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Modern Physics Letters B

Surface Tension of the Induced Magnetization in the 1D Isotropic XY-Model

J.P. de Lima, L.L. Gonçalves

Published Jan 10, 1997• 2 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984997000037

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•Jan 1, 1997•8 citations
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.
The one-dimensional (s = 1/2, open chain) isotropic ferromagnetic XY model in an inhomogeneous transverse field is considered. The exact profile of the induced magnetization is obtained by using the Green function technique. Results are presented for the magnetization profile at different finite temperatures and various strengths of the field. In the scaling region, from the analytical results obtained for the profile of the interface at T = 0, and by a suitable renormalization of the magnetization, it is shown that near the critical point it satisfies a scaling relation identical to the one obtained by van der Waals theory for fluids, provided the correlation length is adequately defined.
The isotropic XYXYXY model (s=1/2)(s=1/2)(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 NNN segments with nnn 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 nthn\text{th}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=0T=0T=0, for the induced magnetization and isothermal susceptibility ${\ensuremath{\chi}}_{T}^{zz}$ as a function of the transverse field.
The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents zzz and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.
We present some exact results for the effect of disorder on the critical properties of an anisotropic XY\mathrm{XY}XY spin chain in a transverse field. The continuum limit of the corresponding fermion model is taken and in various cases results in a Dirac equation with a random mass. Exact analytic techniques can then be used to evaluate the density of states and the localization length. In the presence of disorder the ferromagnetic-paramagnetic or Ising transition of the model is in the same universality class as the random transverse field Ising model solved by Fisher using a real-space renormalization-group decimation technique (RSRGDT). If there is only randomness in the anisotropy of the magnetic exchange then the anisotropy transition (from a ferromagnet in the x direction to a ferromagnet in the y direction) is also in this universality class. However, if there is randomness in the isotropic part of the exchange or in the transverse field then in a nonzero transverse field the anisotropy transition is destroyed by the disorder. We show that in the Griffiths' phase near the Ising transition that the ground-state energy has an essential singularity. The results obtained for the dynamical critical exponent, typical correlation length, and for the temperature dependence of the specific heat near the Ising transition agree with the results of the RSRGDT and numerical work.
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\mathrm{xx}xx correlation function is studied near the critical field.
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems (L ≤ 128). The spin-spin autocorrelation function is found to decay proportional to (log t) −2xm and (log t) −2xs m in the bulk and on the surface, respectively, with xm and x s m the bulk and surface magnetization exponents, respectively. On the other hand the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents ηe ≈ 2.2 in the bulk and η s e ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory. 05.50.+q, 64.60.Ak, 68.35.Rh Typeset using REVTEX 1 The asymptotic behavior of the time-dependent correlation functions for interacting many-body systems turned out a very difficult subject of theoretical research. Exact results in this field are scarce, one can mention the one-dimensional spin 1/2 XY-model [1] and the Ising chain in a transverse-field [2]. Both models can be mapped onto a system of non-interacting fermions, where the equal-position correlation functions are calculated by the Pfaffian method utilizing the theory of Töplitz determinants. In this Letter we consider- at the first time- the critical dynamical correlations of an interacting quantum system in the presence of quenched (i.e. time-dependent) disorder. It has recently become clear that quenched disorder has rather different effects on phase transitions in quantum systems [3] than on those thermally driven phase transitions. For example, in the Griffiths phase, which is situated at the disordered side of the critical point, the susceptibility has an essential singularity in classical systems, whereas in a quantum system the corresponding singularity is stronger, it is in a power law form. Here we consider the prototype of random quantum systems the one-dimensional random transverse-field Ising model defined by the Hamiltonian: