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Journal of Physics A Mathematical and General

Finite-size corrections in the XY model with a uniform magnetic field and a boundary field

Hitoshi Asakawa, Masuo Suzuki

Published Sep 7, 1995• 9 citations
Publisher

Abstract

The one-dimensional XY model with a uniform magnetic field and a boundary field is introduced. The present model is solved analytically at its critical point. Using the present analytical result, the finite-size corrections of the ground-state energy and the low-lying excited energies are evaluated. The partition function of the present model is also evaluated in the scaling limit. Through this calculation, the central charge and the conformal weights of the primary fields in the present model are obtained analytically.

DOI
10.1088/0305-4470/28/17/005

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The one-dimensional XY model with a uniform magnetic field and a boundary field is introduced. The present model is solved analytically at its critical point. Using the present analytical result, the finite-size corrections of the ground-state energy and the low-lying excited energies are evaluated. The partition function of the present model is also evaluated in the scaling limit. Through this calculation, the central charge and the conformal weights of the primary fields in the present model are obtained analytically.
In this chapter we want to refine the analysis of the ground state and low-lying excitations of the Hubbard model in the phases with gapless modes, i.e. phases II, IV and V discussed in Chapters 6, 7, by taking account of corrections which are important when considering Hubbard chains of finite length L . For the generic case, i.e. away from half-filling in a magnetic field, the finite-size corrections to the spectrum of the Hubbard model have been calculated by F. Woynarovich [487]. These results are the basis for our discussion in the following Chapter 9 of the asymptotic behaviour of correlation functions within the conformal approach [6, 51, 62, 75] and thereby will allow us to make contact with Haldane's Luttinger liquid approach for the description of one-dimensional strongly correlated electron systems [189–192] Generic case – the repulsive Hubbard model in a magnetic field To investigate how the thermodynamic limit is approached we have to take into account finite-size corrections in our previous derivation of integral equations from Takahashi's equations. This analysis has to be performed separately for each of the phases with gapless excitations identified before. From a technical point of view the most complex situation is found in phase IV – the partially filled, partially magnetized band with two massless modes. The finite-size scaling behaviour in the phases with a single gapless mode can be studied using the same techniques and we will point out the differences to the ‘generic’ case studied in this section later in this chapter.
The zero-temperature, classical XYXYXY model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice is studied by exploring the distribution ${\mathrm{\ensuremath{\Phi}}}_{L}(y)$ of its centered and normalized magnetization yyy in the large-LLL limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of ${\mathrm{\ensuremath{\Phi}}}_{L}(y)$, and the limit distribution ${\mathrm{\ensuremath{\Phi}}}_{L\ensuremath{\rightarrow}\ensuremath{\infty}}(y)={\mathrm{\ensuremath{\Phi}}}_{0}(y)$ is obtained with high precision. The two leading finite-size corrections ${\mathrm{\ensuremath{\Phi}}}_{L}(y)\ensuremath{-}{\mathrm{\ensuremath{\Phi}}}_{0}(y)\ensuremath{\approx}{a}_{1}(L)\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{1}(y)+{a}_{2}(L)\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{2}(y)$ are also extracted both from numerics and from analytic calculations. We find that the amplitude a1(L){a}_{1}(L)a1​(L) scales as ln(L/L0)/L2ln(L/{L}_{0})/{L}^{2}ln(L/L0​)/L2 and the shape correction function ${\mathrm{\ensuremath{\Phi}}}_{1}(y)$ can be expressed through the low-order derivatives of the limit distribution, ${\mathrm{\ensuremath{\Phi}}}_{1}(y)={[\phantom{\rule{0.16em}{0ex}}y\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{0}(y)+{\mathrm{\ensuremath{\Phi}}}_{0}^{\ensuremath{'}}(y)\phantom{\rule{0.16em}{0ex}}]}^{\ensuremath{'}}$. Thus, ${\mathrm{\ensuremath{\Phi}}}_{1}(y)$ carries the same universal features as the limit distribution and can be used for consistency checks of universality claims based on finite-size systems. The second finite-size correction has an amplitude ${a}_{2}(L)\ensuremath{\propto}1/{L}^{2}$ and one finds that ${a}_{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{2}(y)\ensuremath{\ll}{a}_{1}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Phi}}}_{1}(y)$ already for small system size (L>10L>10L>10). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the XYXYXY model at low temperatures, including T=0T=0T=0.