Competition between the Kondo effect and Ruderman-Kittel-Kasuya-Yosida interaction in the two-impurity Kondo problem can be phenomenologically described by the Rasul-Schlottmann spin model. We revisit this model from the quantum entanglement perspective by calculating both the inter-impurity entanglement and the local Kondo entanglement, the latter being the entanglement between a local magnetic impurity and its spatially nearby conduction electron. A groundstate phase diagram is derived and a discontinuous breakdown of the local Kondo entanglement is found at the singular point, associated concomitantly with a jump in the inter-impurity entanglement. An entanglement monogamy holds in the whole phase diagram. Our results identify the important role of the frustrated cross-coupling and demonstrate the local characteristic of the quantum phase transition in the two-impurity Kondo problem. The implications of these results for Kondo lattices and quantum information processing are also briefly discussed.
We consider the real-time dynamics of the Kondo system after the local probe of the charge state of the magnetic impurity. Using the exactly solvable infinite-degeneracy Anderson model we find explicitly the evolution of the impurity charge after the measurement.
We propose that real-space properties of the two-impurity Kondo model can be obtained from an effective spin model where two single-impurity Kondo spin chains are joined via an RKKY interaction between the two impurity spins. We then use a DMRG approach, valid in all ranges of parameters, to study its features using two complementary quantum-entanglement measures, the negativity and the von Neumann entropy. This non-perturbative approach enables us to uncover the precise dependence of the spatial extent $ξ_K$ of the Kondo screening cloud with the Kondo and RKKY couplings. Our results reveal an…
The entanglement entropy in Kondo impurity systems is studied analytically using conformal field theory. From the impurity contribution to the scaling corrections of the entanglement entropy we extract information about the screening cloud profile for general non-Fermi-liquid fixed points. By also considering the finite-temperature corrections to scaling of the von Neumann entropy we point out a direct connection between the large-distance screening cloud profile and thermodynamic observables such as the specific heat.
The cyclic sudden switching of a magnetic impurity from Kondo to a non-Kondo state and back was recently shown to involve an important dissipation of the order of several $k_BT_K$ per cycle. The possibility to reveal this and other electronic processes through nanomechanical dissipation by e.g., ultrasensitive Atomic Force Microscope (AFM) tools currently represents an unusual and interesting form of spectroscopy. Here we explore the dependence on the switching time of the expected dissipation, a quantity whose magnitude is physically expected to drop from maximum to zero between sudden and sl…
The breakdown of the lattice Kondo effect in local-moment metals can lead to non-trivial forms of quantum criticality and a variety of non-Fermi-liquid phases. Given indications that Kondo-breakdown transitions involve criticality not only in the spin but also in the charge sector, we investigate the interplay of Kondo breakdown and strong valence fluctuations in generalized Anderson lattice models. We employ a parton mean-field theory to describe the transitions between deconfined fractionalized Fermi liquids and various confined phases. We find that rapid valence changes near Kondo breakdown…
Motivated by the observation of light surface states in SmB6, we examine the effects of surface Kondo breakdown in topological Kondo insulators. We present both numerical and analytic results which show that the decoupling of the localized moments at the surface disturbs the compensation between light and heavy electrons and dopes the Dirac cone. Dispersion of these uncompensated surface states are dominated by inter-site hopping, which leads to a much lighter quasiparticles. These surface states are also highly durable against the effects of surface magnetism and decreasing thickness of the s…
We propose that the thermoelectric power distinguishes two competing scenarios for quantum phase transitions in heavy fermions : the spin-density-wave (SDW) theory and breakdown of the Kondo effect. In the Kondo breakdown scenario, the Seebeck coefficient turns out to collapse from the temperature scale $E^{*}$, associated with quantum fluctuations of the Fermi surface reconfiguration. This feature differs radically from the physics of the SDW theory, where no reconstruction of the Fermi surface occurs, and can be considered as the hallmark of the Kondo breakdown theory. We test these ideas, u…
In the first step, experiments on a single cerium or ytterbium Kondo impurity reveal the importance of the Kondo temperature by comparison to other type of couplings like the hyperfine interaction, the crystal field and the intersite coupling. The extension to a lattice is discussed. Emphasis is given on the fact that the occupation number $n_f$ of the trivalent configuration may be the implicit key variable even for the Kondo lattice. Three $(P, H, T)$ phase diagrams are discussed: CeRu$_2$Si$_2$, CeRhIn$_5$ and SmS.
The breakdown of the Kondo effect may be the origin of the anomalous properties of the heavy-fermion compounds at low temperatures. We study the dynamics of one impurity embedded in an antiferromagnetic host at the quantum critical point and show that the impurity is not screened and develops a power law correlation function. This suggests that the breakdown of the Kondo effect can simply be a consequence of the system's proximity to the quantum critical point.
We consider a spin-1/2 Heisenberg chain coupled via a Kondo interaction to two-dimensional Dirac fermions. The Kondo interaction is irrelevant at the \textit{decoupled} fixed-point, leading to the existence of a Kondo-breakdown phase and a Kondo-breakdown critical point separating such a phase from a heavy Fermi liquid. We reach this conclusion on the basis of a renormalization group analysis, large-N calculations as well as extensive auxiliary-field quantum Monte Carlo simulations. We extract quantities such as the zero-bias tunneling conductance which will be relevant to future experiments i…
The single impurity Kondo model is often used to describe metals with dilute concentrations (n_i) of magnetic impurities. Here we examine how dilute the impurities must be for this to be valid by developing a virial expansion in impurity density. The O(n_i^2) term is determined from results on the 2-impurity Kondo problem by averaging over the RKKY coupling. The non-trivial fixed point of the 2-impurity problem could produce novel singularities in the heat capacity of dilute alloys at O(n_i^2).
The unconventional critical behavior near magnetic quantum phase transitions in various heavy-fermion metals, apparently inconsistent with the standard spin-density-wave scenario, has triggered proposals on the breakdown of the Kondo effect at the critical point. Here we investigate, within one specific scenario, the fate of such a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom. We study a Kondo-Heisenberg model with volume-dependent Kondo coupling -- this model displays both Kondo volume collapse and Kondo-breakdown transitions, as well as Lifshitz t…
We show that the scenario of multi-scale Kondo breakdown quantum critical point (QCP) gives rise to a divergent Grüneisen ratio with an anomalous exponent 0.7. In particular, we fit the experimental data of $YbRh_{2}(Si_{0.95}Ge_{0.05})_{2}$ for specific heat, thermal expansion, and Grüneisen ratio based on our simple analytic expressions. A reasonable agreement between the experiment and theory is found for the temperature range between 0.4 K and 10 K. We discuss how the Grüneisen ratio is a key measurement to discriminate between the Kondo breakdown and spin-density wave theories.
The screening of an impurity spin by conduction electrons is associated with the formation of a large Kondo screening cloud, of size xi_K. We study the quantum entanglement between a region of size r surrounding the impurity and the rest of the sample, (of total size R) using Density Matrix Renormalization Group and analytic methods. The corresponding "impurity entanglement entropy", S_{imp}, is shown to be a universal scaling function of r/xi_K and r/R. We calculate this universal function using Fermi liquid theory in the regime xi_K << r.
A spin-1/2 magnetic impurity coupled to a one-dimensional correlated electron system have been studied by applying the density renormalization group method. The Kondo temperature is substantially enhanced by strong repulsive interactions in the chain, but changes non-monotonically in the case of electron attraction. The magnetization of the impurity at zero- temperature shows local Fermi liquid behavior.
We show that a self-assembled phase of potassium (K) doped single-layer para-sexiphenyl (PSP) film on gold substrate is an excellent platform for studying the two-impurity Kondo model. On K-doped PSP molecules well separated from others, we find a Kondo resonance peak near EF with a Kondo temperature of about 30 K. The Kondo resonance peak splits when another K-doped PSP molecule is present in the vicinity, and the splitting gradually increases with the decreased inter-molecular distance, with no signs of phase transition. Our data demonstrate how a Kondo singlet state gradually evolves into a…
The multichannel Kondo model supports effective anyons on the partially screened impurity, as suggested by its fractional impurity entropy. It was recently demonstrated for the multi-impurity chiral Kondo model, that scattering of an electron through the impurities depends on the anyon's total fusion channel. Here we study the correlation between impurity-spins. We argue, based on a combination of conformal field theory, a perturbative limit with a large number of channels $k$, and the exactly solvable two-channel case, that the inter-impurity spin correlation probes the anyon fusion of the pa…
We study both the electrical and thermal transport near the heavy-fermion quantum critical point (QCP), identified with the breakdown of the Kondo effect as an orbital selective Mott transition. We show that the contribution to the electrical conductivity comes mainly from conduction electrons while the thermal conductivity is given by both conduction electrons and localized fermions (spinons), scattered with dynamical exponent $z = 3$. This scattering mechanism gives rise to a quasi-linear temperature dependence of the electrical and thermal resistivity. The characteristic feature of the Kond…
We study how the formation of the Kondo compensation cloud influences the dynamical properties of a magnetic impurity that tunnels between two positions in a metal. The Kondo effect dynamically generates a strong tunneling impurity-conduction electron coupling, changes the temperature dependence of the tunneling rate, and may ultimately result in the destruction of the coherent motion of the particle at zero temperature. We find an interesting two-channel Kondo fixed point as well for a vanishing overlap between the electronic states that screen the magnetic impurity. We propose a number of sy…
We present an analytic universal impurity solver for strongly correlated electrons. We extend the many-body perturbation expansion via suitable two-particle renormalizations from the Fermi-liquid regime to the critical region of the metal-insulator transition. The reliability of the approximation in the strong-coupling limit is demonstrated by reproducing the Kondo scale in the single-impurity Anderson model. We disclose the origin of the Kondo resonance in terms of Feynman diagrams and find criteria for the existence of the proper Kondo asymptotic behavior in approximate theories.
We analyze the consequences of a first-order (in the effective Kondo coupling) inter-impurity correlation present in the Two-Impurity Anderson Hamiltonian. This correlation generates a pair of Kondo Doublet states which are responsible for the first (higher energy) Kondo stage of these systems. Applied to systems of quantum dots in a semiconductor heterolayer, these Doublets predict a mainly ferromagnetic dot-dot spin correlation which is different than the expected from the second-order RKKY interaction. In order to effectively screen both dots, a parallel alignment of the dot-spins is favore…
We consider the non-Fermi liquid quantum critical state of the spin-S two-impurity Kondo model, and its potential realization in a quantum dot device. Using conformal field theory (CFT) and the numerical renormalization group (NRG), we show the critical point to be identical to that of the two-channel Kondo model with additional potential scattering, for any spin-S. Distinct conductance signatures are shown to arise as a function of device asymmetry; with the `smoking gun' square-root behavior, commonly believed to arise at low-energies, dominant only in certain regimes.
The Kondo breakdown scenario has been claimed to allow the $T$-linear resistivity in the vicinity of the Kondo breakdown quantum critical point, two cornerstones of which are the dynamical exponent $z = 3$ quantum criticality for hybridization fluctuations in three dimensions and irrelevance of vertex corrections for transport due to the presence of localized electrons. We revisit the issue of vertex corrections in electrical transport coefficients. Assuming that two kinds of bosonic degrees of freedom, hybridization excitations and gauge fluctuations, are in equilibrium, we derive coupled qua…
We study the real-time dynamics of the Kondo effect after a quantum quench in which a magnetic impurity is coupled to two metallic Hubbard chains. Using an effective field theory approach, we find that for noninteracting electrons the charge current across the impurity is given by a scaling function that involves the Kondo time. In the interacting case, we show that the Kondo time decreases with the strength of the repulsive interaction and the time dependence of the current reveals signatures of the Kondo effect in a Luttinger liquid. In addition, we verify that the relaxation of the impurity…
We study the finite-frequency inter-band transition peak in the optical conductivity of a heavy fermion system close to a Kondo breakdown quantum critical point, where the lattice Kondo temperature vanishes. As the system approaches the phase transition from the heavy Fermi liquid side, we find a new cross-over regime where the peak position is related to, but is not directly proportional to, the lattice Kondo scale. In particular, the position of the peak moves to lower energies, but remains finite at the critical point. On the other hand, the peak value changes non-monotonically and eventual…
The method of continuous unitary transformations (CUTs) is applied to the Anderson impurity and the Kondo model aiming at the systematic derivation of convergent effective models. If CUTs are applied in a conventional way, diverging differential equations occur. Similar to poor man's scaling the energy scale, below which the couplings diverge, corresponds to the Kondo temperature $T_K$. We present a way to apply CUTs to the Kondo and to the Anderson impurity model so that no divergences occur but a converged effective low-energy model is derived with small fnite parameters at arbitrarily small…
We show within the slave boson technique, that the Anderson lattice model exhibits a Kondo breakdown quantum critical point (KB-QCP) where the hybridization goes to zero at zero temper- ature. At this fixed point, the f-electrons experience as well a selective Mott transition separating a local-moment phase from a Kondo-screened phase. The presence of a multi-scale QCP in the An- derson lattice in the absence of magnetism is discussed in the context of heavy fermion compounds. This study is the first evidence for a selective Mott transition in the Anderson lattice.
Motivated by recent Hall-effect experiment in YbRh$_2$Si$_2$, we study ground state properties of a Kondo lattice model in a two-dimensional square lattice using variational Monte Carlo method. We show that there are two types of phase transition, antiferromagnetic transition and topological one (Fermi surface reconstruction). In a wide region of parameters, these two transitions occur simultaneously without the breakdown of Kondo screening, accompanied by a discontinuous change of the Hall coefficient. This result is consistent with the experiment and gives a novel theoretical picture for the…
We consider Dirac electrons on the honeycomb lattice Kondo coupled to spin-1/2 degrees of freedom on the kagome lattice. The interactions between the spins are chosen along the lines of the Balents-Fisher-Girvin model that is known to host a $\mathbb{Z}_2$ spin liquid and a ferromagnetic phase. The model is amenable to sign free auxiliary field quantum Monte Carlo simulations. While in the ferromagnetic phase the Dirac electrons acquire a gap, they remain massless in the $\mathbb{Z}_2$ spin liquid phase due to the breakdown of Kondo screening. Since our model has an odd number of spins per uni…
We reobtain the Kondo-Doublet interaction by means of the Brillouin-Wigner Perturbation theory. By applying the same method to the single impurity case we show that the Kondo-Doublet interaction is a direct consequence of the Kondo screening cloud. We fully confirm our previous results on this novel interaction. In particular, we found that for values of the system parameters typical of semiconductor Quantum Dots heterostructures, the ferromagnetic Kondo-Doublet interaction dominates over the RKKY one.
Probing Collective and Individual Kondo Screening: Multi-Stage, Multi-Channel Kondo Effects in a $C_3$-Symmetric Four-Impurity Model
Chongqing University（重庆大学）；Hangzhou Normal University（杭州师范大学）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
It is believed that the successive antiferromagnetic scattering of the conduction electrons on two magnetic impurities in a metal, induces a magnetic interaction between the impurities which sets an energy scale for the system, in addition to the Kondo temperature. However, it is shown here that this contradicts a scaling law for the two-impurity Anderson model in the magnetic limit which becomes exact for a large bandwidth.
One of the challenges in strongly correlated electron systems, is to understand the anomalous electronic behavior that develops at an antiferromagnetic quantum critical point (QCP), a phenomenon that has been extensively studied in heavy fermion materials. Current theories have focused on the critical spin fluctuations and associated break-down of the Kondo effect. Here we argue that the abrupt change in Fermi surface volume that accompanies heavy fermion criticality leads to critical charge fluctuations. Using a model one dimensional Kondo lattice in which each moment is connected to a separa…
We propose a model of a strongly-interacting two-impurity Kondo system based on the AdS/CFT correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature $T$. The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the RKKY interaction. For spin-$1/2$ impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one a…
We study the system of two magnetic impurities described by a two-impurity Kondo model where only the first impurity couples directly to the conduction band, while the second impurity interacts with the first through Heisenberg exchange coupling in a "side-coupled" configuration. We consider various choices of the impurity spins (S_1<S_2, S_1=S_2, and S_1>S_2) and we contrast the regimes where the inter-impurity exchange coupling J is either lower or higher than the Kondo temperature T_K^(0) of the first impurity in the absence of the second. This model is a high-spin generalization of the two…
We consider the spatial spin correlations around a partially screened spin-1 magnetic moment in a metal exhibiting the underscreened Kondo effect. We find that the underscreening of the impurity spin results in spatial spin correlations that are more pronounced as compared to the fully screened Kondo effect; their power-law decay is weaker because of characteristic logarithmic corrections at large distances. The spin correlator also changes sign as a function of distance to the impurity allowing for ferromagnetic correlations between conduction electron spin density and the local moment. The n…
A generalized Anderson model for a magnetic ion in a harmonic potential is formulated. The model is investigated by the numerical renormalization group(NRG) method. In addition to the conventional s-wave screening, the model exhibits phonon assisted p-wave Kondo effect as well as Yu and Anderson type Kondo effect. It is shown that the s-wave Kondo and the Yu-Anderson Kondo belong to the same fixed point. At the boundary between the s-wave and p-wave Kondo regions line of fixed points of the two channel Kondo effect is identified.
A Bethe Ansatz solution of a (modified) Hubbard chain with a Kondo impurity of arbitrary spin S at a highly symmetric line of parameter space is proposed and explored. Our results confirm the existence of a strong-coupling (line of) fixed-point(s) with ferromagnetic Kondo coupling as first hypothetized by Furusaki and Nagaosa on the basis of perturbative renormalization group calculations. For on-site Hubbard repulsion and ferromagnetic Kondo exchange, the ground state has spin S-1/2, i.e., is a singlet when S=1/2. The contributions of the impurity to the magnetic susceptibility and low-temper…
In his renormalization paper of the Kondo effect Wilson replaced the full band of s-electrons by a small number of ''Wilson states''. He started from a rather artificial symmetric band with constant density of states and constant interaction with the impurity. It is shown in the present paper that with a minor modification the Wilson states are optimally suited to treat the interaction of an impurity with an arbitrary s-band. Each Wilson state represents electrons of a whole energy range. It carries the interaction of all these electrons with the impurity. All the other electron states in this…
The Kondo resonance at the Fermi level is well-established for the electronic structure of Ce (f1 electron) and Yb (f1 hole) based systems. In this work, we report complementary experimental and theoretical studies on the Kondo resonance in Pr-based f2 system, PrTi2Al20. Using Pr 3d-4f resonant photoemission spectroscopy and single impurity Anderson model (SIAM) calculations including the full multiplets of Pr ions, we show that an f2 system can also give rise to a Kondo resonance at the Fermi level. The Kondo resonance peak is experimentally observed through a final-state-multiplet dependent …
The behavior of two magnetic impurities coupled to correlated electrons in one dimension is studied using the DMRG technique for several fillings. On-site Coulomb interactions among the electrons lead to a small Kondo screening cloud and an overall suppression of magnetic order. For arbitrary electronic correlations and large inter-impurity distances R, we find a 1/R^2 decay of magnetic correlations.
A magnetic impurity in the spin-1/2 chain is a simple realization of the two-channel Kondo problem since the field theoretical descriptions in the spin-sector are identical. The correlation functions near the impurity can be calculated. Using a modified version of the numerical transfer matrix DMRG, we are able to accurately determine local properties close to the impurity in the thermodynamic limit. The local susceptibilities (Knight-shifts) show an interesting behavior in a large range around the impurities. We are able to make quantitative experimental predictions which would allow to obser…
Accurate numerical results are derived for transport properties of Kondo impurity systems with potential scattering and orbital degeneracy. Using the continuous-time quantum Monte Carlo (CT-QMC) method, static and dynamic physical quantities are derived in a wide temperature range across the Kondo temperature T_K. With strong potential scattering, the resistivity tends to decrease with decreasing temperature, in contrast to the ordinary Kondo effect. Correspondingly, the quasi-particle density of states obtains the antiresonance around the Fermi level. Thermopower also shows characteristic dev…
We use two different methods, the Hirsch-Fye quantum Monte Carlo simulation and the slave-boson mean field analysis to estimate the effect of spin-orbit coupling on the Kondo temperature in a quantum wire. In quantum Monte Carlo simulation, we calculate the product of spin susceptibility and temperature for different spin-orbit couplings and impurity energies. The variation of the Kondo temperature is estimated via the low temperature universal curves. In the mean field analysis, the Kondo temperature is estimated as the condensation temperature of slave-boson. Both the two methods demonstrate…
A recently developed continuous time solver based on an expansion in hybridization about an exactly solved local limit is reformulated in a manner appropriate for general classes of quantum impurity models including spin exchange and pair hopping terms. The utility of the approach is demonstrated via applications to the dynamical mean field theory of the Kondo lattice and two-orbital models. The algorithm can handle low temperatures and strong couplings without encountering a sign problem.
In a Kondo lattice, the spin exchange coupling between a local spin and the conduction electrons acquires nonlocal contributions due to conduction electron scattering from surrounding local spins and the subsequent RKKY interaction. It leads to a hitherto unrecognized interference of Kondo screening and the RKKY interaction beyond the Doniach scenario. We develop a renormalization group theory for the RKKY-modified Kondo vertex. The Kondo temperature, $T_K(y)$, is suppressed in a universal way, controlled by the antiferromagnetic RKKY coupling parameter $y$. Complete spin screening ceases to e…
We show that the inclusion of nonlocal correlation effects in a variational wave function for the ground state of a topological Anderson lattice Hamiltonian is capable of describing both topologically trivial insulating phases and nontrivial ones characterized by an indirect gap, as well as its closure at the transition into a metallic phase. The method, though applied to an oversimplified model, thus captures the metallic and insulating states that are indeed observed in a variety of Kondo semiconductors, while accounting for topologically nontrivial band structures.
We study a spin-1/2 Kondo impurity coupled to an unconventional host in which the density of band states vanishes either precisely at (``gapless'' systems) or on some interval around the Fermi level (``gapped''systems). Despite an essentially nonlinear band dispersion, the system is proven to exhibit hidden integrability and is diagonalized exactly by the Bethe ansatz.
Kondo breakdown induced by non-Hermitian complex hybridization
Research Institute for Innovation and Co-Creation, Osaka Metropolitan University（大阪公立大学创新与共创研究所）；Department of Physics, Osaka Metropolitan University（大阪公立大学物理系）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南部阳一郎理论实验物理研究所）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Department of Physics, University of Tokyo（东京大学物理系）；Fundamental Quantum Science Program, TRIP Headquarters, RIKEN（理化学研究所TRIP总部基础量子科学项目）；Department of Physics, Ritsumeikan University（立命馆大学物理系）；Department of Materials Engineering Science, University of Osaka（大阪大学材料工程科学系）
We have carried out a generalized Schrieffer-Wolff transformation of an Anderson two-impurity Hamiltonian to study the low-energy spin interactions of the system. The second-order expansion yields the standard Kondo Hamiltonian for two impurities with additional scattering terms. At fouth-order, we get the well-known RKKY interaction. In addition, we also find an antiferromagnetic superexchange coupling and a correlated Kondo coupling between the two impurities.
We consider the Kondo effect arising from a hydrogen impurity in graphene. As a first approximation, the strong covalent bond to a carbon atom removes that carbon atom without breaking the $C_{3}$ rotation symmetry, and we only retain the Hubbard interaction on the three nearest neighbors of the removed carbon atom which then behave as magnetic impurities. These three impurity spins are coupled to three conduction channels with definite helicity, two of which support a diverging local density of states (LDOS) $\propto 1/ [ | ω\ | \ln ^{2}( Λ/| ω\ | \ ) \ ] $ near the Dirac point $ω\rightarrow …
We study the Kondo screening of a magnetic impurity in a two-dimensional superconductor with Rashba spin-orbit coupling (SOC). It is found that the Rashba interaction generates a novel Kondo screening channel, in which the local moment is screened by the exchange coupling with conduction electrons in different spin and orbital states. The Kondo temperature associated with this process is determined by the interplay between the Rashba SOC and superconducting energy gap. As a result, the quantum phase transition between the magnetic doublet and Kondo singlet ground states is significantly affect…
An one dimensional model of a Kondo impurity with arbitrary spin in the presence of electronic correlations is solved exactly. The Bethe ansatz equations are obtained for the case of quadratic dispersion. The ground state is studied in the thermodynamic limit, and the magnetic susceptibility at zero temperature is calculated.
