I am learning the 1D ising model (spin 1/2), without external field and considering the nearest site interaction, the hamiltonian for 1D chain is simple [tex]H = -J\sum_i S_iS_{i+1}[/tex] Since each spin can only take either +1 or -1, we can write the transition matrix as [tex] T = \left( \begin{matrix} e^{K} & e^{-K} \\ e^{-K} & e^{K} \end{matrix} \right) [/tex] where [tex]K=\beta J[/tex] Now I try to learn 2D case, I read some book on it but seems quite complicated, so I started with the simplest case (no external field, only nearest interaction, rectangular lattics with only 2 rows and N columns and perodic boundary condition). The hamiltonian is [tex]H = -J\sum_{i=1}^{2}\sum_{j=1}^N S_{ij}S_{i, j+1}[/tex] right? What I am really consfuing is how to find the transition matrix? Now each site has four nearest neighbor (of course, to avoid double counting, we only need to count two one at a time, let's say we count the one next to and below the current site) and each spin can take 2 values, so what's the dimension of the transition matrix? and what does [tex]T_{ij}[/tex] means?
Here is my idea. Since we have to sum over all possible sites, so consider the site (1, 1), and (2, 1) (i.e. the current site and the site below it), the energy is [tex]-J\left[\left(S_{11}S_{12}+S_{11}S_{21}\right) + \left(S_{21}S_{22} + S_{21}S{11}\right)\right][/tex] All other sites has the same case as this specific site (1, 1), and in this case, we only need to consider four spin [tex]S_{11}, S_{12}, S_{21}, S_{22}[/tex]. Note that each of them can take two values so there are totally 16 possible values. The transition matrix would be 4x4, right?
Your Hamiltonian is incorrect- it's missing a term like S_ij S_i+1,j .The 2-D ising model is significantly more complicated than the 1-D. There's a brief treatment in Landau and Lifgarbagez, vol. 5 (pp 498-506) and Chaikin and Lubensky "Principles of condensed matter physics" has additional detail along with the O_n model.
I checked it, now the Hamiltonian is modified as [tex]H = -J\sum_{i=1}^{2}\sum_{j=1}^N S_{ij}S_{i, j+1} + S_{ij}S_{i+1,j}[/tex] So, the for site (i, j), the energy could take 16 possible values. For [tex]S_{ij}, S_{i+1, j}, S_{i, j+1}, S_{i+1, j+1} \Longrightarrow E[/tex] ++++: 4 +++-: 2 ++-+: 2 ++--: 0 +-++: 2 +-+-: 0 +--+:-2 +---: -2 -+++: 2 -+-+: 0 -+--: -2 --++: 0 --+-: -2 ---+: -2 ----: 4 So could I just use this values to setup the transition matrix?
The exact solution of the 2D Ising model was done by Onsager. See here for a brief outline: http://www.nyu.edu/classes/tuckerman/stat.mech/lectures/lecture_26/node2.html
Thanks. I read that before, but I found this quite confusing. The above outline is about n x n lattices, now let's apply his result to 2 x n, we get the matrix element [tex] T_{jk} = \exp\left[\beta J\left(\sigma_{1j}\sigma_{1k} + \sigma_{1j}\sigma_{2j} + \sigma_{2j}\sigma_{2k} + \sigma_{2j}\sigma_{1j}\right)\right] [/tex] For the last term, we apply the periodic boundary condition. My doubt is [tex]\sigma_{ij}[/tex] is the eigenvalue and could be -1 or +1, so each terms in above expression couble be either +1 or -1, how come do we get a certain value for specific matrix element [tex]T_{jk}[/tex] ?
Solvng the 2D Ising model is almost trivial by transforming it to a close packed dimer model. The dimer model can be solved using very simple combinatorial techiques, you don't need complicated transfer matrix techniques. See e.g. here: http://arxiv.org/abs/cond-mat/0212363
Thank you very much. I will read that later. I am learning this problem because there is one chapter about transfer matrix in my text and this method is useful in some other place. So I want to learn it by studying 2D ising model as an example.
Then you should read this book: http://tpsrv.anu.edu.au/Members/baxter/book You can download it free of charge. The transfer matrix technique that most textbooks explain for solving the Ising model is of no use for most other models. The Ising model is a so-called "free fermion model", the transfer matrix can then be diagonalized using a Bogoliubov transformation. This won't work for the vast class of integrable models. So, if you want to learn about solving models, you should learn about the Bethe Ansatz, the Yang-Baxter equation etc. etc. This is explained in detail in the book by Baxter.
Using the 2D Ising model to learn about the Transfer matrix is probably not the best approach. First, the Transfer Matrices are somewhat obscure (although certainly doable), and second, it doesn't even get you half way there in solving the model (i.e. you still need perform some other steps as well - steps that are quite specific to the Ising model). But just to throw in another book, I know that this one solves the Ising model using Transfer matrices: http://www.amazon.com/Equilibrium-Statistical-Physics-Michael-Plischke/dp/9810216424 (Chapter 5.1)