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One-dimensional periodic Anderson model
Abstract
In this paper, we study the one-dimensional periodic Anderson model via bosonization. In the Toulouse limit, an effective Hamiltonian for the correlated electrons is derived. The effective Hamiltonian provides new information in understanding the emergence of ferromagnetism in the model.
... We derive the effective Hamiltonian for the f spins for the partially filled PAM and provide a justification of results briefly reported elsewhere. 18 The obtained phase diagram shows two magnetic transitions, which we analyze in details. Ferromagnetism was known to exist in KLM. 4 As mentioned before, the two models are connected by the well-known Schrieffer-Wolff transformation. ...
... In the following we set e 0 F as the zero of energy. The bosonization of the noninteracting (U = 0) Hamiltonian is essentially identical to the procedure presented in Ref. 18: ...
Low dimensional rare-earth alloys reveal a rich phase diagram which always incorporate a ferromagnetic (FM) phase. Here we show that rare-earth ferromagnetism in low dimensions is due to double-exchange mechanism. We use the bosonized version of the one-dimensional Anderson lattice model in Toulouse limit to characterize the properties of this emerging FM phase. We give a comprehensive description of the FM ordering of the correlated electrons which appears at intermediate couplings and doping. Determine the critical properties of the phase transitions into the quantum disordered paramagnetic phases. The obtained phase transitions have been identified to be an order–disorder transition of the quantum random transverse-field Ising type.
The emergence of ferromagnetism in conducting polymers is well known, however, the influence of lattice vibrations has not been analyzed yet. Hereafter, we fill this gap by studying the common effect of strong correlations and lattice vibrations in hole doped conjugated polymers. The results underline that away from system half-filling, the lattice vibrations have negligible effect on the studied ordered phase.
We construct a class of exact ground states of three-dimensional periodic Anderson models (PAMs) -- including the conventional PAM -- on regular Bravais lattices at and above 3/4 filling, and discuss their physical properties. In general, the f electrons can have a (weak) dispersion, and the hopping and the non-local hybridization of the d and f electrons extend over the unit cell. The construction is performed in two steps. First the Hamiltonian is cast into positive semi-definite form using composite operators in combination with coupled non-linear matching conditions. This may be achieved in several ways, thus leading to solutions in different regions of the phase diagram. In a second step, a non-local product wave function in position space is constructed which allows one to identify various stability regions corresponding to insulating and conducting states. The compressibility of the insulating state is shown to diverge at the boundary of its stability regime. The metallic phase is a non-Fermi liquid with one dispersing and one flat band. This state is also an exact ground state of the conventional PAM and has the following properties: (i) it is non-magnetic with spin-spin correlations disappearing in the thermodynamic limit, (ii) density-density correlations are short-ranged, and (iii) the momentum distributions of the interacting electrons are analytic functions, i.e., have no discontinuities even in their derivatives. The stability regions of the ground states extend through a large region of parameter space, e.g., from weak to strong on-site interaction U. Exact itinerant, ferromagnetic ground states are found at and below 1/4 filling.
Thermodynamical properties of the “resonance level model” are calculated for arbitrary spin and magnetic field. Assuming this model to describe the basic physics of the Kondo problem we find a satisfactory agreement with recent magnetization data on Fe.
We determine the bounday of the fully polarized ferromagnetic states in, the one dimensional Kondo lattice model at partial conduction electron band filling by using a newly developed infinite size DMRG method which conserves the total spin quantum numbers. The obtained paramagnetic to ferromagnetic phase bounday is bellow J approximate to 3.5 for the whole range of band filling. By this we solve the controversy in the phase diagram over the extent of the ferromagnetic region close to half filling.
A comprehensive theory of electron spin resonance (ESR) for a Luttinger liquid state of correlated metals is presented. The ESR measurables such as the signal intensity and the linewidth are calculated in the framework of Luttinger liquid theory with broken spin rotational symmetry as a function of magnetic field and temperature. We obtain a significant temperature dependent homogeneous line broadening which is related to the spin-symmetry breaking and the electron-electron interaction. The result crosses over smoothly to the ESR of itinerant electrons in the noninteracting limit. These findings explain the absence of the long-sought ESR signal of itinerant electrons in single-wall carbon nanotubes when considering realistic experimental conditions.
We present a class of exact ground states of a three-dimensional periodic Anderson model at 3/4 filling. Hopping and hybridization of d and f electrons extend over the unit cell of a general Bravais lattice. Employing novel composite operators combined with 55 matching conditions the Hamiltonian is cast into positive semidefinite form. A product wave function in position space allows one to identify stability regions of an insulating and a conducting ground state. The metallic phase is a non-Fermi liquid with one dispersing and one flat band.
The spin-1/2 quantum Ising chain in a transverse random magnetic field is studied by means of the density-matrix renormalization group. The system evolves from an ordered to a paramagnetic state as the amplitude of the random field is increased. The dependence of the magnetization on a uniform magnetic field in the z direction and the spontaneous magnetization as a function of the amplitude of the transverse random magnetic field are determined. The behavior of the spin-spin correlation function both above and at criticality is studied. The scaling laws for magnetization and correlation functions are tested against previous numerical and renormalization-group results. Comment: 5 pages with 7 figures inside them, proper format of authors' names used
We find a solvable limit to the problem of the 1D electron gas interacting with a lattice of Kondo scattering centers. In this limit, we present exact results for the problems of incommensurate filling, commensurate filling, impurity vacancy states, and the commensurate-incommensurate transition. Comment: 4 pages, two columns, Latex file
We establish a mapping of a general spin-fermion system in one dimension into a classical generalized Coulomb gas. This mapping allows a renormalization group treatment of the anisotropic Kondo chain both at and away from half-filling. We find that the phase diagram contains regions of paramagnetism, partial and full ferromagnetic order. We also use the method to analyze the phases of the Ising-Kondo chain.
Using a non-Abelian density matrix renormalization group method we determine the phase diagram of the Kondo lattice model in one dimension, by directly measuring the magnetization of the ground-state. This allowed us to discover a second ferromagnetic phase missed in previous approaches. The phase transitions are found to be continuous. The spin-spin correlation function is studied in detail, and we determine in which regions the large and small Fermi surfaces dominate. The importance of double-exchange ordering and its competition with Kondo singlet formation is emphasized in understanding the complexity of the model. Comment: Revtex, 4 pages, 4 eps figures embedded
Presenting exact solutions for the two dimensional periodic Anderson model with finite and nonzero on-site interaction U>0, we are describing a rigorous non-Fermi liquid phase in normal phase and 2D. This new state emerges in multi-band interacting Fermi systems above half filling, being generated by a flat band effect. The momentum distribution function n_k together with its derivatives of any order is continuous. The state possesses a well defined Fermi energy, but the Fermi momentum concept is not definable, so the Fermi surface in k-space is missing. The state emerges in the vicinity of a Mott insulating phase when lattice distortions are present, is highly degenerated and paramagnetic. A gap is present at high U in the density of low lying excitations. During low lying excitations, quasi-particles are not created above the Fermi level, only the number of particles at the Fermi energy increases. Comment: 46 pages, 2 ps files, to be published in Phys. Rev. B
We report the discovery of a new ferromagnetic phase in the one-dimensional Kondo lattice model for intermediate values of the coupling constant. This new ferromagnetic phase was observed using a non-Abelian density-matrix renormalization group algorithm, which allowed us to measure directly the magnetization of the ground state with high accuracy.
The effective s-d spin interaction is derived exactly for the one-impurity Anderson model via a unitary transformation. The unitary transformation has been calculated up to infinite order, thus an exact transformation was performed in the strict mathematical sense.
Substantial improvements in the computational effort in a density matrix
renormalisation program can be made by utilising symmetries of the
Hamiltonian. Extra quantum numbers are always desirable to include in the
calculation, since it allows the Hilbert space of the superblock to be
refined. Since the speed of the calculation is approximately
O(n 3 ) in
superblock states, the speed increase in targeting a specific total spin state
can be considerable. In this paper a new density matrix renormalisation
algorithm is presented which conserves the total spin. The general procedure
obtained works for any operator, even operators that do not commute with the
Hamiltonian.
Exact static nondegenerate stripe and checkerboard ground states are obtained in a two-dimensional generalized periodic Anderson model, for a broad concentration range below quarter filling. The random droplet states, also present in the degenerate ground state, are eliminated by extending the Hamiltonian with terms of different physical origin such as dimerization, periodic charge displacements, density waves, or distortion lines.
Presenting exact solutions for the two-dimensional periodic Anderson
model with finite and nonzero on-site interactions U>0, we describe a
rigorous non-Fermi-liquid phase in normal phase and two dimensions. This
new state emerges in multiband interacting Fermi systems above half
filling, being generated by a flat-band effect. The momentum
distribution function nk--> together with its derivatives
of any order is continuous. The state possesses a well-defined Fermi
energy (eF), but the Fermi momentum concept is not definable,
so the Fermi surface in k--> space is missing. The state emerges in
the vicinity of a Mott insulating phase when lattice distortions are
present and is highly degenerated and paramagnetic. A gap is present at
high U in the density of low-lying excitations. During low-lying
excitations, quasiparticles are not created above the Fermi level, only
the number of particles at eF increases.
Recent experiments demonstrate that at the Curie temperature the specific heat may be a smooth function of the temperature. We propose that this effect can be due to random impurities and substantiate our proposal by a study of an Ising model containing such impurities. We modify the usual rectangular lattice by allowing each row of vertical bonds to vary randomly from row to row with a prescribed probability function. In the case that this probability is a particular distribution with a narrow width, we find that the logarithmic singularity of Onsager's lattice is smoothed out into a function which at Tc is infinitely differentiable but not analytic. This function is expressible in terms of an integral involving Bessel functions and is computed numerically.
The periodic Anderson and Kondo lattice model describe the physics of conduction electrons in extended orbitals interacting with strongly correlated electrons in localized orbitals. These models are relevant for the so-called heavy-fermion and related systems such as the Kondo insulators. In this review we summarize recent progress in the understanding of these models, in particular, the one-dimensional Kondo lattice model. The ground-state phase diagram for the one-dimensional Kondo lattice model is determined and shows three distinct phases: a ferromagnetic metallic, an insulating spin liquid, and a paramagnetic metallic state. We present results on these phases obtained from rigorous and approximate analytical calculations supported by various extensive numerical studies on finite-size systems. The ferromagnetic phase appears in the limit of low density of conduction electrons and for strong Kondo coupling away from half filling. On the other hand, the half-filled Kondo lattice has a gap in both spin and charge excitations, i.e., it has a spin-liquid ground state. The paramagnetic phase may be considered as the generic heavy-fermion state and appears in the weak-coupling limit away from half filling. While the former two phases are well understood, the physics of the paramagnetic phase is not worked out in detail yet. In this context various questions will be considered here: Does the Fermi surface include conduction electrons only or also the localized electrons? Does the concept of Luttinger liquid apply in this case? The extension of these results to higher dimensions is also discussed. It is important to notice that the ground states of the Kondo lattice and the periodic Anderson model involve complicated effects, which cannot be understood by simple extension of the single- or two-impurity problem.
The current understanding of the theory of quantum liquids is presented. The properties of the three- and one-dimensional conductors are compared. A comprehensive description of the bosonization technique is presented for the one-dimensional systems. The two-dimensional systems are also analysed, and the theoretical and experimental results for this case are briefly presented.
The analogy between the thermodynamics of the s-d model and those of a one-dimensional classical Coulomb gas is exploited to calculate the impurity-spin susceptibility on the computer. The numerical method is a special Monte Carlo procedure first used by Metropolis et al. We find a Curie-Weiss form for the static susceptibility with a Néel temperature of about one-third of the Kondo temperature. We discuss the connection between our results and a recent scaling theory of Anderson, Yuval, and Hamann.
We study a nonuniversal contribution to the dephasing rate of conduction electrons due to local vibrational modes. The inelastic-scattering rate is strongly influenced by multiphonon excitations, exhibiting oscillatory behavior. For higher frequencies, it saturates to a finite, coupling dependent value. In the strong-coupling limit, the phonon is almost completely softened and the inelastic cross section reaches its maximal value. This represents a magnetic-field insensitive contribution to the dephasing time in mesoscopic systems, in addition to magnetic impurities.
The local spin order in the one-dimensional Kondo lattice model is studied for the conduction-electron band filling n=1/2 and 1/3 in a special parameter case. The local spin-dimerization ground state is confirmed for the quarter-filling case. And the spin order is studied for n=1/3. The spin and charge gaps are given for different band-filling cases.
We carefully consider the interplay between ferromagnetism and the Kondo
screening effect in the conventional Kondo lattice systems at finite
temperatures. Within an effective mean-field theory for small conduction
electron densities, a complete phase diagram has been determined. In the
ferromagnetic ordered phase, there is a characteristic temperature scale to
indicate the presence of the Kondo screening effect. We further find two
distinct ferromagnetic long-range ordered phases coexisting with the Kondo
screening effect: a spin polarized phase and a non-polarized phase. A
continuous phase transition exists to separate the spin non-polarized
ferromagnetic ordered phase from the paramagnetic heavy Fermi liquid phase.
These results may be used to explain the weak ferromagnetism observed recently
in the Kondo lattice materials.
The effective s–d spin interaction is derived exactly for the single-impurity Anderson model via a unitary transformation. This unitary transformation was calculated up to infinite order and no restrictions were imposed upon the coefficients of the Hubbard interaction and the hybridization. We also discuss briefly the impact of the obtained result on the magnetic properties of several Kondo compounds. This will shed new light on the understanding of the competition between the Kondo effect and the Ruderman–Kittel–Kasuya–Yosida interaction and reinterpret the Doniach diagram.
A new density matrix renormalization group (DMRG) algorithm is presented which conserves the total spin. This is the first time that standard DMRG is extended to exploit fully the symmetries of the Hamiltonian. The obtained general procedure works for any operator, even operators that do not commute with the Hamiltonian. The new method gives substantial improvements over the standard DMRG.
Quantum Ising models in a transverse field are related to continuous-time percolation processes whose oriented percolation versions are contact processes. We study such models in the presence of quasiperiodic disorder and prove localization in the ground state, no percolation, and extinction, respectively, for sufficiently large disorder.
We present the results of an infinite order unitary transformation applied to a multiband Hubbard Hamiltonian by which it can be shown rigorously that an attractive interaction term appears at the oxygen ion sites as a result of oxygen–copper virtual charge excitations. This exact result yields convincing evidence that the pairing mechanism in two and/or three band Hubbard models results precisely from this attraction.
We compare different approximation schemes for investigating ferromagnetism in the periodic Anderson model. The use of several
approximations allows for a detailed analysis of the implications of the respective methods, and also of the mechanisms driving
the ferromagnetic transition. For the Kondo limit, our results confirm a previously proposed mechanism leading to ferromagnetic
order, namely an RKKY exchange mediated via the formation of Kondo screening clouds in the conduction band. The contrary case is found in the intermediate-valence regime.
Here, the bandshift correction ensuring a correct high-energy expansion of the self-energy is essential. Inclusion of damping
effects reduces stability of the ferromagnetic phase.
It is shown exactly that for an N-site cyclic chain with hamiltonian H = −ΣNi=1(γiSxi + JiSziSziSzi+1), the gap in the excitation spectrum goes to zero when N → ∞ at the “critical point” given by the relation ΠNi=1Γi = ΠNi=1Ji.
The infinite-size version of the density-matrix renormalization-group approach in real space is applied to the Kondo lattice model in one dimension, providing a more accurate determination of the zero-temperature magnetic properties for electron densities . The paramagnetic state of the free model evolves into a state with ferromagnetic correlations, by increasing the strength of the Kondo coupling, in agreement with previous numerical results. The change in the magnetic properties is produced by the competition between the RKKY mechanism, which favours the paramagnetic state in the weak-coupling region, and the Kondo screening, which leads to the formation of tightly bound singlets in the strong-coupling region, enforcing ferromagnetic correlations in the system.
The coexistence of spin-density waves and superconductivity has been theoretically analyzed in a two-band model, a case for which the commonly used descriptions for the ternary rare-earth compounds and highly anisotropic organic solids are unapplicable. The phase diagram at T=0, the temperature dependence of the order parameters, and the critical transition temperatures have been obtained. The theoretical results are in good agreement with the experiments concerning the Cr alloys.
The first theoretical description of the itinerant antiferromagnetic state in heavy-fermion systems is presented in detail. We analyze the phase diagram, the stability of the phases, the magnetic susceptibility, and the specific heat. For the case in which the gap vanishes in points on the Fermi surface, the deduced results are in good agreement with the experimental data.
We study a d = 2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d = 1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index ..nu.. = 1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.
It is shown that interlayer pairing interaction can enhance the superconducting critical temperature of layered compounds in comparison with the critical temperature of the intralayer processes. Possible implications for high-Tc oxidic superconductors are also discussed.
We study quantum spin chains exhibiting long-range order in the presence of quasiperiodic interactions which are modulating functions of sites. The magnetic phase transition to long-range order is shown to accompany a transition from critical to localized states. The presence of more than one harmonic in the modulating interactions results in a cascade of transitions characterized by the vanishing of the gap and sharp peaks in the total bandwidth and free energy.
Within the ice-type models, the solution of the five-vertex model is obtained with the use of the Bethe ansatz. Since the allowed number of vertex types is odd, the arrow-reversal symmetry of the system is broken by construction. Due to this, the exact solution obtained and the phase diagram are very different from those of the symmetric six-vertex model. A connection to the asymmetric six-vertex model (of which the five-vertex model is an extreme case) is made. The different regions of the phase diagram are described and the transitions between them are analyzed. Several aspects of the phase diagram are unusual, i.e., the ordered phases (both ferroelectric and antiferroelectric) are frozen-in phases and the disordered phase is replaced by a ferrielectric phase. In the free-fermion case, the known results of the modified KDP model are recovered.
A real-space renormalization-group treatment of random transverse-field
Ising spin chains that was introduced previously is developed and
extensively analyzed. It yields results that are asymptotically exact in
the critical region near the zero-temperature para-to-ferromagnetic
quantum phase transition. In particular, the exact scaling function is
obtained for the magnetization as a function of a uniform applied
magnetic field and the distance to the critical point, and up to the
solution of a linear ordinary differential equation whose solution can
be exhaustively analyzed, the scaling function of the average spin-spin
correlation function is also obtained. Thus more exact information is
obtainable about the critical behavior for this random model than is
known for the pure version which is equivalent to the two-dimensional
Ising model. The basic reason for this is the extremely broad
distribution of energy scales that occurs at low energies near the
critical point of the random system. For the random chain the
distribution of the magnetization of the first spin in a semi-infinite
system is also studied and the results found to agree in the scaling
limit with results of McCoy obtained from the exact solution of the
closely related McCoy-Wu Ising model; this provides strong justification
for the validity of the present approach. The singular properties of the
weakly ordered and weakly disordered ``Griffiths' phases'' that occur at
zero temperature near the critical point are also studied, as well as
the behavior at low but nonzero temperature. Possible extensions of the
results and general lessons drawn from them for other random systems are
briefly discussed.
We study numerically the critical region and the disordered phase of the random transverse-field Ising chain. By using a mapping of Lieb, Schultz and Mattis to non-interacting fermions, we can obtain a numerically exact solution for rather large system sizes, . Our results confirm the striking predictions of earlier analytical work and, in addition, give new results for some probability distributions and scaling functions. Comment: 16 pages with 23 embedded postscript figures. A uuencoded, compressed tar file. The postscript file is available by anonymous ftp from ftp://chopin.ucsc.edu/pub/one-d.ps
The quantum Ising model in a transverse field is studied where the exchange interaction is a modulating function of sites with many harmonics. The magnetically disordered phase is found to exhibit a ``mixed'' spectrum containing both critical and localized states, while the magnetically ordered phase has a pure spectrum with all states localized. Therefore, the spectral transition, although broadened, occurs simulataneously with the magnetic transition, and the magnetic disorder is accompanied by spectral disorder.
We analyse the role which the distance scale plays in the single-impurity Kondo problem using renormalization group improved perturbation theory. We derive the scaling functions for the local spin susceptibility in various limiting cases. In particular, we demonstrate exactly that the non-oscillating part of it should be short-range, i.e., vanish for distances and show explicitly that the interior of the screening cloud {\it does not} exhibit weak coupling behavior. Comment: 4 pages, 2 EPS figures, submitted to PRL; a missing term has been added to Eq. (9); this additional term doesn't effect our main conclusions
We describe the transition from a ferromagnetic phase, to a disordered para- magnetic phase, which occurs in one-dimensional Kondo lattice models with partial conduction band filling. The transition is the quantum order-disorder transition of the transverse-field Ising chain, and reflects double-exchange ordered regions of localized spins being gradually destroyed as the coupling to the conduction electrons is reduced. For incommensurate conduction band filling, the low-energy properties of the localized spins near the transition are dominated by anomalous ordered (disordered) regions of localized spins which survive into the paramagnetic (ferromagnetic) phase. Many interesting properties follow, including a diverging susceptibility for a finite range of couplings into the paramagnetic phase. Our critical line equation, together with numerically determined transition points, are used to determine the range of the double-exchange interaction. Models we consider are the spin 1/2 Kondo lattices with antiferromagnetic (Kondo) coupling, with ferromagnetic (Hund's rule) coupling, and the Kondo lattice with repulsive interactions between the conduction electrons. Comment: 18 pages, 6 embedded eps figures. To appear in Phys Rev B
The Kondo lattice model introduced in 1977 describes a lattice of localized magnetic moments interacting with a sea of conduction electrons. It is one of the most important canonical models in the study of a class of rare earth compounds, called heavy fermion systems, and as such has been studied intensively by a wide variety of techniques for more than a quarter of a century. This review focuses on the one dimensional case at partial band filling, in which the number of conduction electrons is less than the number of localized moments. The theoretical understanding, based on the bosonized solution, of the conventional Kondo lattice model is presented in great detail. This review divides naturally into two parts, the first relating to the description of the formalism, and the second to its application. After an all-inclusive description of the bosonization technique, the bosonized form of the Kondo lattice hamiltonian is constructed in detail. Next the double-exchange ordering, Kondo singlet formation, the RKKY interaction and spin polaron formation are described comprehensively. An in-depth analysis of the phase diagram follows, with special emphasis on the destruction of the ferromagnetic phase by spin-flip disorder scattering, and of recent numerical results. The results are shown to hold for both antiferromagnetic and ferromagnetic Kondo lattice. The general exposition is pedagogic in tone.
An effective Hamiltonian for the localized spins in the one-dimensional Kondo lattice model is derived via a unitary transformation involving a bosonization of delocalized conduction electrons. The effective Hamiltonian is shown to reproduce all the features of the model as identified in various numerical simulations, and provides much new information on the ferro- to paramagnetic phase transition and the paramagnetic phase. Comment: 11 pages Revtex, 1 Postscript figure. To appear in Phys. Rev. Lett
The asymmetric infinite-dimensional periodic Anderson model is examined with a quantum Monte Carlo simulation. For small conduction band filling, we find a severe reduction in the Kondo scale, compared to the impurity value, as well as protracted spin screening consistent with some recent controversial photoemission experiments. The Kondo screening drives a ferromagnetic transition when the conduction band is quarter-filled and both the RKKY and superexchange favor antiferromagnetism. We also find RKKY-driven ferromagnetic and antiferromagnetic transitions. Comment: 5 pages, LaTeX and 4 PS figures
We compute the magnetic structure factor, the singlet correlation function and the momentum distribution of the one-dimensional Kondo lattice model at the density . The density matrix-renormalization group method is used. We show that in the weak-coupling regime, the ground state is paramagnetic. We argue that a Luttinger liquid description of the model in this region is consistent with our calculations . In the strong-coupling regime, the ground state becomes ferromagnetic. The conduction electrons show a spinless-fermion like behavior. Comment: 8 pages, Latex, 5 figures included, to be published in PRB (Rapid Communications)
We describe here the extension of the density matrix renormalization group algorithm to the case where Hamiltonian has a non-Abelian global symmetry group. The block states transform as irreducible representations of the non-Abelian group. Since the representations are multi-dimensional, a single block state in the new representation corresponds to multiple states of the original density matrix renormalization group basis. We demonstrate the usefulness of the construction via the one-dimensional Hubbard model as the symmetry group is enlarged from , up to . Comment: Revised version discusses the Hubbard model with SO(4) symmetry
R9757; I. I. Satija, ibid
- I I Satija
- M M Doria