We explore short-distance static correlation functions in the infinite XXZ chain using previously derived formulae which represent the correlation functions in factorized form. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field in the massive regime ∆> 1, extending our previous results to the full parameter plane of the antiferromagnetic chain ( ∆> −1 and arbitrary field h). The factorized formulae are numerically efficient and allow for taking the isotropic limit ( ∆ = 1) and the Ising limit ( ∆ = ∞). At the critical field separating the fully polarized phase from the Néel phase, the Ising chain possesses exponentially many ground states. The residual entropy is lifted by quantum fluctuations for large but finite ∆ inducing unexpected crossover phenomena in the correlations. 1
We have investigated the zero-and finite-temperature behaviors of the anisotropic antiferromagnetic Heisenberg XXZ spin-1/2 chain in the presence of a transverse magnetic field (h). The attention is concentrated on an interval of magnetic field between the factorizing field (h(f)) and the critical one (h(c)). The model presents a spin-flop phase for 0 < h < h(f) with an energy scale which is defined by the long range antiferromagnetic order while it undergoes an entanglement phase transition at h = h(f). The entanglement estimators clearly show that the entanglement is lost exactly at h = h(f), which justifies different quantum correlations on both sides of the factorizing field. As a consequence of zero entanglement (at h = h(f)) the ground state is known exactly as a product of single-particle states which is the starting point for initiating a spin wave theory. The linear spin wave theory is implemented to obtain the specific heat and thermal entanglement of the model in the interested region. A double-peak structure is found in the specific heat around h = h(f), which manifests the existence of two energy scales in the system as a result of two competing orders before the critical point. These results are confirmed by the low temperature Lanczos data which we have computed.
Using the Jordan-Wigner transformation we calculate exactly the ground state and low-temperature thermodynamic properties of the spin
S=1/2 isotropic
XX chain with four spin interaction. In terms of the equivalent spinless fermion (SF) representation the system is viewed as a lattice fermion gas with nearest-neighbor (
J) and next-next-next-neighbor (
J∗/4) hopping. It is shown that with the increase of four spin coupling, at
Jc∗=4J/3 the system experiences the Lifshitz type topological phase transition characterized by the tripling of Fermi points. The quantum phase transition (QPT) point marks transition from a gapless spin-liquid phase of standard
XX chain into again a gapless spin-liquid phase with different character of power-low decay of spin correlations. At the transition point the free fermion dispersion relation shows flattering at Fermi points, what determines singular character of density of states
ρ(ω)∼(ω/J)−2/3 and as a consequence unconventional temperature dependence of heat capacity of the system
C∼(T/J∗)1/3, and singular magnetic susceptibility of the system
χ(H)∼(H/J∗)−2/3. In the case of alternating magnetic field the system is characterized by the rich ground state phase diagram which contains fully polarized (ferromagnetic), gapped antiferromagnetic (AFM) and spin liquid phases. At the transition point from the gapped AFM phase into the gapless polarized spin liquid phase the system shows rapid increase of magnetization
m∼(H−Hc)1/6 and magnetic susceptibility a singular behavior as
χ(H)∼(H−Hc)−5/6.
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.
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
The spin-1/2 Ising-Heisenberg branched chain composed of regularly alternating Ising spins and Heisenberg dimers involving an additional side branching is rigorously solved in a magnetic field by the transfer-matrix approach. The ground-state phase diagram, the magnetization process and the concurrence measuring a degree of bipartite entanglement within the Heisenberg dimers are examined in detail. Three different ground states were found depending on a mutual interplay between the magnetic field and two different coupling constants: the modulated quantum antiferromagnetic phase, the quantum ferrimagnetic phase, and the classical ferromagnetic phase. Two former quantum ground states are manifested in zero-temperature magnetization curves as intermediate plateaus at zero and one-half of the saturation magnetization, whereas the one-half plateau disappears at a triple point induced by a strong-enough ferromagnetic Ising coupling. The ground-state phase diagram and zero-temperature magnetization curves of the analogous spin-1/2 Heisenberg branched chain were investigated using density-matrix renormalization group calculations. The latter fully quantum Heisenberg model involves, besides two gapful phases manifested as zero and one-half magnetization plateaus, gapless quantum spin-liquid phase. The intermediate one-half plateau of the spin-1/2 Heisenberg branched chain vanishes at Kosterlitz-Thouless quantum critical point between gapful and gapless quantum ground states unlike the triple point of the spin-1/2 Ising-Heisenberg branched chain.