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Acta Physica Sinica (Overseas Edition)

Critical exponents of the one-dimensional continuum Heisenberg-Ising model

Yumei Zhang, Xu Bo-wei

Published Nov 1, 1995• 1 citation
Publisher

Abstract

The continuum generalization of the one-dimensional Heisenberg-Ising model is studied by the bosonization method. With reference to the results of the renormalization group theory, we are able to obtain the critical exponents by considering the influence of the large momentum transfer and improve Luther and Peschel's results to the next order. A comparison of the exact result of the lattice model with that of the continuum model by various treatments are also given.

DOI
10.1088/1004-423x/4/11/007

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The continuum generalization of the one-dimensional Heisenberg-Ising model is studied by the bosonization method. With reference to the results of the renormalization group theory, we are able to obtain the critical exponents by considering the influence of the large momentum transfer and improve Luther and Peschel's results to the next order. A comparison of the exact result of the lattice model with that of the continuum model by various treatments are also given.
We construct a relationship between the Baxter model in two dimensions and the Luttinger model in one, and use it to calculate critical exponents for the Baxter model from appropriate Luttinger-model correlation functions. An important part of this work involves the generalization of the Jordan-Wigner transformation to provide a representation for continuum spin operators. With this generalization, we are also able to calculate spin correlation functions for a continuum generalization of the spin-\textonehalf{} Heisenberg-Ising chain. We discuss the difference between the continuum and discrete lattice models, and with the help of a new scaling law, use previous results for the Baxter model to calculate new exponents for the Baxter and Heisenberg-Ising model on a lattice.
The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.
The critical properties of the spin-density-wave/semimetal phase transition in Dirac materials are still under debate despite all the efforts from continuum and lattice methods. Using state-of-the-art functional renormalization group methods, the author provides improved estimates of the leading critical exponents. They are in line with earlier results obtained with the same method and in disagreement with recent quantum Monte Carlo estimates. The study supports the conjecture the only adequate nonperturbative treatment of the different universality classes, related to this phase transition, is via the functional renormalization group.
The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.
Sarker's real-space renormalization-group method, applied previously to the one-dimensional dimerized XY model, is used to study the one-dimensional pure and dimerized Heisenberg models. Comparison of critical exponents (the dynamical and the gap exponents) with exact or conjectured values indicates that the method, in its original version, does not yield an improved real-space technique, at least for the models studied for which the opening of the gap leads to essential singularity or logarithmic corrections.
A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.