Phase Transition in a Quantum Ising Model with Long-Range Interaction

Abstract

Among the many models of Statistical Physics, Ising model [1–3] is perhaps the most ubiquitous. Till date it is being applied in varied context with reasonable success. In the “classical” version of this model, one has in general a set of lattice sites and to each site is associated a variable called spin which can assume the values $\pm \frac{1}{2}$. For a system of N sites, there can hence be 2^N possible configurations.

12.1 Appendix: Working Formulas for Degenerate Perturbation Theory

We present here the working formulas for the eigenvalues in the case of degenerate perturbation theory for ready reference.Footnote

All text books on Quantum Mechanics (and there is no dearth of it) cover this topic [20],[21],[22].

Consider a Hamiltonian

$${\mathcal H} = {\mathcal H}_0 +V.$$

The eigenstates of ${\mathcal H}_0$ are known and we want to solve for the eigenvalues of ${\mathcal H}$ when the effect of V is small. Let ${\mathcal H}_0$ have eigenvalues $E_1^{(0)}$, $E_2^{(0)}, \ldots$ and one of them, say $E^{(0)}_n$ be g-fold degenerate with (one choice of orthogonal) eigenvectors

$$\mid \!\uppsi^{(n)}_1 \rangle, \mid \!\uppsi^{(n)}_2 \rangle, \ldots, \mid \!\uppsi^{(n)}_g \rangle.$$

The first-order perturbation corrections to the eigenvalue $E_n^{(0)}$ are the eigenvalues of the $g \times g$ matrix M defined by

$$M_{ij} = \langle \uppsi^{(n)}_i \mid V \mid \uppsi^{(n)}_j \rangle.$$

((12.44))

Let us call the eigenvalues of this matrix as λ ₁, $\lambda_2, \ldots, \lambda_g$. If no two eigenvalues of this matrix are equal, then the degeneracy is said to be completely removed in the first order. To obtain the second-order correction, one needs to calculate the eigenvectors $\mid \!e_1 \rangle, \mid \!e_2 \rangle, \cdots, \mid \!e_g \rangle$ of the matrix M and construct therefrom the set of vectors

$$\mid \!\phi^{(n)}_i \rangle = \sum_{j=1}^g e_i (j) \mid \!\uppsi^{(n)}_j \rangle , \,\,\,\,\, i = 1, 2, \cdots, g,$$

((12.45))

which can be identified as the correct choice of unperturbed eigenvectors for the eigenvalue $E_n^{(0)}$. In this equation, $e_i (j)$ stands for the jth component of the vector $\mid \!e_i \rangle$. The second-order corrections to the eigenvalue $E_n^{(0)} + \lambda_i$ is now given by

$$\sum_{m, m \ne n} \frac{\mid \langle \phi^{(n)}_i \mid V \mid \uppsi^{(m)} \rangle \mid^2} {E_n^{(0)} - E_m^{(0)}},$$

((12.46))

where $\uppsi^{(m)}$ is the eigenvector corresponding to an eigenvalue $E_m^{(0)}$. Note that the sum is here over all the (degenerate or non-degenerate) eigenstates of ${\mathcal H}_0$ whose eigenvalue is different from the one under consideration, namely $E_n^{(0)}$.

If all the eigenvalues of the matrix M are equal

$$\lambda_1 = \lambda_2 = \cdots = \lambda_g = \lambda \textrm{(say)},$$

then the first-order correction is still λ, but the degeneracy is not lifted (even partially). To find the second-order perturbation correction here, one needs to construct the $g \times g$ matrix P defined by

$$P_{ij} = \sum_{m, m \ne n} \frac{ \langle \uppsi^{(n)}_i \mid V \mid \uppsi^{(m)} \rangle \langle \uppsi^{(m)} \mid V \mid \uppsi^{(n)}_j \rangle } {E_n^{(0)} - E_m^{(0)}}.$$

((12.47))

Here also, the sum is over all the eigenstates of ${\mathcal H}_0$ satisfying $E_m^{(0)} \ne E_n^{(0)}$. The second-order correction to $E_n^{(0)}$ are now the eigenvalues of P.

When some of the eigenvalues of M are degenerate, the degeneracy of the eigenvalue $E_n^{(0)}$ is partially lifted, and one has to perform the degenerate perturbation theory (i.e. construct P and diagonalise it) over each degenerate subspace of M.

The condition of validity of these perturbation equations are that (as mentioned above) the change in eigenstates due to the introduction of V must be small. In quantitative terms, this means

$$\hbox{second-order correction} \ll \hbox{first-order correction} \ll {\textit d},$$

where d stands for the separation of eigenvalues at $E_n^{(0)}$, that is, the smaller one of the quantities $\mid E_n^{(0)} - E_{n-1}^{(0)} \mid$ and $\mid E_n^{(0)} - E_{n+1}^{(0)} \mid$, $E_{n \pm1}^{(0)}$ being the eigenvalues of ${\mathcal H}_0$ just below and above $E_n^{(0)}$.