First, a quick remark: I'm a mathematician, now working on some problems coming from physics (in particular Ising models on quasiperiodic chains). A few things I find rather mysterious. I would appreciate your help.

For the purpose of generality, let's consider the following Ising model on a chain of $N$ nodes.

$$H_N = - \sum_{i = 1}^N J_i\sigma_i^{(x)}\sigma_{i+1}^{(x)} - \sum_{i = 1}^N\sigma_i^{(z)},$$

with $J_i$ depending on the node $i$ (we assume no particular order for generality), and $\sigma_i^{(x),(z)}$ the Pauli matrices. By Jordan-Wigner, we can consider the corresponding Fermionic operator given by

$$\widehat{H}_N = \sum_{i,j}\left[c_i^{\dagger}A_{ij}c_j + \frac{1}{2}\left(c_i^{\dagger}B_{ij}c_j^{\dagger}+ H.c.\right)\right],$$

where $c_i$, $1\leq i \leq N$ are anticommuting Fermionic operators and $\left\{A_{ij}\right\},\left\{B_{ij}\right\}$, $1\leq i, j \leq N$ are the elements of appropriately chosen matrices $A, B$, which depend on $\left\{J_i\right\}_{1\leq i \leq N}$.

We use periodic boundary conditions. Now we can extend $\widehat{H}_N$ to a lattice of infinite size, by gluing the unit cell of size $N$ infinitely many times. Let us call this new extension $\tilde{H}_N$. Now the questions:

1) What is $H.c.$?

2) I am interested in the thermodynamic limit $N\rightarrow\infty$. Is it obvious whether the sequence of operators $\left\{\tilde{H}_N\right\}$ converges, say in strong operator topology, to some well-defined operator $\tilde{H}$ as $N\rightarrow\infty$?

Let me motivate the second question: For a certain sequence $\left\{J_i\right\}$, constructed deterministically with certain properties (so-called quasi-periodic sequence), I believe I can say something about what Physicists call the "energy-spectrum in the thermodynamic limit". I'm interested to know whether this energy spectrum is the spectrum (in the usual functional-analytic sense) of some operator $\tilde{H}$.

Thanks for any help!

  • $\begingroup$ H.c. stands for Hermitian conjugate, meaning the Hermitian conjugate of whatever directly precedes H.c. It's just a way to save space $\endgroup$ – Greg P May 21 '11 at 6:36
  • $\begingroup$ Thank you, Greg. I am still puzzled by the second question. It shouldn't be too difficult to check, but I want to know first whether this is something standard. $\endgroup$ – user3657 May 21 '11 at 7:12
  • $\begingroup$ The second question is really a math question rather than a physics question. $\endgroup$ – Peter Shor May 21 '11 at 9:36
  • $\begingroup$ The "thermodynamic limit" usually means that one solves the problem (in principle at least) for finite N, thus obtaining thermodynamic quantities of interest, and then lets N go to infinity. In that case one never actually discusses an infinite lattice Ising model, and rather than issues of convergence of operators (which physicists are unlikely to worry about anyway) we are just dealing with convergence of functions. $\endgroup$ – Greg P May 21 '11 at 16:26
  • $\begingroup$ If you're just concerned about the existence of a well defined $N\rightarrow\infty$ limit, I can't tell you how to prove it -- but TFIMs with quasiperiodic couplings have been studied in the literature in the past, so if you have something neat to say, don't let that concern stop you! $\endgroup$ – wsc May 21 '11 at 16:37

Well, if it is not too immodest to answer my own question by referring to my own papers, I'd like to suggest reading of the following papers, where the answer is essentially spelled out for a specific model (i.e., the sequence of interactions $\{J_i\}$ following a quasiperiodic substitution). This, I suspect, can be extended to the general case.

I'd like to note that these papers were written some months after I posed the question above.




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