# Quantum Ising/Heisenberg model, states representation

I am working with a Hamiltonian which looks like this (Heisenberg model) $$\hat{H} = -\frac{1}{2}\sum_{j=1}^N \left( J_x\sigma_j^x\sigma_{j+1}^x +J_y\sigma_j^y\sigma_{j+1}^y +J_z\sigma_j^z\sigma_{j+1}^z + h\sigma_j^z \right).$$

I have made a program which computes this Hamiltonian using Pauli matrices (spin 1/2). My working space is then the tensor product ($$N$$ times) of $$\mathbb{C}^2$$. I know that the canonical base of my space can be expressed as a tensor product of base vectors of $$\mathbb{C}^2$$, for example: $$(1,0,0,0) = (1,0)\otimes (1,0)$$

This works fine when I am working with only $$J_z$$ not null (classical Ising model) because all the eigenstates can be expressed this way (all eigenstates are vectors of the canonical base). When I work with, for example, only $$J_x$$ not null (quantum Ising) I get eigenstates which are a bit more messy, for example $$(0,1/\sqrt{2},-1/\sqrt{2},0)$$.

This eigenstate can be expressed as a linear combination of canonical base vectors and those as a tensor product of the spin 1/2 Z base.

My problem is that I seek a "visual representation" of all states (or eigenspaces), I believe that any two level system can be represented in polar coordinates in $$\mathbb{R}^3$$ (Bloch sphere) but I fail at doing so, how should I proceed? Let's say I wanted to represent the state I used as a example before, which is non-degenerate, in a visual way, that is, in polar coordinates (it corresponds to the case of two 1/2 spins, so two points in polar coordinates would be required).

Maybe you would like to trace over other base. For example, your interest eigenstate is $N-$ body state $|\psi\rangle$, then you have density matrix $\rho=|\psi\rangle\langle\psi|$, and to derive a single site's information you only have to trace over other state, i.e., $\rho_j=\prod_{i\neq j}\sum_{\sigma_i}\langle \sigma_i|\rho|\sigma_i\rangle$, and get a $2\times2$ matrix.
For your case as an example, say $|\psi\rangle = (0, 1/\sqrt2,-1\sqrt2,0)^T$,
$\rho=\left(\begin{matrix}0&0&0&0\\0&1/2&-1/2&0\\0&-1/2&1/2&0\\0&0&0&0\end{matrix}\right)$,
and to get information about the first site, we trace over the second by $|2_\uparrow\rangle=(1,0)^T,|2_\downarrow\rangle=(0,1)^T$, then every matrix element of $\rho_1$ is given by (take $\langle1_\uparrow|\rho_1|1_\uparrow\rangle$ as example)
$\langle1_\uparrow|\rho_1|1_\uparrow\rangle=(|1_\uparrow\rangle\otimes|2_\uparrow\rangle)^\dagger|\psi\rangle\cdot\langle\psi|(|1_\uparrow\rangle\otimes|2_\uparrow\rangle)+(|1_\uparrow\rangle\otimes|2_\downarrow\rangle)^\dagger|\psi\rangle\cdot\langle\psi|(|1_\uparrow\rangle\otimes|2_\downarrow\rangle)$