Finite Heisenberg Quantum Spin Chain

2019 ◽  
pp. 667-686
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Conformal Symmetry ◽  
Finite Size ◽  
Quantum Spin ◽  
Ansatz Method ◽  
Finite Sums ◽  
Mathematical Techniques ◽  
Higher Order Corrections ◽  
Finite Size Corrections ◽  
Maclaurin Formula

The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.

Mathematical Tools

2019 ◽  
pp. 657-666
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Finite Size ◽  
Summation Formula ◽  
Bernoulli Numbers ◽  
Quantum Spin ◽  
Mathematical Technique ◽  
Finite Sums ◽  
Mathematical Techniques ◽  
Linear Integral ◽  
Linear Integral Equations

Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the Bernoulli numbers are needed as a prerequisite for the Euler– Maclaurin summation formula, which allows to transform finite sums into integrals plus, in a systematic way, corrections taking into account the finite size of the system. Applying this mathematical technique to the Bethe ansatz equations results in linear integral equations of the Wiener–Hopf type for the solution of which an elaborate mathematical technique exists, the Wiener–Hopf technique.

scholarly journals A distribution approach to finite-size corrections in Bethe Ansatz solvable models

2018 ◽  
Vol 934 ◽  
pp. 96-117 ◽  
Author(s):  
Etienne Granet ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Bethe Ansatz ◽  
Finite Size ◽  
Solvable Models ◽  
Finite Size Corrections

scholarly journals ASYMMETRY AND SPIN-ORBIT EFFECTS IN BINDING ENERGY IN THE EFFECTIVE NUCLEAR SURFACE APPROXIMATION

2009 ◽  
Vol 18 (04) ◽  
pp. 885-891 ◽  
Author(s):  
A. G. MAGNER ◽  
A. I. SANZHUR ◽  
A. M. GZHEBINSKY
Keyword(s):  
Binding Energy ◽  
Finite Size ◽  
Nuclear Surface ◽  
Spin Orbit ◽  
Surface Binding ◽  
Collective Variables ◽  
Higher Order Corrections ◽  
Analytical Expressions ◽  
Finite Size Corrections ◽  
Local Energy Density

Isoscalar and isovector particle densities are derived analytically by using the approximation of a sharp edged nucleus within the local energy density approach with the proton-neutron asymmetry and spin-orbit effects. Equations for the effective nuclear-surface shapes as collective variables are derived up to the higher order corrections in the form of the macroscopic boundary conditions. The analytical expressions for the isoscalar and isovector tension coefficients of the nuclear surface binding energy and the finite-size corrections to the β stability line are obtained.

EXACTLY SOLVABLE MODELS AND FINITE SIZE CORRECTIONS

1992 ◽  
Vol 06 (08) ◽  
pp. 1119-1180 ◽  
Author(s):  
JUNJI SUZUKI ◽  
TARO NAGAO ◽  
MIKI WADATI
Keyword(s):  
Finite Size ◽  
Spin Model ◽  
Quantum Spin ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
Soliton Theory ◽  
Exactly Solvable ◽  
Recent Developments ◽  
Quantum Spin Models ◽  
Finite Size Corrections

Recent developments in the theory of exactly solvable models are reviewed. Particular attention is paid to the finite size corrections to the Bethe ansatz equations. Baxter’s formula which relates a 2-dimensional statistical model with a 1-dimensional spin model is extended into the finite temperature case. A combination of this extension and the theory of finite size corrections gives a systematic method to evaluate low temperature expansions of physical quantities. Applications of the method to 1-dimensional quantum spin models are discussed. Throughout this paper, the usefulness of the soliton theory should be observed.

FINITE SIZE CORRECTIONS FOR MULTIPARTICLE SPIN-½ NONLINEAR SCHRÖDINGER MODEL

1991 ◽  
Vol 05 (23) ◽  
pp. 1603-1606
Author(s):  
YI-MIN LIU ◽  
FU-CHO PU ◽  
HANG SU
Keyword(s):  
Critical Point ◽  
Ground State ◽  
Finite Size ◽  
Conformal Anomaly ◽  
Nonlinear Schrödinger ◽  
Size Correction ◽  
Finite Size Correction ◽  
Finite Size Corrections ◽  
Maclaurin Formula

Applied Euler-Maclaurin formula, we compute the finite size correction to the energy of the ground state for the Spin-½ Nonlinear Schrödinger model. We get conformal anomaly for this model at the critical point (T=0).

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