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Physical review. B, Condensed matter

One-dimensional transverse-field Ising model in a complex longitudinal field from a real-space renormalization-group method at T=0

Katarina Uzelac, Rémi Jullien, Pierre M. Pfeuty

Published Jul 1, 1980• 15 citations
Publisher

Abstract

The ground state of the quantum one-dimensional transverse-field Ising model in a longitudinal field is studied using a real-space renormalization-group method. The longitudinal magnetization in the ground state is calculated. Considering a real longitudinal field, this study provides an indirect investigation of the equation of state of the equivalent classical two-dimensional Ising model. Considering then the case of a purely imaginary longitudinal field, a new critical behavior is found which corresponds to the Yang-Lee edge singularity of the classical equivalent. In this last case the details of calculations and results are given, published earlier in a letter form.

DOI
10.1103/physrevb.22.436

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The ground state of the quantum one-dimensional transverse-field Ising model in a longitudinal field is studied using a real-space renormalization-group method. The longitudinal magnetization in the ground state is calculated. Considering a real longitudinal field, this study provides an indirect investigation of the equation of state of the equivalent classical two-dimensional Ising model. Considering then the case of a purely imaginary longitudinal field, a new critical behavior is found which corresponds to the Yang-Lee edge singularity of the classical equivalent. In this last case the details of calculations and results are given, published earlier in a letter form.
A T=0 real-space renormalization-group method is presented and applied to the quantum 1D transverse field Ising model in the presence of a longitudinal field. This is an indirect way to study the equation of state of the equivalent 2D classical Ising model at finite temperature. When the longitudinal field is purely imaginary we obtain a new critical behavior which is related to the Yang-Lee edge singularity of the corresponding classical model. We extract the edge exponent σ in agreement with previous series expansion estimates.
The quantum one-dimensional transverse Ising model with an imaginary longitudinal field is studied at T=0T=0T=0 using a real-space renormalization-group method. Evaluation is made of the real part of the ground-state longitudinal magnetization which is related to the Yang-Lee distribution of zeros of the partition function for the two-dimensional classical Ising model. The edge exponents are found as $\ensuremath{\sigma}=\ensuremath{-}0.25\ifmmode\pm\else\textpm\fi{}0.05$ and $\ensuremath{\eta}=\ensuremath{-}1.0\ifmmode\pm\else\textpm\fi{}0.02$, in agreement with series expansion estimates.
A quantum renormalization-group technique is applied to the one-dimensional transverse field Ising model in a complex longitudinal field. The Yang-Lee edge singularity is analysed in this model in view of its analogy with the 2d classical Ising model. Previous two-level calculations are improved by retaining four levels in the blocking procedure. A more precise edge exponent σ is found in agreement with the infinite block extrapolation of the two-level calculation, but still larger in absolute value than the latest series analysis results. The occurrence of oscillations in the real part of the magnetization appears to be an artifact of the real-space method, which introduces finite size effects.
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.
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.