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Journal of Statistical Physics

Transfer Matrix Spectrum and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

Ricardo S. Schor, Michael O’Carroll

Published Jun 1, 2000• 29 citations
Publisher
DOI
10.1023/a:1018632604807

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We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d -dimensional lattice ferromagnetic classical N -component vector spin systems at high temperature (0 < β << 1). Each system is characterized by a single site a priori spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe–Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe–Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.

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Bound states in the transfer matrix spectrum for general lattice ferromagnetic spin systems at high temperature

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We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 2000•18 citations
We obtain the low-lying energy-momentum spectrum for the imaginary-time lattice quantum field model associated with d -dimensional lattice ferromagnetic classical N -component vector spin systems at high temperature (0 < β << 1). Each system is characterized by a single site a priori spin probability distribution. The energy-momentum spectrum exhibits isolated dispersion curves which are identified as single particles and multi-particle bands. Our two-particle bound state analysis is restricted to a ladder approximation of the Bethe–Salpeter equation, and the existence of bound states depend on whether or not Gaussian domination for the four-point function is verified. It is known that two-particle bound states appear below the two-particle band if Gaussian domination does not hold. Here, we show that two two-particle bound states appear above the two-particle band if Gaussian domination is verified. We also show how the complete two-particle spectral pattern for these models can be understood by making a correspondence between the Bethe–Salpeter equation and a two-particle lattice Schrödinger Hamiltonian operator with attractive or repulsive spin-dependent delta potentials at the origin. A staggering transformation is used to relate the attractive and repulsive potential cases, as well as their associated Hamiltonians spectrum and eigenfunctions.
We obtain different properties of general d dimensional lattice ferromagnetic spin systems with nearest neighbor interactions in the high temperature region (beta<<1). Each model is characterized by a single site a priori spin distribution, taken to be even. We state our results in terms of the parameter alpha=<s(4)>-3<s(2)>(2) where <s(k)> denotes the kth moment of the a priori distribution. Associated with the model is a lattice quantum field theory that is known to contain particles. We show that for alpha>0, beta small, there exists a bound state with mass below the two-particle threshold. For alpha<0, bound states do not exist. The existence of the bound state has implications on the decay of correlations, i.e., the four-point function decays at a slower rate than twice that of the two-point function. These results are obtained using a lattice version of the Bethe-Salpeter equation in the ladder approximation. The existence and nonexistence results generalize to N-component models with rotationally invariant a priori spin distributions.