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Physical review. B, Condensed matter

Renormalization of the anisotropic linearXYmodel

A. Drzewiński, Raf Dekeyser

Published Jun 1, 1995• 11 citations
Publisher

Abstract

The renormalization scheme recently proposed by White is applied to the d=1 anisotropic XY model in a transverse field (AXY). It is found that this scheme offers a distinct improvement over standard techniques as far as the computation of the ground state is concerned. However, compared to the Ising model in a transverse field, on account of more complicated symmetries the AXY demands more precautions during the construction of a renormalization-group transformation. The new method predicts definitely better the location of the phase transition in the XY-like region than in the Ising-like region, but only in the Ising-like region is there any progress for the critical exponent \ensuremath{\alpha}.

DOI
10.1103/physrevb.51.15218

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The renormalization scheme recently proposed by White is applied to the d=1 anisotropic XY model in a transverse field (AXY). It is found that this scheme offers a distinct improvement over standard techniques as far as the computation of the ground state is concerned. However, compared to the Ising model in a transverse field, on account of more complicated symmetries the AXY demands more precautions during the construction of a renormalization-group transformation. The new method predicts definitely better the location of the phase transition in the XY-like region than in the Ising-like region, but only in the Ising-like region is there any progress for the critical exponent \ensuremath{\alpha}.
The renormalization scheme recently proposed by White is applied to the d=1 anisotropic XY model in a transverse field (AXY). A flow diagram, critical exponents and energies have been calculated. It is found that this scheme is a distinct improvement over the standard technique as far as the computation of the ground state is concerned. The accuracy increases rapidly, when we keep more states in each renormalization step, but the errors in the ground state energy are always the largest in the neighborhood of the phase transitions. Comparing with the Ising model in a transverse field, on account of more complicated symmetries, the AXY demands more precautions during constructing a renormalization group transformation.
We derive several closed-form expressions for the fidelity susceptibility (FS) of the anisotropic XY model in the transverse field. The basic idea lies in a partial fraction expansion of the expression so that all the terms are related to a simple fraction or its derivative. The critical points of the model are reiterated by the FS, demonstrating its validity for characterizing the phase transitions. Moreover, the critical exponents ν associated with the correlation length in both critical regions are successfully extracted by the standard finite-size scaling analysis.
We have adjusted the density-matrix renormalization method to handle two-dimensional systems of limited width. The key ingredient for this extension is the incorporation of symmetries in the method. The advantage of our approach is that we can force certain symmetry properties to the resulting ground-state wave function. Combining the results obtained for system sizes up to $30\ifmmode\times\else\texttimes\fi{}6$ and finite-size scaling, we derive the phase-transition point and the critical exponent for the gap in the Ising model in a transverse field on a two-dimensional square lattice.
A zero-temperature real-space renormalization-group method is presented and applied to the quantum Ising model with a transverse field in one dimension. The transition between the low-field and high-field regimes is studied. Magnetization components, spin correlation functions, and critical exponents are derived and checked against the exact results. It is shown that increasing the size of the blocks in the iterative procedure yields more accurate results, especially for the critical "magnetic" exponents near the transition.
The nature of phase transitions involving the questions why and how phase transitions take place has not been sufficiently touched in the literature. In contrast, the current attention to certain extent still focus on the description of critical phenomena and the classification of the associated phase transition along with the Ginzburg-Landau-Wilson paradigm, where the key issue is to identify phenomenologically order parameters and related symmetries. This brings the question to topological phase transitions (TPTs), where no obvious order parameter and the broken symmetry are identified. Here we present a unified pattern description of the second-order quantum phase transition (QPT) and TPT, both involved in the one-dimensional anisotropic quantum XY model in a transverse field, which contains the transverse Ising model (TIM) as a limit case. Away from the TIM, the XY model enters the ferromagnetic phase (marked by a second-order QPT or a direct TPT) as increasing ferromagentic exchange coupling, a series of TPTs occur, which are absent in the TIM. The TPTs behave like the first-order QPTs. In the isotropic and large exchange coupling cases, the ground state of the XY model is dominated by two topologically different vortices along positive and negative direction of the transverse field. We confirm the above conclusion by analyzing the energy contributions of the patterns to the ground state and calculating the ground state pattern occupations of the XY model. The results have been obtained in a unified and self-evident way and answer the questions why and how the QPT and TPTs take place in the XY model.