Physics Asked by RicknJerry on January 15, 2021
I’m reading Quantum Field Theory in Strongly Correlated Electronic Systems, Nagaosa.
Consider 1D Ising model,
$$H=J_zsum_i S^z_iS^z_{i+1}.$$
on page 3, it says
The groud stae is 2-fold degenerate because the Hamiltonian is invariant
under the transformation $S^i_z rightarrow -S^i_z$, performed at all sites $i$.
Calling these two ground states $A$ and $B$ and assuming that the system at the right-hand side is in state $A$, and at the left-hand side in state $B$, then somewhere there
must exist a boundary between region $A$ and region $B$. This boundary is called a kink or soliton. Because at finite temperature this excitation occurs
with a finite density, the spin correlation function $F(r) =langle S^z_iS^z_{i+r}rangle$ will decay exponentially with a correlation length $xi$.
I know how to directly calculate the correlation function, but I wonder how the argument for exponential decay of correlation function is made here and how to understand it.
Any help would be highly appriciated!!
Let me write the Hamiltonian $$ H = -J sum_i S_i^z S_{i+1}^z. $$ This choice will avoid some annoying (and irrelevant) signs.
One way to formulate the statement in the OP precisely is as follows.
Consider the variables $delta_i=S_i^zS_{i+1}^z$. Since $delta_i=1$ when the spins at $i$ and $i+1$ agree and $delta_i=-1$ when the spins at $i$ and $i+1$ disagree, you can identify them with the kinks in your question (that is, there is a kink between $i$ and $i+1$ when $delta_i=-1$).
Introducing the variables $delta_i=S_i^zS_{i+1}^z$, the Hamiltonian becomes $$ H = J^z sum_i delta_i. $$ It follows that the random variables $delta_i$ are independent and identically distributed. One can easily compute their expectation: since $$ P(delta_i = 1) = frac{e^{beta J^z}}{e^{beta J^z} + e^{-beta J^z}}, $$ one has $$ langle delta_i rangle = frac{e^{beta J^z} - e^{-beta J^z}}{e^{beta J^z} + e^{-beta J^z}} = tanh(beta J^z). $$ Finally, noting that $S_i^zS_{i+r}^z = delta_idelta_{i+1}cdotsdelta_{i+r-1}$, we obtain $$ langle{S_i^zS_{i+r}^z}rangle = langledelta_idelta_{i+1}cdotsdelta_{i+r-1}rangle = langle delta_i rangle^r = (tanh(beta J^z))^r. $$
In words, the fact that kinks proliferate in the system (at each $i$, there is a positive probability that a kink is present, so there will be a positive density of them in the system) prevents the ordering of the spins.
Correct answer by Yvan Velenik on January 15, 2021
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