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• W2181822858 abstract "In the previous chapter, the Nonlinear Sigma Model (NLSM) in d dimensions has emerged as the continuum approximation to the quantum Heisenberg antiferromagnet in d-1 dimensions with additional Berry phases. The partition function is $${{Z}_{{NLSM}}} = int_{Lambda } {mathcal{D}hat{n} exp left( { - frac{{{{Lambda }^{{d - 2}}}}}{{2f}}int {{{d}^{d}}xsumlimits_{{mu = 1}}^{d} {{{partial }_{mu }}hat{n} cdot } {{partial }_{mu }}hat{n}} } right)} .$$ (13.1) (left| {hat n(x)} right| = 1) is a unit vector and x ∈ R d. Λ is the momentum cutoff, i.e., the shortest wavelength included in the (Dhat n). The path integral (13.1) is not well defined until we specify a regularization procedure. We consider a classical Heisenberg model on a d-dimensional cubic lattice, with N sites and lattice constant a. Its partition function is $${{Z}_{{HM}}} = int {prodlimits_{{i = 1}}^{mathcal{N}} {d{{{hat{Omega }}}_{i}} exp left( {frac{{bar{J}}}{T}sumlimits_{{langle ijrangle }} {{{{hat{Omega }}}_{j}} cdot } {{{hat{Omega }}}_{j}}} right)} .}$$ (13.2) The continuum limit of Z HM is given by the substitutions $$ hat Omega _i to eta _i hat n(x_i ), $$ (13.3) where ηi is 1 (((e^{ix_i bar pi } ))) for the ferromagnet (antiferromagnet). Sums and differences are substituted by $$begin{gathered} begin{array}{*{20}{c}} hfill {sumlimits_{i} {F({{x}_{i}}) to {{a}^{{ - d}}}int {{{d}^{d}}xF(x),} } } hfill {hat{Omega }({{x}_{i}} + a{{{hat{x}}}_{mu }}) - hat{Omega }({{x}_{i}}) to a{{partial }_{mu }}hat{n},} end{array} hfill begin{array}{*{20}{c}} {sumlimits_{i} {F({{x}_{i}}) to {{a}^{{ - d}}}int {{{d}^{d}}xF(x),} } } {hat{Omega }({{x}_{i}} + a{{{hat{x}}}_{mu }}) - hat{Omega }({{x}_{i}}) to a{{partial }_{mu }}hat{n},} end{array} hfill end{gathered}$$ (13.4) and a dimensionless coupling constant is given by $$frac{T}{{bar{J}}} leftrightarrow f.$$ (13.5) The measure is replaced by $$mathcal{D}hat{Omega } to prodlimits_{{|q| leqslant Lambda }} {d{{{hat{n}}}_{q}},}$$ (13.6) where η is the radius of a spherical (sph) Brillouin zone that has the same number of degrees of freedom as the cubic zone (cube) of the lattice $${{mathcal{N}}^{{ - 1}}}sumlimits_{{|{{k}_{mu }}| leqslant tfrac{pi }{a}}}^{{cube}} { = {{mathcal{N}}^{{ - 1}}}} sumlimits_{{|k| leqslant Lambda }}^{{sph}} { = 1.}$$ (13.7) This determines the spherical Brillouin zone’s radius to be $$Lambda = left{ {begin{array}{*{20}{c}} {pi /a} hfill & {d = 1} hfill {2sqrt {pi } /a} hfill & {d = 2.} hfill {{{{(6{{pi }^{2}})}}^{{1/3}}}/a} hfill & {d = 3} hfill end{array} } right.$$ (1) Expanding the Hamiltonian in (13.2) in terms of gradients of (hat n) and keeping up to quadratic terms, the relation $$ Z_{HM} (T/bar J,a) approx Z_{NLSM} (f,Lambda ^{ - 1} )e^{dN/f} $$ (13.9) is established at low temperatures (f << 1). The factor e dN/f is the Boltzmann weight of the classical ground state. The continuum approximation holds also for the generating functional (see Appendix B) and for the longwavelength spin correlations, as long as the spin correlation length ξ is much larger than η-1. In this regime, the NLSM properties are weakly dependent on the regularization procedure, and therefore it serves as a model for diverse Heisenberg models. The crucial features of these models are their O(3) symmetry of the ground state manifold and their short-range interactions. Other details will primarily affect the choice of f and η.KeywordsPartition FunctionWeak CouplingSpin WaveNonlinear Sigma ModelWeak Coupling ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves." @default.
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• W2181822858 date "1994-01-01" @default.
• W2181822858 modified "2023-09-27" @default.
• W2181822858 title "Nonlinear Sigma Model: Weak Coupling" @default.
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• W2181822858 doi "https://doi.org/10.1007/978-1-4612-0869-3_13" @default.
• W2181822858 hasPublicationYear "1994" @default.
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