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SZFKI Annual Report 1995

Published Jan 1, 1995• 0 citations
OpenAlex

Abstract

Low dimensional magnetic models.-The one-dimensional Heisenberg model is known to be soluble by the Bethe Ansatz.The spectrum of a ferromagnet consists of magnons and bound state of magnons.The problem of finding the bound state of a macroscopically large number of magnons poses a special problem.These states are described by macroscopically long string solutions of the Bethe Ansatz equations.To find such solutions numerically we developed a program by which even chains of several hundred sites can be treated.Our results show that, as expected, the structure of the macroscopically long strings is not of the Bethe-type.Recently several new materials have been discovered in which the localized magnetic moments are situated in a chain like structure.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of coupled chains.We started to study these models using the density matrix renormalization group.One-dimensional fermionic models.-The Hubbard model is the simplest model of the interacting electron system.It is usually formulated on a discrete lattice.Sometimes it is more appropriate to work in the continuum limit.We have shown, that the continuum limit of the Hubbard chain, in which the particle number per site is kept constant leads to a relativistic model describing both finite and zero mass particles.By constructing the relativistic limit of the half-filled band we have shown, that also the relativistic particles are described by Bethe-Ansatz type equations, while in the relativistic limit of the non-half-filled chain this is true only for the massive particles, and the equations of the massless particles are of somewhat different structure.Theory of dissipative motion of heavy particles.-We studied the low energy fixed point structure of an N f -flavor two level system (TLS) model, where the spin of the conduction electrons is replaced by a flavor quantum number, f= 1, ... , N f .Using the multiplicative renormalization group we constructed the (1/N f ) 2 order scaling equations and we analyzed their fixed point structure.We showed that the fixed point structure of this Hamiltonian is equivalent to that of the N f -channel Kondo model and we found that the exponent of the leading irrelevant operators is also the same.However, for a TLS an infinite number of leading irrelevant operators appear which give a contribution to the different physical quantities but do not change their critical exponents.

Related papers

SZFKI Annual Report 1997

Low dimensional magnetic models.-We continued our study of the physical properties of the so-called magnetic ladder systems.We have considered anisotropic two-leg ladder models with S=1/2 and S=1 spins on the rungs, using the density matrix renormalization group (DMRG) method.We have generalized the numerical procedure to allow to take into account the symmetries of the models in order to reduce the dimension of the matrix to be diagonalized or to achieve better accuracy.We have determined the boundaries of the extended region in the parameter space, where the gapped Haldane phase exists.In our study of the scaling limit of the one dimensional XXZ Heisenberg chain we have found, that the limiting model possesses spin 1/2 massive particles, and possibly can be identified with the Sine-Gordon chain.One-dimensional fermionic models.-We continued the study of the models defined through the relativistic limit of the Hubbard chain by investigating the scaling limit of the one dimensional attractive Hubbard model at less than half filling.Like in the half filled band case, also this model possesses both massive and massless dressed particles.Now, unlike the half filled case, only the massive particles are described by Bethe Ansatz type equations, nevertheless the massless excitations still fit into the framework of a conform field theory.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in various models and approximations.We continued the study of the region below the Kondo temperature by the use of the model with electron spin degeneracy N f > 2. The only stable fixed point and the exponents of the leading irrelevant operators are equivalent to those of the spin 1/2 Kondo model, but there are essentially more irrelevant operators in our case.We studied the role of the electron-hole symmetry in the low temperature behaviour of TLS, moreover the differences between the scaling methods starting from path integrals and Matsubara technique.We showed that the breaking of the electron-hole symmetry results in electron assisted tunneling, but the contribution cannot bring the system to the two-channel Kondo fixed point.

•Jan 1, 1997•0 citations

SZFKI Annual Report 1996

Low dimensional magnetic models.-Recently several new materials have been discovered in which the localized magnetic moments are coupled in a way resembling ladders.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of legs of the ladder.We studied the phase diagram of such ladder models using the density matrix renormalization group (DMRG).First in a rather technical work we have checked the accuracy of the DMRG procedure on the example of the Ising model in transverse field, as the simplest quantum mechanical spin problem, and have shown how the so-called truncation error is related to the real error in the energy values.With this background we have determined the phase diagram of a two leg ladder model including also plaquette couplings, and have found indication for the existence of several different massive phases.One-dimensional fermionic models.-We investigated in detail the model defined through the relativistic limit of the one dimensional attractive Hubbard model.This model possesses both massive and massless dressed particles which are described just like the particles in the Hubbard chain, by Bethe-Ansatz type equations.We analysed the eigenstates of the system.We have found that the limiting model has higher symmetry than the original Hubbard chain; it acquires in addition also the chiral symmetry.We determined the scattering properties of the particles, and also the conformal properties of the field corresponding to the massless sector.All these results confirm, that the scaling limit of the Hubbard chain can be identified with the chiral invariant Gross-Neveu model.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in amorphous metals.To approach the temperature region below the Kondo temperature we generalized the model to electron spin degeneracies: N f > 2. The solution of the scaling equations written in N f 2 order confirmed that the only stable fixed point of the system is equivalent to the Kondo-model of spin 1/2, moreover we found the limits of this equivalency.We also studied what conditions can make a commutative TLS model -not showing Kondo-effect in simple casesto be non-commutative.The asymmetry of the electron band even in leading order, the splitting of the levels in higher orders result in logarithmic divergencies in the couplings, which were originally invariant in the scaling.

•Jan 1, 1996•0 citations

Bethe strings in the spin dynamical structure factor of the Mott-Hubbard phase in the one-dimensional fermionic Hubbard model

J. M. P. Carmelo, Tilen Čadež

The spectra and role in the spin dynamical properties of bound states of elementary magnetic excitations named Bethe strings that occur in some integrable spin and electronic one-dimensional models have recently been identified and realized in several materials by experiments. Corresponding theoretical studies have usually relied on the one-dimensional spin-$1/2$ Heisenberg antiferromagnet in a magnetic field. At the isotropic point, it describes the large onsite repulsion $U$ limit of the spin degrees of freedom of the one-dimensional fermionic Hubbard model with one electron per site in a magnetic field $h$. In this paper we consider the thermodynamic limit and study the effects of lowering the latter quantum problem ratio $u=U/4t$, where $t$ is the first-neighbor transfer integral, on the line-shape singularities in $(k,\ensuremath{\omega})$-plane regions at and just above the lower thresholds of the transverse and longitudinal spin dynamical structure factors. The most significant spectral weight contribution from Bethe strings leads to a gapped continuum in the spectrum of the spin dynamical structure factor ${S}^{+\ensuremath{-}}(k,\ensuremath{\omega})$. Our study focuses on the line shape singularities at and just above the gapped lower threshold of that continuum, which have been identified in experiments. Our results are consistent with the contribution of Bethe strings to ${S}^{zz}(k,\ensuremath{\omega})$ being small at low spin densities and becoming negligible upon increasing that density. Our results provide physically important information about how electron itinerancy affects the spin dynamics.

arXiv:2101.06198•Physical review. B./Physical review. B•Jan 15, 2021•2 citations

The half-filled Hubbard chain in the Composite Operator Method: A comparison with Bethe Ansatz

M. M. Sánchez, A. Avella, F. Mancini

PACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest model

•Jan 1, 1998•8 citations

Dynamical correlations of S=1/2 quantum spin chains

Rodrigo G. Pereira

Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.

•Jan 1, 2008•0 citations

Hubbard chain with reflecting ends

Hermann Schulz

The one-dimensional Hubbard model bounded by infinitely high potential walls is exactly diagonalised by Bethe-Yang ansatz techniques. The 'surface' energy is studied for the half-filled band and in the Fermi gas limit. The analysis starts with a solution of a more general model in which the particles also interact with their mirror-image sites. In the linear band limit the states of this system become the Bethe eigenstates of the Kondo model at zero coupling.

Journal of Physics C Solid State Physics•Jan 30, 1985•108 citations
Low dimensional magnetic models.-The one-dimensional Heisenberg model is known to be soluble by the Bethe Ansatz.The spectrum of a ferromagnet consists of magnons and bound state of magnons.The problem of finding the bound state of a macroscopically large number of magnons poses a special problem.These states are described by macroscopically long string solutions of the Bethe Ansatz equations.To find such solutions numerically we developed a program by which even chains of several hundred sites can be treated.Our results show that, as expected, the structure of the macroscopically long strings is not of the Bethe-type.Recently several new materials have been discovered in which the localized magnetic moments are situated in a chain like structure.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of coupled chains.We started to study these models using the density matrix renormalization group.One-dimensional fermionic models.-The Hubbard model is the simplest model of the interacting electron system.It is usually formulated on a discrete lattice.Sometimes it is more appropriate to work in the continuum limit.We have shown, that the continuum limit of the Hubbard chain, in which the particle number per site is kept constant leads to a relativistic model describing both finite and zero mass particles.By constructing the relativistic limit of the half-filled band we have shown, that also the relativistic particles are described by Bethe-Ansatz type equations, while in the relativistic limit of the non-half-filled chain this is true only for the massive particles, and the equations of the massless particles are of somewhat different structure.Theory of dissipative motion of heavy particles.-We studied the low energy fixed point structure of an N f -flavor two level system (TLS) model, where the spin of the conduction electrons is replaced by a flavor quantum number, f= 1, ... , N f .Using the multiplicative renormalization group we constructed the (1/N f ) 2 order scaling equations and we analyzed their fixed point structure.We showed that the fixed point structure of this Hamiltonian is equivalent to that of the N f -channel Kondo model and we found that the exponent of the leading irrelevant operators is also the same.However, for a TLS an infinite number of leading irrelevant operators appear which give a contribution to the different physical quantities but do not change their critical exponents.
Low dimensional magnetic models.-We continued our study of the physical properties of the so-called magnetic ladder systems.We have considered anisotropic two-leg ladder models with S=1/2 and S=1 spins on the rungs, using the density matrix renormalization group (DMRG) method.We have generalized the numerical procedure to allow to take into account the symmetries of the models in order to reduce the dimension of the matrix to be diagonalized or to achieve better accuracy.We have determined the boundaries of the extended region in the parameter space, where the gapped Haldane phase exists.In our study of the scaling limit of the one dimensional XXZ Heisenberg chain we have found, that the limiting model possesses spin 1/2 massive particles, and possibly can be identified with the Sine-Gordon chain.One-dimensional fermionic models.-We continued the study of the models defined through the relativistic limit of the Hubbard chain by investigating the scaling limit of the one dimensional attractive Hubbard model at less than half filling.Like in the half filled band case, also this model possesses both massive and massless dressed particles.Now, unlike the half filled case, only the massive particles are described by Bethe Ansatz type equations, nevertheless the massless excitations still fit into the framework of a conform field theory.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in various models and approximations.We continued the study of the region below the Kondo temperature by the use of the model with electron spin degeneracy N f > 2. The only stable fixed point and the exponents of the leading irrelevant operators are equivalent to those of the spin 1/2 Kondo model, but there are essentially more irrelevant operators in our case.We studied the role of the electron-hole symmetry in the low temperature behaviour of TLS, moreover the differences between the scaling methods starting from path integrals and Matsubara technique.We showed that the breaking of the electron-hole symmetry results in electron assisted tunneling, but the contribution cannot bring the system to the two-channel Kondo fixed point.
Low dimensional magnetic models.-Recently several new materials have been discovered in which the localized magnetic moments are coupled in a way resembling ladders.The energy spectrum and the behaviour of these systems depends strongly on the value of the spin and the number of legs of the ladder.We studied the phase diagram of such ladder models using the density matrix renormalization group (DMRG).First in a rather technical work we have checked the accuracy of the DMRG procedure on the example of the Ising model in transverse field, as the simplest quantum mechanical spin problem, and have shown how the so-called truncation error is related to the real error in the energy values.With this background we have determined the phase diagram of a two leg ladder model including also plaquette couplings, and have found indication for the existence of several different massive phases.One-dimensional fermionic models.-We investigated in detail the model defined through the relativistic limit of the one dimensional attractive Hubbard model.This model possesses both massive and massless dressed particles which are described just like the particles in the Hubbard chain, by Bethe-Ansatz type equations.We analysed the eigenstates of the system.We have found that the limiting model has higher symmetry than the original Hubbard chain; it acquires in addition also the chiral symmetry.We determined the scattering properties of the particles, and also the conformal properties of the field corresponding to the massless sector.All these results confirm, that the scaling limit of the Hubbard chain can be identified with the chiral invariant Gross-Neveu model.Theory of dissipative motion of heavy particles.-We studied the low temperature behaviour of two level systems (TLS) interacting with the conduction electrons in amorphous metals.To approach the temperature region below the Kondo temperature we generalized the model to electron spin degeneracies: N f > 2. The solution of the scaling equations written in N f 2 order confirmed that the only stable fixed point of the system is equivalent to the Kondo-model of spin 1/2, moreover we found the limits of this equivalency.We also studied what conditions can make a commutative TLS model -not showing Kondo-effect in simple casesto be non-commutative.The asymmetry of the electron band even in leading order, the splitting of the levels in higher orders result in logarithmic divergencies in the couplings, which were originally invariant in the scaling.
The spectra and role in the spin dynamical properties of bound states of elementary magnetic excitations named Bethe strings that occur in some integrable spin and electronic one-dimensional models have recently been identified and realized in several materials by experiments. Corresponding theoretical studies have usually relied on the one-dimensional spin-1/21/21/2 Heisenberg antiferromagnet in a magnetic field. At the isotropic point, it describes the large onsite repulsion UUU limit of the spin degrees of freedom of the one-dimensional fermionic Hubbard model with one electron per site in a magnetic field hhh. In this paper we consider the thermodynamic limit and study the effects of lowering the latter quantum problem ratio u=U/4tu=U/4tu=U/4t, where ttt is the first-neighbor transfer integral, on the line-shape singularities in $(k,\ensuremath{\omega})$-plane regions at and just above the lower thresholds of the transverse and longitudinal spin dynamical structure factors. The most significant spectral weight contribution from Bethe strings leads to a gapped continuum in the spectrum of the spin dynamical structure factor ${S}^{+\ensuremath{-}}(k,\ensuremath{\omega})$. Our study focuses on the line shape singularities at and just above the gapped lower threshold of that continuum, which have been identified in experiments. Our results are consistent with the contribution of Bethe strings to ${S}^{zz}(k,\ensuremath{\omega})$ being small at low spin densities and becoming negligible upon increasing that density. Our results provide physically important information about how electron itinerancy affects the spin dynamics.
PACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest model
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.