Can we have a spin glass in the one-dimensional Heisenberg hamiltonian with nearest neighbours only?
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Consider the one dimensional Heisenberg Hamiltonian of the form
H=−∑<i,j>JijSi⋅Sj
with nearest neighbour interactions. Assume that we have a chain with N spins and that the distance from site to site is a=1. Let us assume that Jij is randomly distributed and can only take the integer values {−1,1}. Furthermore, let us say that the spin at site i=1 and at i=N is pointing in the same direction (up).
Now, I would naively think that the ground state of this Hamiltonian would be an alternating chain of spins. That is each spin will either point up or down depending on the coupling Jij.
However, in literature on spin glasses people often say that spin glasses arise due to competing ferromagnetic and antiferromagnetic interactions. I can imagine that this competition also exists in the above Hamiltonian, perhaps because we have two boundary conditions.
To put my question(s) bluntly:
Can the Hamiltonian above have a spin glass ground state? Or is it simply alternating up and down?
If there is a spin glass ground state: is there a way to determine this ground state given that we know all of the exchange couplings Jij
H=−∑<i,j>JijSi⋅Sj
with nearest neighbour interactions. Assume that we have a chain with N spins and that the distance from site to site is a=1. Let us assume that Jij is randomly distributed and can only take the integer values {−1,1}. Furthermore, let us say that the spin at site i=1 and at i=N is pointing in the same direction (up).
Now, I would naively think that the ground state of this Hamiltonian would be an alternating chain of spins. That is each spin will either point up or down depending on the coupling Jij.
However, in literature on spin glasses people often say that spin glasses arise due to competing ferromagnetic and antiferromagnetic interactions. I can imagine that this competition also exists in the above Hamiltonian, perhaps because we have two boundary conditions.
To put my question(s) bluntly:
Can the Hamiltonian above have a spin glass ground state? Or is it simply alternating up and down?
If there is a spin glass ground state: is there a way to determine this ground state given that we know all of the exchange couplings Jij
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One-Dimensional Spin Glass 13 Thus, one can expect that thermal properties of all these models in zero external fleld and at low temperatures will be similar.
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