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Even though the explicit commutator you wrote is wrong--you should not have conjugated $\Pi$ in the second term-- your conclusion is sound that you cannot possibly satisfy the Born-Heisenberg commutation relation with 2x2 matrices. In fact, there is a general Theorem: The Heisenberg algebra does not admit faithful finite-dimensional (matrix) representations....

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Yes, yes and yes. If you cool a spin glass rapidly (or, more likely to happen in a computer model: start it in a completely random state, which corresponds to infinite temperature, and immediately set the temperature to a low value) then it will indeed end up frozen into a higher-energy state than the ground state. This is known as "quenching." There would ...

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Perhaps this isn't quite the answer you're seeking, but you may be interested in the phenomena of thermoremanent and isothermal remanent magnetization. Basically, if you perform a deep quench on a spin glass (i.e., freeze the spins) in a uniform external magnetic field and then, after some time, switch off the magnetic field, (or alternatively, quench ...

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In spin liquids, the ordered state is broken by zero-point fluctuations even at $T=0$. Even though it is common for spin liquids to be frustrated, it is not necessarily so. The $S=1$ Heisenberg spin chain (AFM), for example, is a spin liquid without being frustrated. The name spin liquid comes (I believe), from the exponentially decaying correlation (like ...

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I think you have the wrong idea when you ask how specific heat is "defined". In computational physics, the starting point is an experimental measurement that one could measure, or at least, a physical quantity that one might care about ... and then the question is, "how do I compute it?" The wrong approach is to have in mind a certain formula. You should ...

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Regardless of the system, Cv will be proportional to the variance of energy. If you have peaks at higher energies, that will increase its value. But at high enough energies the occupation of those states will be so low they won't significantly affect the variance. In this case the variance of the distribution isn't just the width² of one of the peaks, you ...

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One could interpret the update steps as possible discrete steps in a fictitious time and in that case the transitions represent dynamics on the state space of a Markov chain. As an example, there is the relaxational non-conservative Glauber dynamics and the magnetization conserving Kawasaki dynamics which are used to simulate Ising and related systems. The ...

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Try to look at Introduction to the Replica Theory of Disordered Statistical Systems by V. Dotsenko. In the following, I've written a possible answer to your question: $$f=-\lim_{N\rightarrow\infty}\frac{1}{\beta N}\mathbb{E}\left[\ln Z_{J}\right]$$ where: \$\mathbb{E}\left[\mathcal{O}\right]=\left(\prod_{\left\{ i,j\right\} }\...

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