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1.
We analyse birational mappings generated by transformations on q × q matrices which correspond respectively to two kinds of transformations: the matrix inversion and a permutation of the entries of the q × q matrix. Remarkable factorization properties emerge for quite general involutive permutations. It is shown that factorization properties do exist, even for birational transformations associated with noninvolutive permutations of entries of q × q matrices, and even for more general transformation which are rational transformations but no longer birational. The existence of factorization relations independent of q, the size of the matrices, is underlined. The relations between the polynomial growth of the complexity of the iterations, the existence of recursions in a single variable and the integrability of the mappings, are sketched for the permutations yielding these properties. All these results show that permutations of the entries of the matrix yielding factorization properties are not so rare. In contrast, the occurrence of recursions in a single variable, or of the polynomial growth of the complexity are, of course, less frequent but not completely exceptional. 相似文献
2.
We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters ( and ), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q × q matrix. This mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q × q matrices and permuting some of the entries of these matrices. It has been seen to yield weak chaos and integrability. We have found new integrable cases of this Cremona transformation, corresponding to the values of = 0 when
, besides the already known values = 0 and = −1, and also arbitrary when = 0. For these cases, one has a foliation of the parameter space in elliptic curves. We give the equations of these elliptic curves. Based on this very example we show how one can find these integrability cases of the Cremona transformation and actually integrate it using a method based on the systematic study of the finite-order conditions of the Cremona transformation. The method is shown to be efficient and straightforward. The various integrability cases are revisited using many different representations of this very mapping (birational transformations, recursion in one variable, …). 相似文献
3.
We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations on q× q matrices: the inversion of the q× q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensional q
4-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties. 相似文献
4.
We consider the one-dimensional t
-
J model, which consists of electrons with spin S on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J
=
t. Without spoiling the integrability we introduce an Anderson-like impurity of spin S (degenerate Anderson model in the limit), which interacts with the correlated conduction states of the host. The lattice model is defined by the scattering
matrices via the Quantum Inverse Scattering Method. We discuss the general form of the interaction Hamiltonian between the impurity and
the itinerant electrons on the lattice and explicitly construct it in the continuum limit. The discrete Bethe ansatz equations
diagonalizing the host with impurity are derived, and the thermodynamic Bethe ansatz equations are obtained using the string
hypothesis for arbitrary band filling as a function of temperature and external magnetic field. The properties of the impurity
depend on one coupling parameter related to the Kondo exchange coupling. The impurity can localize up to one itinerant electron
and has in general mixed valent properties. Groundstate properties of the impurity, such as the energy, valence, magnetic
susceptibility and the specific heat coefficient, are discussed. In the integer valent limit the model reduces to a Coqblin-Schrieffer impurity.
Received: 31 December 1997 / Accepted: 17 March 1998 相似文献
5.
We investigate under which circumstances extended Hubbard models, including bond-charge, exchange, and pair-hopping terms,
are invariant under gl
(2,1) superalgebra. This happens for a two-parameter Hamiltonian which includes as particular cases the t
-
J, the EKS and the one-parameter BGLZ Hamiltonians, all integrable in one dimension. We show that the two parameter Hamiltonian
can be recasted as the sum of the BGLZ Hamiltonian plus the graded permutation operator of electronic states on neighbouring
sites. The integrability of the corresponding one-dimensional model is discussed.
Received: 17 February 1998 / Received in final form: 6 March 1998 / Accepted: 17 April 1998 相似文献
6.
The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model
consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory.
In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators.
We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice ( N=3) and a simple-cubic lattice ( N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features
resulting from the fluctuation of the crystal field interactions.
Received: 13 February 1998 / Accepted: 17 March 1998 相似文献
7.
A new model for correlated electrons is presented which is integrable in one-dimension. The symmetry algebra of the model
is the Lie superalgebra gl(2|1) which depends on a continuous free parameter. This symmetry algebra contains the pairing algebra as a subalgebra which is used to show that the model exhibits Off-Diagonal Long-Range Order in any number
of dimensions.
Received: 9 December 1997 / Revised: 12 February 1998 / Accepted: 17 March 1998 相似文献
8.
Using the density matrix renormalization group method (DMRG) we calculate the magnetization of frustrated S=1/2 Heisenberg chains for various modulation patterns of the nearest neighbour coupling: commensurate, incommensurate with
sinusoidal modulation and incommensurate with solitonic modulation. We focus on the order of the phase transition from the
commensurate dimerized phase (D) to the incommensurate phase (I). It is shown that the order of the phase transition depends
sensitively on the model. For the solitonic model in particular, a k-dependent elastic energy modifies the order of the transition. Furthermore, we calculate gaps in the incommensurate phase
in adiabatic approximation.
Received: 9 March 1998 / Accepted: 17 April 1998 相似文献
9.
We point out the existence of an alternative algebraic structure in Yang-Baxter algebra with trigonometric R-matrix, which appears to be the generalization of the Yangian in Yang-Baxter algebras with rational R-matrix and which is described most naturally by q-commutators. Some properties are presented, in particular in the case of the well-known symmetric six-vertex model.
Received: 13 February 1998 / Revised: 16 March 1998 / Accepted: 17 April 1998 相似文献
10.
We present a pseudopotential method to study the absorption spectroscopy of NO in an argon matrix modeled by a large albeit
finite cluster. The excited states of NO are described with the virtual orbitals of a NO + Hartree-Fock calculation plus a core-polarization operator to account for the electron-NO + correlation. The argon atoms of the matrix are replaced by pseudopotentials for the repulsive contributions and core-polarization
operators to account for matrix polarization and correlation with the excited electron. The model is shown to account for
the matrix-induced transition shifts and also for the cut-off of the Rydberg series for n >3 reported in absorption experiments from the ground state.
Received: 6 March 1998 / Revised: 1st June 1998 / Accepted: 16 June 1998 相似文献
11.
In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable
quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle
spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy
is obtained as the derivative with respect to n of the trace of the nth power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are
naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the nth power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function
in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation.
Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations
satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide
solutions, which we specialize both to the Ising and sinh-Gordon models. 相似文献
12.
We investigate the Peierls transition in finite chains by exact (Lanczos) diagonalization and within a seminumerical method
based on the factorization of the electron-phonon wave function (Adiabatic Ansatz, AA). AA can be applied for mesoscopic chains
up to micrometer sizes and its reliability can be checked self-consistently. Our study demonstrates the important role played
for finite systems by the tunneling in the double well potential. The chains are dimerized only if their size N exceeds a critical value N c which increases with increasing phonon frequency. Quantum phonon fluctuations yield a broad transition region. This smooth Peierls transition contrasts not only to the sharp mean field transition, but also with the sharp RPA soft mode instability,
although RPA partially accounts for quantum phonon fluctuations. For weak coupling the dimerization disappears below micrometer
sizes; therefore, this effect could be detected experimentally in mesoscopic systems.
Received: 3 January 1998 / Revised: 13 March 1998 / Accepted: 3 April 1998 相似文献
13.
We derive the effective low-energy theory for single-wall carbon nanotubes including the Coulomb interactions among electrons.
The generic model found here consists of two spin-fermion chains which are coupled by the interaction. We analyze the theory using bosonization, renormalization-group techniques,
and Majorana refermionization. Several experimentally relevant consequences of the breakdown of Fermi liquid theory observed
here are discussed in detail, e.g., magnetic instabilities, anomalous conductance laws, and impurity screening profiles.
Received: 12 December 1997 / Revised: 9 March 1998 / Accepted: 12 March 1998 相似文献
14.
We derive the jump in the specific heat at T= T
c
for a superconductor in a non-Fermi liquid model. We took into consideration the two possible limits in this problem: the
spin-charge separation model for a Fermi liquid and the usual non-Fermi liquid model which satisfies the homogeneity relation
for the spectral function , ). We also derive the order parameter behavior for these two cases in the vecinity of the critical temperature.
Received: 25 January 1998 / Revised: 25 March 1998 / Accepted: 25 March 1998 相似文献
15.
We associate to an arbitrary ℤ-gradation of the Lie algebra of a Lie group a system of Riccati-type first order differential
equations. The particular cases under consideration are the ordinary Riccati and the matrix Riccati equations. The multidimensional
extension of these equations is given. The generalisation of the associated Redheffer–Reid differential systems appears in
a natural way. The connection between the Toda systems and the Riccati-type equations in lower and higher dimensions is established.
Within this context the integrability problem for those equations is studied. As an illustration, some examples of the integrable
multidimensional Riccati-type equations related to the maximally nonabelian Toda systems are given.
Received: 3 August 1998 / Accepted: 21 December 1998 相似文献
16.
We discovered a simple quadratic equation, which relates scattering phases of particles on Fermi surface. We consider one-dimensional
Bose gas and XXZ Heisenberg quantum spin chain.
Received: 4 December 1997 / Accepted: 17 March 1998 相似文献
17.
The two-dimensional Ising films with bulk H and surface H 1 fields of opposite sign are studied above and close to bulk criticality by the density matrix renormalization group method.
This technique, applied recently to d=2 Ising films, allows for very accurate results for the adsorption as a function of the reduced deviation from the critical temperature .For strong H 1 three distinct classes of shapes of ,determined by the value of the parameter ,where L is the width of the film, are found in agreement with earlier predictions [A. Macioek, A. Ciach, R. Evans, J. Chem. Phys. 108, 9765 (1998)]. For strong and for weak bulk fields is a monotonic function, increasing for strong H and decreasing for weak H, in agreement with scaling analysis and earlier mean-field results. For H between these extreme cases assumes a maximum for and for a depletion occurs, as in recent experiments for critical adsorption in porous materials. For a limited range of H a qualitatively new behavior of is found. In addition to a maximum, a minimum of for appears, which in the mean-field analysis was absent.
Received: 11 February 1998 / Received in final form: 16 February 1998 / Accepted: 17 March 1998 相似文献
18.
We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical
potential in the random-phase approximation (RPA). Calculating the unperturbed response functions for current- and charge-density
exactly, without any expansion with respect to frequency or wave vector, we find that the integral for the ground-state energy
converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at small k. This divergence originates in the k -2 singularity of the Chern-Simons interaction and it is already present in lowest-order perturbation theory. A similar divergence
appears in the chemical potential. Beyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally
crossed, or a combination of both) which diverge with powers of the logarithm. We expand our result for the RPA ground-state
energy in the strength of the Coulomb interaction. The linear term is finite and its value compares well with numerical simulations
of interacting electrons in the lowest Landau level.
Received: 19 February 1998 / Revised: 25 March 1998 / Accepted: 17 April 1998 相似文献
19.
A two dimensional antiferromagnetic spin-1 Ising model with negative next- nearest neighbour interaction ( J
2
<0) and under an external magnetic field is investigated by two methods: The mean-field theory and Finite-Size-Scaling based
on transfer matrix (TMFSS) calculations. The ground state diagrams exhibit several new phases including frustrated ones. At
finite temperature we obtain by these two methods quite rich phase diagrams, with several multicritical points. While Mean
field approximation yields phase diagrams which are sometimes even qualitatively incorrect, accurate results are obtained
from transfer matrix finite size scaling calculations. For a certain range of interaction parameters, the model is shown to
violate the ordinary universality hypothesis.
Received: 3 November 1997 / Revised: 31 March 1998 / Accepted: 7 April 1998 相似文献
20.
The theory presented by Gerardy
and Ausloos for the calculation of the linear optical response of aggregates of spherical particles is analytically continued for absorbing
embedding media. The method is based on the calculation of the extinction rate by a single particle embedded in an absorbing
matrix. Explicit expressions for the extinction and scattering cross-sections are given. The method is applied to calculate
the energy losses in several organic matrices with embedded silver clusters. Comparison with experimental data shows a very
good agreement.
Received: 21 December 1998 相似文献
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