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I am reading this book about numerical methods in physics. It has the following question:

Consider the Ising Hamiltonian defined as following $$H=-\sum_ {i=1}^{N-1} \sigma_i^x \sigma_ {i+1} ^x + h \sum_ {i=1} ^N \sigma_i^z$$

Write a program that computes the $2^N \times 2^N$ matrix for different $N$

From quantum mechanics, I know that any operator can be expressed in matrix form as follows $$H_{rs}\dot{=}\langle r |H|s\rangle$$ where $|i\rangle$ are any basis.

My question is what basis can I take for this Hamiltonian? Is there any other way to write this Hamiltonian in matrix form?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – rob Dec 30 '18 at 14:15
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For completeness I'll summarize the answer here. After a fun conversation in the comments, we saw that it will be more illuminating to write

$$H=-\sum_ {i=1}^{N-1} \sigma_i^x \sigma_ {i+1} ^x + h \sum_ {i=1} ^N \sigma_i^z $$

as

$$ H=-\sum_ {i=1}^{N-1} \sigma_i^x \sigma_ {i+1} ^x + h\left( \sum_ {i=1} ^{N-1} 1_1 \otimes \cdots 1_{i-1} \otimes \sigma_i^z \otimes 1_{i+1} \otimes \cdots \otimes 1_{N-1} \right) + h ( 1_1 \otimes \cdots \otimes 1_{N-1} \otimes \sigma^z_N) $$

where it is understood that (to prevent clutter)

$$ \sigma_i^x \sigma_{i+1}^x = 1_1 \otimes \cdots\otimes 1_{i-1} \otimes \sigma^x_i \otimes \sigma^x_{i+1} \otimes 1_{i+2} \otimes \cdots \otimes 1_{N} $$

and the subscripts are there to denote nothing more than the position of insertion (i.e. they are all $2\times 2$ identity)

We then create a matrix from the direct product of matrices $A,B$ where $[A] = m \times n$ and $[B] = p \times q$ matrices

$$ \mathbf {A} \otimes \mathbf {B} ={\begin{bmatrix}a_{11}\mathbf {B} &\cdots &a_{1n}\mathbf {B} \\\vdots &\ddots &\vdots \\a_{m1}\mathbf {B} &\cdots &a_{mn}\mathbf {B} \end{bmatrix}}. $$

We apply this to the $i^{th}$ term and then loop over all $i$ to achieve the desired matrix.

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  • $\begingroup$ Your tensor product notation doesn't make sense: $\sigma_N^z$ is $\sigma^z$ acting on site $N$. You can't tensor this with $I_N$, which is $I$ acting on site $N$. You want to rather tensor it with $I$ on all other sites -- that's the definition of $\sigma_N^z$. $\endgroup$ – Norbert Schuch Dec 29 '18 at 22:59
  • $\begingroup$ You're right. Thank you. I've made an edit but the notation is a bit sloppy still. $\endgroup$ – InertialObserver Dec 29 '18 at 23:13
  • $\begingroup$ Much better, but for consistency you should also add the identities in the first term! $\endgroup$ – Norbert Schuch Dec 29 '18 at 23:13
  • $\begingroup$ Will do, gonna do that now $\endgroup$ – InertialObserver Dec 29 '18 at 23:14
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    $\begingroup$ I see. Yes, that would screw up things dimensionally. Thanks a lot for clearing everything up. $\endgroup$ – InertialObserver Dec 29 '18 at 23:31
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$\textbf{Quick hint :}$

Interpret spin operators (for chain of length-N) this way (spin operators act on tensor product space) : $$\sigma_{i}^{\alpha} \rightarrow\underset{N_{}^{\text{th}}\text{order direct/kronecker product of identity and pauli matrices}}{\mathbb{I}_{}^{}\otimes\cdots\otimes\underset{i_{}^{\text{th}} \text{position}}{\sigma_{}^{\alpha}}\otimes\cdots\otimes\mathbb{I}_{}^{}}$$ which makes (for $i < j$):

$$\sigma_{i}^{\alpha}\sigma_{j}^{\beta} \stackrel{i < j}{\rightarrow} \mathbb{I}_{}^{}\otimes\cdots\otimes\underset{i_{}^{\text{th}} \text{position}}{\sigma_{}^{\alpha}}\otimes\cdots\otimes\underset{j_{}^{\text{th}} \text{position}}{\sigma_{}^{\beta}}\otimes \cdots\otimes\mathbb{I}_{}^{}$$ with standard definition of direct/Kronecker product of matrices

Note also that Kronecker product is associative.

$\mathbb{I}$ is $2 \times 2$ identity matrix and $\sigma_{}^{}$'s are standard Pauli matrices.

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