The surface tension of the induced magnetization in the one-dimensional isotropic ferromagnetic XY-model (s = 1/2, open chain) is considered. The magnetization domains are induced by an inhomogeneous transverse field, and the model can be exactly solved by using the Green function technique. At T = 0 the system undergoes a quantum transition in the bulk driven by the field which is associated to the induced renormalized magnetization. In this temperature limit, the surface tension is determined, and it is shown that, when properly renormalized, it follows a scaling law near the critical point, with a critical exponent μ equal to 1. A brief review is presented for the calculations of the critical exponents and it is also shown that they satisfy the known relations, provided the effective classical dimension is equal to 3. Since the dynamic exponent z is equal to 2, this result confirms the dynamic shift in the dimensionality of the system, d+z, as expected for the quantum transitions. It is also shown that, consistently with this result, the dimension d in the relation μ = (d-1)ν for classical systems should be replaced by the effective dimension d+z for quantum transitions.
A zero-temperature real-space renormalization-group method is applied to the one-dimensional quantum spin-\textonehalf{} isotropic $X\ensuremath{-}Y$ model in a transverse field. A line of unstable fixed points is found and is attributed to the oscillatory behavior of the correlation functions. Magnetization components and critical indices are in good agreement with the exact results. The behavior of the
xx correlation function is studied near the critical field.
The static and dynamic properties of the anisotropic
XY model
(s=1∕2) on the inhomogeneous periodic chain, composed of
N cells with
n different exchange interactions and magnetic moments, in a transverse field
h, are determined exactly at arbitrary temperatures. The properties are obtained by introducing the Jordan-Wigner fermionization and by reducing the problem to a diagonalization of a finite matrix of
nth order. The quantum transitions are determined exactly by analyzing, as a function of the field, the induced magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}{\ensuremath{\mu}}_{m}⟨{S}_{j,m}^{z}⟩$ (
j denotes the cell,
m the site within the cell, ${\ensuremath{\mu}}_{m}$ the magnetic moment at site
m within the cell) and the spontaneous magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}⟨{S}_{j,{m}^{\ensuremath{'}}}^{x}⟩$ which is obtained from the correlations
⟨Sj,mxSj+r,mx⟩ for large spin separations. These results, which are obtained for infinite chains, correspond to an extension of the ones obtained by Tong and Zhong [Physica B 304, 91 (2001)]. The dynamic correlations, $⟨{S}_{j,m}^{z}(t){S}_{{j}^{\ensuremath{'}},{m}^{\ensuremath{'}}}^{z}(0)⟩$, and the dynamic susceptibility, ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$, are also obtained at arbitrary temperatures. Explicit results are presented in the limit
T=0, where the critical behavior occurs, for the static susceptibility ${\ensuremath{\chi}}_{q}^{zz}(0)$ as a function of the transverse field
h, and for the frequency dependency of dynamic susceptibility ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$.
The isotropic
XY model
(s=1/2) in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of
N segments with
n different exchange interactions and magnetic moments, is exactly solved by introducing the integral Gaussian transformation and the generalized Jordan--Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of
nth order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at
T=0, for the induced magnetization and isothermal susceptibility ${\ensuremath{\chi}}_{T}^{zz}$ as a function of the transverse field.
Abstract The dynamics of the spin 1/2 one‐dimensional XY‐model in a transverse field is considered. The short‐time behaviour of the auto‐correlation function 〈S (t) S (0)〉 for the closed chain is obtained numerically by considering the exact results known for the open chain which are given in terms of Pfaffians. The bulk auto‐correlation is identified with the correlation 〈S (t) S (0)〉 for the open chain, and this identification is shown to be valid up to a characteristic time Jt ≅ 20. It is also shown, as expected, that the boundary effects are more pronounced for fields smaller than the exchange constant (h < |J|) and in the low temperature limit. The results are obtained for different temperatures and field strengths, and the numerical convergence is discussed in each case.
The anisotropic XY-model in a transverse field (s=1/2) on the one-dimensional alternating superlattice (closed chain) is considered. The solution of the model is obtained by introducing a generalized Jordan-Wigner transformation which maps the system onto a non-interacting fermion gas. The exact excitation spectrum is determined by reducing the problem to a diagonalization of a block matrix, and it is shown that it is numerically identical to the one obtained by using the approximate transfer matrix method . The induced magnetization and the susceptibility
χzz are determined as a function of the transverse field, and it is shown that, at T=0, the susceptibility presents multiple singularities. It is also shown, as expected, that this critical behaviour driven by transverse field belongs to the same universality class of the model on the alternating chain.