The recent discovery of ferromagnetic single-layer CrI creates ample opportunities for studying the fundamental properties and the spintronic applications of atomically thin magnets. Through first-principles calculations and model Hamiltonian simulations, here we build for the first time a substantial magnetic phase diagram under lateral strain and charge doping, the two factors that are easily modulated in single-layer CrIvia substrate and gating controls. We demonstrate that both lateral strain and charge doping efficiently change the coupling between the local spins and thus have unexpected effects on the magnetic properties of CrI. In particular, the strain tunes the magnetic order and anisotropy: a compressive strain leads to a phase transition from a ferromagnetic insulator to an antiferromagnetic insulator, while a tensile strain can flip the magnetic orientation from off-plane to in-plane. Furthermore, we find that the phase transition under compressive strain is insensitive to charge doping, whereas the phase transition under tensile strain is modulated by electron doping significantly. Our predicted magnetic phase diagram and rationalized analysis indicate the single-layer CrI to be an ideal system to harness both basic magnetic physics and building blocks for magnetoelastic applications.
We have investigated the physical effects of the Dzyaloshinskii-Moriya (DM) interaction in copper benzoate. In the low field limit, the spin gap is found to vary as H 2/3 ln 1/6 (J/µBHs) (Hs: an effective staggered field induced by the external field H) in agreement with the prediction of conformal field theory, while the staggered magnetization varies as H 1/3 and the ln 1/3 (J/µBHs) correction predicted by conformal field theory is not confirmed. The linear scaling behavior between the momentum shift and the magnetization is broken. We have determined the coupling constant of the DM interaction and have given a complete quantitative account for the field dependence of the spin gaps along all three principal axes, without resorting to additional interactions like interchain coupling. A crossover to strong applied field behavior is predicted for further experimental verification. . In these materials, the Dzyaloshinskii-Moriya (DM) interaction [7,8,9] plays an important role, especially in an applied magnetic field. This has stimulated extensive investigation on the physical properties of the DM interaction. However, this interaction is rather difficult to handle analytically, which has brought much uncertainty in the interpretation of experimental data and has limited our understanding of many interesting quantum phenomena of low-dimensional magnetic materials.For copper benzoate, Dender et al [1,2] found that the spin excitation gap shows a peculiar field dependence, ∆ ∼ H 0.65 , in low fields. On the contrary, excitations remain gapless in the S=1/2 Heisenberg model below a critical field. Oshikawa and Affleck (OA) suggested that this field dependence of the gap is due to a staggered magnetic field induced by the DM interaction in addition to the staggered g-factor in a uniform field [9,10]. However, a satisfactory explanation for the field-dependence of the energy gaps in all three directions is still lacking [11,12]. It was argued that the inconsistency between the experimental data and theoretical results might be due to the neglect of the interchain coupling and/or anisotropic interaction terms in the low-field effective model used by Oshikawa and Affleck [9,11]. We believe this issue can be clarified by a thorough study of the DM interaction and a direct comparison with experiments.Copper benzoate is a quasi-1D spin-1/2 antiferromagnetic Heisenberg system. The chain direction is the caxis. It contains two types of alternating and slightly tilted CuO 8 octahedra. This leads to two inequivalent Cu ++ ions and an alternating DM coupling [13]. In an applied field, copper benzoate can be modeled by the following Hamiltonian,( 1) where the three terms in the summation are the antiferromagnetic Heisenberg, DM and Zeeman splitting interactions, respectively. The exchange coupling constant J, determined from the neutron scattering measurements, is about 1.57meV. The DM interaction is much weaker than the Heisenberg term. The D-vector, primarily aligned along the a ′′ axis, will be determined numerically. g u and ...
Resonant tunneling through quantum dot under a finite bias voltage at zero temperature is investigated by using the adaptive time-dependent density matrix renormalization group (TdDMRG) method. Quantum dot is modeled by the Anderson Hamiltonian with the one-dimensional nearest-neighbor tightbinding leads. Initially the ground state wave function is calculated with the usual DMRG method. Then the time evolution of the wave function due to the slowly changing bias voltage between the two leads is calculated by using the TdDMRG technique. Even though the system size is finite, the expectation values of current operator show steady-like behavior for a finite time interval, in which the system is expected to resemble the real nonequilibrium steady state of the infinitely long system. We show that from the time intervals one can obtain quantitatively correct results for differential conductance in a wide range of bias voltage. Finally we observe an anomalous behavior in the expectation value of the double occupation operator at the dot hn " n # i as a function of bias voltage.
We study the entanglement properties of anisotropic open spin one-half Heisenberg chains with a modified central bond. The entanglement entropy between the two half-chains is calculated with the density-matrix renormalization method (DMRG). We find a logarithmic behaviour with an effective central charge c ′ varying with the length of the system. It flows to one in the ferromagnetic region and to zero in the antiferromagnetic region of the model. In the XX case it has a non-universal limit and we recover previous results.The consideration of entanglement properties has brought a new element into the study of manyparticle quantum states. Entanglement is related to a division of the system into two parts and can be quantified via the reduced density matrix ρ and the entropy S = −tr(ρ ln ρ) connected with it. This quantity is particularly interesting for the ground state of critical one-dimensional systems. For example, if one cuts an open chain into two halves of length L, conformal invariance predicts the universal formwhere c is the central charge in the conformal classification. An analogous formula with c/6 replaced by c/3 holds for a segment of length L in an infinite chain. The logarithmic divergence is a particular signature of the criticality and can be related to analogous universal contributions to the free energy of critical two-dimensional systems with conical shape [2,3,4]. It has been verified numerically for a number of quantum chains [5,6] and derived analytically for free fermions hopping on a chain [7]. For this system it can also be proven by putting proper bounds on S [8, 9]. Since the entanglement entropy measures the mutual coupling of the two parts of a system in the wave function, a defect at their boundary should have a strong influence. In one dimension, defects show particularly interesting features in Luttinger liquids, i.e. in strongly correlated fermionic systems. Their effective strength then depends on the sign of the interaction [10,11] and goes to zero for attraction while it diverges for repulsion, as the system size increases. This has been checked in various further studies, see e.g. [12,13,14,15,16,17]. One then wonders how the entanglement behaves in such a case and how the logarithmic law (1) is affected. This question was first raised by Levine [4] who used bosonization and found results to lowest order in the impurity strength which are consistent with the general picture. The simpler case of a free-fermion hopping model, corresponding to the XX spin chain, was treated numerically in Ref. [18]. In this case, the logarithmic behaviour was found to persist, but with a prefactor c ef f which depends continuously on the defect strength.In the present paper we present a numerical study of this problem for the case of a planar XXZ spin chain, which has c = 1 and is the lattice version of a Luttinger model. We treat open chains with one modified bond in the middle and use density-matrix renormalization (DMRG) [19,20]. This method is ideally suited for such a study since the calc...
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