Heisenberg spin chain at the continuum limit and corresponding field theory

The Heisenberg Spin Chain system, in the continuum limit, can be represented by the non-linear Schr\odinger equation through the Hashimoto map. Inhomogeneity induced through localizing the nearest neighbor interaction strength can also be mapped similarly to an integro -differential generalization of the non-linear Schrodinger system [J. Phys. C: Solid State Phys. 15, L1305 (1982)] which is integrable. We show that the latter system is a particular non-holonomic deformation of the usual non-linear Schrodinger equation, aided by generalized parameterizations. General non-holonomic deformations correspond only to temporal inhomogeneity, with the additional spatial inhomogeneity corresponding to particular spectral orders, identical for both continuum Heisenberg spin chain and non-linear schrodinger system. Being semi-Classical in nature, these continuum limits of spin system display amplitude-phase entanglement.The Heisenberg Spin Chain system, in the continuum limit, can be represented by the non-linear Schr\odinger equation through the Hashimoto map. Inhomogeneity induced through localizing the nearest neighbor interaction strength can also be mapped similarly to an integro -differential generalization of the non-linear Schrodinger system [J. Phys. C: Solid State Phys. 15, L1305 (1982)] which is integrable. We show that the latter system is a particular non-holonomic deformation of the usual non-linear Schrodinger equation, aided by generalized parameterizations. General non-holonomic deformations correspond only to temporal inhomogeneity, with the additional spatial inhomogeneity corresponding to particular spectral orders, identical for both continuum Heisenberg spin chain and non-linear schrodinger system. Being semi-Classical in nature, these continuum limits of spin system display amplitude-phase entanglement.

arXiv:1703.02353Mar 7, 20170 citations

Numerical investigation of correlation functions for the UqSU(2) invariant spin-1/2 Heisenberg chain

P F Arndt, T Heinzel

We consider the U q SU(2) invariant spin-1/2 XXZ quantum spin chain at the roots of unity q=exp(i pi /(m+1)), corresponding to different minimal models of conformal field theory. We conduct a numerical investigation of the correlation functions of U q SU(2) scalar two-point operators in order to find which operators in the minimal models they correspond to. Using graphical representations of the Temperley-Lieb algebra we are able to deal with chains of up to 28 sites. Depending on q, the correlation functions show different characteristics and finite-size behaviour. For m=2/3, which corresponds to the Lee-Yang edge singularity, we find the surface and bulk critical exponent -1/5. Together with the known result in the case m=3 (Ising model) this indicates that in the continuum limit the two-point operators involve conformal fields of spin-m-1/m+1. For other roots of unity q the chains are too short to determine the surface and bulk critical exponents.We consider the U q SU(2) invariant spin-1/2 XXZ quantum spin chain at the roots of unity q=exp(i pi /(m+1)), corresponding to different minimal models of conformal field theory. We conduct a numerical investigation of the correlation functions of U q SU(2) scalar two-point operators in order to find which operators in the minimal models they correspond to. Using graphical representations of the Temperley-Lieb algebra we are able to deal with chains of up to 28 sites. Depending on q, the correlation functions show different characteristics and finite-size behaviour. For m=2/3, which corresponds to the Lee-Yang edge singularity, we find the surface and bulk critical exponent -1/5. Together with the known result in the case m=3 (Ising model) this indicates that in the continuum limit the two-point operators involve conformal fields of spin-m-1/m+1. For other roots of unity q the chains are too short to determine the surface and bulk critical exponents.

arXiv:cond-mat/9501035Journal of Physics A Mathematical and GeneralJul 7, 19953 citations

SU(n) Heisenberg spin chain

Sophocles J. Orfanidis

Physics Letters AJan 1, 198035 citations

Elementary excitations in dimerized and frustrated Heisenberg chains

G. Bouzerar, A. P. Kampf, G. I. Japaridze

We present a detailed numerical analysis of the low-energy excitation spectrum of a frustrated and dimerized spin $S=1/2$ Heisenberg chain. In particular, we show that in the commensurate spin-Peierls phase the ratio of the singlet and triplet excitation gap is a universal function which depends on the frustration parameter only. We identify the conditions for which a second elementary triplet branch in the excitation spectrum splits from the continuum. We compare our results with predictions from the continuum limit field theory. We discuss the relevance of our data in connection with recent experiments on ${\mathrm{CuGeO}}_{3},$ ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5},$ and $(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}$ .We present a detailed numerical analysis of the low-energy excitation spectrum of a frustrated and dimerized spin $S=1/2$ Heisenberg chain. In particular, we show that in the commensurate spin-Peierls phase the ratio of the singlet and triplet excitation gap is a universal function which depends on the frustration parameter only. We identify the conditions for which a second elementary triplet branch in the excitation spectrum splits from the continuum. We compare our results with predictions from the continuum limit field theory. We discuss the relevance of our data in connection with recent experiments on ${\mathrm{CuGeO}}_{3},$ ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5},$ and $(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}$.

arXiv:cond-mat/9801046Physical review. B, Condensed matterAug 1, 199881 citations

We present a detailed numerical analysis of the low-energy excitation spectrum of a frustrated and dimerized spin

S=1/2

Heisenberg chain. In particular, we show that in the commensurate spin-Peierls phase the ratio of the singlet and triplet excitation gap is a universal function which depends on the frustration parameter only. We identify the conditions for which a second elementary triplet branch in the excitation spectrum splits from the continuum. We compare our results with predictions from the continuum limit field theory. We discuss the relevance of our data in connection with recent experiments on

{\mathrm{CuGeO}}_{3},

{\mathrm{NaV}}_{2}{\mathrm{O}}_{5},

and

(\mathrm{VO}{)}_{2}{\mathrm{P}}_{2}{\mathrm{O}}_{7}