The transmission of quantum states in the anisotropic Heisenberg XXZ chain model with three-spin exchange interaction is studied. The average fidelity is used to evaluate the state transfer. It is found that quantum communication can be enhanced by the anisotropic coupling and multiple spin interaction. Such spin model can reduce the time required for the perfect state transmission where the fidelity is unity. The maximally entangled Bell states can be generated and separated from the whole quantum systems.     

Integrable Lattice Spin Models from Supersymmetric Dualities

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang–Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories and make clear notes on the special functions used by several authors.     

Higher Toda Mechanics and Spectral Curves

For each of the Lie algebras gln and g~ln we construct a family of integrable generalizations of the Toda chains characterized by two integers m and m_. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m =m_,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m ,m_)-Toda chains, which turns out to be very different for different pairs of m and m_. Finally we also obtain the nonabelian generalizations of the (m ,m_)-Toda chains in an explicit form.     

Higher Toda Mechanics and Spectral Curves

For each of the Lie algebras gln and gl～n., we construct a family of integrable generalizations of the Toda chains characterized by two integers m and m-. The Lax matrices and the equations of motion are given explicitly, and the integrals of motion can be calculated in terms of the trace of powers of the Lax matrix L. For the case of m = m-,we find a symmetric reduction for each generalized Toda chain we found, and the solution to the initial value problems of the reduced systems is outlined. We also studied the spectral curves of the periodic (m ,m-)-Toda chains, which turns out to be very different for different pairs of m and m-. Finally we also obtain thenonabelian generalizations of the (m , m-)-Toda chains in an explicit form.     

A Farey Fraction Spin Chain

We introduce a new number-theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation-invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices. We prove that the free energy exists and a phase transition occurs for positive inverse temperature β= 2. The free energy is the same as that of related, non-translation-invariant number-theoretic spin chain. Using a number-theoretic argument, the low-temperature (β > 3) state is shown to be completely magnetized for long chains. The number of states of energy E= log(n) summed over chain length is expressed in terms of a restricted divisor problem. We conjecture that its asymptotic form is (n log n), consistent with the phase transition at β= 2, and suggesting a possible connection with the Riemann ζ-function. The spin interaction coefficients include all even many-body terms and are translation invariant. Computer results indicate that all the interaction coefficients, except the constant term, are ferromagnetic. Received: 20 August 1998/ Accepted: 17 December 1998     

Non-Universality in Ising Models with Four Spin Interaction

We consider two bidimensional classical Ising models, coupled by a weak interaction bilinear in the energy densities of the two systems; the model contains, as limiting cases, the Ashkin–Teller and the Eight-vertex models for certain values of their parameters. We write the energy–energy correlations and the specific heat as Grassman integrals formally describing Dirac 1+1 dimensional interacting massive fermions on a lattice, and an expansion based on Renormalization Group is written for them, convergent up to temperatures very close to the critical temperature for small coupling. The asymptotic behaviour is determined by critical indices which are continuous functions of the coupling.     

Ising Models with Four Spin Interaction at Criticality

We consider two bidimensional Ising models coupled by an interaction quartic in the spins, like in the spin representation of the Eight vertex or the Ashkin-Teller model. By Renormalization Group methods we write a convergent perturbative expansion for the specific heat and for the energy-energy correlation up to the critical temperature. A form of nonuniversality is proved, in the sense that the critical behaviour is described in terms of critical indices which are non trivial functions of the coupling. The logarithmic singularity of the specific heat of the Ising model is removed or changed in a power law (with a non universal critical index) depending on the sign of the interaction.     

Spin Coherent State Representation of the Crow-Kimura and Eigen Models of Quasispecies Theory

We present a spin coherent state representation of the Crow-Kimura and Eigen models of biological evolution. We deal with quasispecies models where the fitness is a function of Hamming distances from one or more reference sequences. In the limit of large sequence length N, we find exact expressions for the mean fitness and magnetization of the asymptotic quasispecies distribution in symmetric fitness landscapes. The results are obtained by constructing a path integral for the propagator on the coset SU(2)/U(1) and taking the classical limit. The classical limit gives a Hamiltonian function on a circle for one reference sequence, and on the product of 2 ^m −1 circles for m reference sequences. We apply our representation to study the Schuster-Swetina phenomena, where a wide lower peak is selected over a narrow higher peak. The quadratic landscape with two reference sequences is also analyzed specifically and we present the phase diagram on the mutation-fitness parameter phase space. Furthermore, we use our method to investigate more biologically relevant system, a model of escape from adaptive conflict through gene duplication, and find three different phases for the asymptotic population distribution.     

Decoherence Induced by Spin Chain Environment

We study the decoherence process of an exact solvable model that consists of a central spin-1/2 coupling to the surrounding anisotropy spin-1/2 chain in transverse fields. The Loschrnidt echo is calculated to study the character of decoherence with different degree of anisotropy. Our results show that the degree of anisotropy γ greatly affects the decoherence process of the central spin-system when the spin chain is in weak transverse fields, but it gives weak effect in the strong transverse field. The decoherence process of the central system changed dramatically along the line of the critical points, and this may be explained as the reflection of quantum phase transitions.     

Villain Transformation for Ferrimagnetic Spin Chain

By using the Villain transformation, the Heisenberg ferrimagnetic spin chain is calculated. Two branches of the low-lying excitation in both the absence and presence of magnetic field are obtained. The thermodynamic quantities （such as free energy, magnetization, specific heat and static magnetic susceptibility） are also evaluated at finite temperature. This ＆ the first time to calculate the Ferrimagnetic spin chain by using Villain＇s method, and we find that the results at a low temperature are quite similar to the previous calculation. The results of free energy and magnetization in zero temperature suggest that the Villain transformation has a good efficiency.     

Twisted Wess-Zumino-Witten Models on Elliptic Curves

Investigated is a variant of the Wess-Zumino-Witten model called a twisted WZW model, which is associated to a certain Lie group bundle on a family of elliptic curves. The Lie group bundle is a non-trivial bundle with flat connection and related to the classical elliptic r-matrix. (The usual (non-twisted) WZW model is associated to a trivial group bundle with trivial connection on a family of compact Riemann surfaces and a family of its principal bundles.) The twisted WZW model on a fixed elliptic curve at the critical level describes the XYZ Gaudin model. The elliptic Knizhnik-Zamolodchikov equations associated to the classical elliptic r-matrix appear as flat connections on the sheaves of conformal blocks in the twisted WZW model. Received: 21 January 1997 / Accepted: 1 April 1997     

Entanglement in One-Dimensional Random XY Spin Chain with Dzyaloshinskii--Moriya Interaction

The impurities of exchange couplings, external magnetic fields and Dzyaloshinskii-Moriya （DM） interaction considered as Gaussian distribution, and the entanglement in one-dimensional random XY spin systems is investigated by the method of solving the different spin-spin correlation functions and the average magnetization per spin. The entanglement dynamics at central locations of ferromagnetic and antiferromagnetic chains have been studied by varying the three impurities and the strength of DM interaction. （i） For the ferromagnetic spin chain, the weak DM interaction can improve the amount of entanglement to a large value, and the impurities have the opposite effect on the entanglement below and above critical DM interaction. （ii） For the antiferromagnetic spin chain, DM interaction can enhance the entanglement to a steady value. Our results imply that DM interaction strength, the impurity and exchange couplings （or magnetic field） play competing roles in enhancing quantum entanglement.     

Controlling Quantum State Transfer in Spin Chain with Confined Field

As a demonstration of the spectrum-parity matching condition (SPMC) for quantum state transfer, we investigate the propagation of single-magnon state in the Heisenberg chain in the confined external tangent magnetic field analytically and numerically. It shows that the initial Gaussian wave packet can be retrieved at the counterpart location near-perfectly over a longer distance if the dispersion relation of the system meets the SPMC approximately.     

Controlling Quantum State Transfer in Spin Chain with Confined Field

As a demonstration of the spectrum-parity matching condition （SPMC） for quantum state transfer, we investigate the propagation of single-magnon state in the Heisenberg chain in the confined external tangent magnetic field analytically and numerically. It shows that the initial Gaussian wave packet can be retrieved at the counterpart location near-perfectly over a longer distance if the dispersion relation of the system meets the SPMC approximately.     

Spectral Artifacts Resulting from Multiple-Quantum Coherence Transfers by Spin Decoupling during Acquisition

Off-resonance coherent decoupling of spin X in a heteronuclear SX_N spin system can result in a multiplet with overlapping lines of spin Sand an overall shape similar to that of a doublet. Such lineshape distortions occur if certain multiple-spin coherences are present when the decoupler is switched on. The distortion could cause an apparent change in the multiplicity of an M + 1 multiplet of an SX_NY_M spin system when a selective decoupling of spins X is applied after a polarization transfer from spins X to spin S by INEPT or DEPT. Possibilities for suppressing the distortions are discussed.