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Consider the model $$ H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z) $$ with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.

Equation 33 in this paper says the reduced ground state for a single spin (in the $N\to\infty$) is $$ \rho_1 = \frac{I+\langle \sigma^x \rangle \sigma^x + \langle \sigma^z \rangle \sigma^z}{2}. $$ I'm trying to do verify this numerically using exact diagonalization for finite $N$, but I'm getting something completely different. In particular, my off-diagonal entries are nearly zero. I am really lost as to what's going on. Does anyone have experinece with this?

Consider the model $$ H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z) $$ with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.

Equation 33 in this paper says the reduced ground state for a single spin (in the $N\to\infty$ and zero temperature limits) is $$ \rho_1 = \frac{I+\langle \sigma^x \rangle \sigma^x + \langle \sigma^z \rangle \sigma^z}{2}. $$ I'm trying to do verify this numerically using exact diagonalization for finite $N$, but I'm getting something completely different. In particular, my off-diagonal entries are nearly zero. I am really lost as to what's going on. Does anyone have experinece with this?

Consider the model $$ H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z) $$ with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.

Equation 33 in this paper says the reduced state for a single spin (in the $N\to\infty$ limit) is $$ \rho_1 = \frac{I+\langle \sigma^x \rangle \sigma^x + \langle \sigma^z \rangle \sigma^z}{2}. $$ I'm trying to do verify this numerically using exact diagonalization for finite $N$, but I'm getting something completely different. In particular, my off-diagonal entries are nearly zero. I am really lost as to what's going on. Does anyone have experinece with this?

Consider the model $$ H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z) $$ with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.

Equation 33 in this paper says the reduced ground state for a single spin (in the $N\to\infty$) is $$ \rho_1 = \frac{I+\langle \sigma^x \rangle \sigma^x + \langle \sigma^z \rangle \sigma^z}{2}. $$ I'm trying to do verify this numerically using exact diagonalization for finite $N$, but I'm getting something completely different. In particular, my off-diagonal entries are nearly zero. I am really lost as to what's going on. Does anyone have experinece with this?

Consider the model $$ H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z) $$ with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.

This paper says that when $\lambda=0$, the ground state is $|0\rangle\langle 0|$. I also obtain this when I diagonalize $H$ numerically.

What I am confused about is that Equation 33 says the reduced state for a single spin is $$ \rho_1 = \frac{I+\langle \sigma^x \rangle \sigma^x + \langle \sigma^z \rangle \sigma^z}{2}. $$ I thought that I should be able to obtain the reduced state by taking the partial trace of $|0\rangle \langle 0|$ over all but a single site, which (I think) is $|0\rangle \langle 0| = \begin{bmatrix}1&0 \\ 0 & 0\end{bmatrix}$.

Am I computing the reduced state incorrectly?

Consider the model $$ H = - \sum_{i=1}^{N} (\lambda \sigma_i^x \sigma_{i+1}^x + \sigma_z) $$ with periodic boundary conditions $\sigma_1^x=\sigma_{N+1}$.

Equation 33 in this paper says the reduced state for a single spin (in the $N\to\infty$ limit) is $$ \rho_1 = \frac{I+\langle \sigma^x \rangle \sigma^x + \langle \sigma^z \rangle \sigma^z}{2}. $$ I'm trying to do verify this numerically using exact diagonalization for finite $N$, but I'm getting something completely different. In particular, my off-diagonal entries are nearly zero. I am really lost as to what's going on. Does anyone have experinece with this?