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Journal of Physics C Solid State Physics

Excitation spectrum of an antiferromagnetic model

Chanchal K. Majumdar, Kanwar Krishan, Vinod Mubayi

Published Oct 17, 1972• 19 citations
Publisher

Abstract

Short chains of up to 12 spins are studied exactly to obtain the excitation spectrum of an antiferromagnetic hamiltonian H=1/2J Sigma i=1N sigma i, sigma i+1+1/4J Sigma i=1N sigma i, sigma i+2 (J>0, even, N+1 identical to 1, N+2 identical to 2). As well as antiferromagnetic spin waves of total spin S=1, there are S=0 bound magnon states lying partly below the spin wave branch. The susceptibility shows the usual peak at low temperatures, and the zeros of the partition function lie on the negative real axis in the complex mu =exp(-mHe/kBT) plane.

DOI
10.1088/0022-3719/5/20/009

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The exact excitation spectrum of the 3 \ifmmode\times\else\texttimes\fi{} 3 square and the 3 \ifmmode\times\else\texttimes\fi{} 4 rectangular lattices interacting via the nearest-neighbor anisotropic Heisenberg interaction $H=J\ensuremath{\Sigma}{(\mathrm{ij})}^{}[S_{i}^{}{}_{}{}^{z}S_{j}^{}{}_{}{}^{z}+\ensuremath{\gamma}(S_{i}^{}{}_{}{}^{x}S_{j}^{}{}_{}{}^{x}+S_{i}^{}{}_{}{}^{y}S_{j}^{}{}_{}{}^{y})]$ has been obtained for various values of the anisotropy constant $\ensuremath{\gamma}$. For $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, the zeros of the partition function in the complex $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{m{H}_{e}}{{k}_{B}T})$ plane obey the generalized Lee-Yang theorem for ferromagnetic coupling. The usual ferromagnetic transition seems to persist up to $\ensuremath{\gamma}\ensuremath{\simeq}0.6$. For $0.6\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\le}1$, nothing definite about a ferromagnetic transition can be inferred from the behavior of the zeros. When $\ensuremath{\gamma}>1$, the ferromagnetic zeros do not obey the generalized Lee-Yang theorem at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the negative real axis for all values of $\ensuremath{\gamma}$ and of the temperature of the studied lattices.

Physical review. B, Solid state•Oct 1, 1973•2 citations
Short chains of up to 12 spins are studied exactly to obtain the excitation spectrum of an antiferromagnetic hamiltonian H=1/2J Sigma i=1N sigma i, sigma i+1+1/4J Sigma i=1N sigma i, sigma i+2 (J>0, even, N+1 identical to 1, N+2 identical to 2). As well as antiferromagnetic spin waves of total spin S=1, there are S=0 bound magnon states lying partly below the spin wave branch. The susceptibility shows the usual peak at low temperatures, and the zeros of the partition function lie on the negative real axis in the complex mu =exp(-mHe/kBT) plane.
The exact excitation spectrum of a 3×3 two dimensional square lattice interacting via the nearest neighbour anisotropic Heisenberg Hamiltonian H = J ∑ < i j > { S i z S j z +γ( S i x S j x + S i y S j y )}, has been obtained for various values of the anisotropy constant γ. The zeros of the partition function in the complex µ=exp (- m H e / k B T ) plane are calculated for different γ for both ferromagnetic ( J <0) and antiferromagnetic ( J >0) coupling. For 0⩽γ⩽10{\leqslant}\gamma{\leqslant}10⩽γ⩽1, the ferromagnetic zeros obey the generalized Lee-Yang theorem at all temperatures, but, for γ>1, they violate it at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the (unphysical) negative real axis for all values of γ and temperature. The zero-field susceptibility, for antiferromagnetic coupling, does not display a local maximum as a function of temperature in contrast to Kawabata's results on the same lattice.
With the commutation relations of the spin operators, we first write out the equations of motion of the spin susceptibility and related correlation functions that have a hierarchical structure, then under the “soft cut-off” approximation, we give a set of equations of motion of spin susceptibilities for a spin [Formula: see text] antiferromagnetic Heisenberg model that is independent of whether or not the system has a long-range order in the low energy/temperature limit. Applying for a chain, a square lattice and a honeycomb lattice, respectively, we obtain the upper and the lowest boundaries of the low-lying excitations by solving this set of equations. For a chain, the upper and the lowest boundaries of the low-lying excitations are the same as that of the exact ones obtained by the Bethe ansatz, where the elementary excitations are the spinon pairs. For a square lattice, the spin wave excitation (magnons) resides in the region close to the lowest boundary of the low-lying excitations, and the multispinon excitations take place in the high-energy region close to the upper boundary of the low-lying excitations. For a honeycomb lattice, we have one kind of “mode” of the low-lying excitation. The present results obey the Lieb–Schultz–Mattis theorem, and they are also consistent with recent neutron scattering observations and numerical simulations for a square lattice.
The authors investigate the properties of the two-dimensional quantum S=1/2 Heisenberg antiferromagnetic model on a square lattice by means of a block approach. The lattice is divided into four spin square blocks and the Hamiltonian is expressed in terms of low-energy block states. In the framework of the mean-field theory based on the 'auxiliary-boson' formalism; they show that the spin excitation spectrum contains one singlet mode with the temperature-dependent gap and three degenerate triplet modes. The asymptotic low-temperature spin-spin correlations are proportional to R -3/2 exp(-R(1/ xi 1 +1/ xi 2 )), where xi 1 =exp(A/T), xi 2 =constant and A approximately=0.5J. The uniform susceptibility tends to zero in agreement with numerical results; the energy per site is close to -0.57J. Beyond the meanfield theory the authors find gapless collective excitations analogous to the gapless mode in the neutral Fermi gas with attraction.
We study the properties of the ground state and the low-lying excited states of an alternating spin-1 and spin-12\frac{1}{2}21​ antiferromagnetic chain with ferromagnetic next-nearest-neighbor coupling with a variety of methods, namely, spin-wave analysis, density-matrix renormalization-group (DMRG) method, and exact-diagonalization method. The ground state of this model with 2N2N2N spins is ferrimagnetic with the total spin SG=N∕2{S}_{G}=N∕2SG​=N∕2 for all parameter values. In the long chain limit, a gapless excitation is found with characteristics of goldstone mode of the ferromagnetic order with total spin $S={S}_{G}\ensuremath{-}1$. The lowest gapped excitation, however, exists in the total spin S=SG+1S={S}_{G}+1S=SG​+1 sector. Interestingly, we find that with the increase in ferromagnetic coupling, spin-wave gapped excitation becomes lower than the massless mode for the large wave numbers. Correspondingly, from DMRG analysis we find that the excitation in the total spin S=SG+1S={S}_{G}+1S=SG​+1 is lower than that in the total spin $S={S}_{G}\ensuremath{-}1$ when the size of the chain is short and the ferromagnetic interaction is large. Characteristics of these excitations are also reflected in the low temperature thermodynamic quantities, where the low-energy properties are strongly affected by the system size and the coupling strength.
The exact excitation spectrum of the 3 \ifmmode\times\else\texttimes\fi{} 3 square and the 3 \ifmmode\times\else\texttimes\fi{} 4 rectangular lattices interacting via the nearest-neighbor anisotropic Heisenberg interaction $H=J\ensuremath{\Sigma}{(\mathrm{ij})}^{}[S_{i}^{}{}_{}{}^{z}S_{j}^{}{}_{}{}^{z}+\ensuremath{\gamma}(S_{i}^{}{}_{}{}^{x}S_{j}^{}{}_{}{}^{x}+S_{i}^{}{}_{}{}^{y}S_{j}^{}{}_{}{}^{y})]$ has been obtained for various values of the anisotropy constant $\ensuremath{\gamma}$. For $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, the zeros of the partition function in the complex $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{m{H}_{e}}{{k}_{B}T})$ plane obey the generalized Lee-Yang theorem for ferromagnetic coupling. The usual ferromagnetic transition seems to persist up to $\ensuremath{\gamma}\ensuremath{\simeq}0.6$. For $0.6\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\le}1$, nothing definite about a ferromagnetic transition can be inferred from the behavior of the zeros. When $\ensuremath{\gamma}>1$, the ferromagnetic zeros do not obey the generalized Lee-Yang theorem at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the negative real axis for all values of $\ensuremath{\gamma}$ and of the temperature of the studied lattices.