Ground state of the one-dimensional spin-1/2 Ising model

In summary, the conversation discusses the ground state of the spin-1/2 Ising model and the process of deriving it from the spin hamiltonian without solving the system. The speaker mentions attempting to compute the derivative of the hamiltonian with respect to the spin variable, but ultimately finding the solution by using the minimum of the hamiltonian.
  • #1
William Crawford
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TL;DR Summary
How to derive the low energy ground state for the one-dimensional spin-1/2 Ising model on either a periodic or an infinite chain.
Hi,

I know that the ground state of the spin-1/2 Ising model is the ordered phase (either all spin up or all spin down). But how do I actually go about deriving this from say the one-dimensional spin hamiltonian itself, without having to solve system i.e. finding the partition function? $$ \mathcal{H} = -J\sum_n s_{n}s_{n+1}, \qquad s_n=\pm1 $$ I've tried computing the derivative of ## \mathcal{H} ## w.r.t. the spin variable ## s_i ##, but this leaves me with the trivial difference equation ## s_n + s_{n+1} = 0 ## yielding the high energy solution ## s_n = (-1)^ns_0 ## and not the low energy solution that I was searching for (assuming ##J>0##).
 
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  • #2
Never mind, I've solved it myself. Simply using that
$$ \min_{\lbrace s_n\rbrace}\mathcal{H} = \sum_n\min\left(-Js_ns_{n+1}\right). $$
 

Related to Ground state of the one-dimensional spin-1/2 Ising model

1. What is the ground state of the one-dimensional spin-1/2 Ising model?

The ground state of the one-dimensional spin-1/2 Ising model refers to the lowest energy state of the system, where all the spins are aligned in a specific direction. This state is also known as the ferromagnetic state.

2. How is the ground state determined in the one-dimensional spin-1/2 Ising model?

The ground state is determined by minimizing the energy of the system using mathematical techniques such as the mean-field approximation or Monte Carlo simulations. The ground state can also be determined experimentally by measuring the magnetic properties of the system.

3. What factors affect the ground state of the one-dimensional spin-1/2 Ising model?

The ground state of the one-dimensional spin-1/2 Ising model is affected by the strength of the interaction between the spins, the temperature of the system, and the external magnetic field. These factors can cause the ground state to change from ferromagnetic to paramagnetic or even to a spin glass state.

4. Can the ground state of the one-dimensional spin-1/2 Ising model be changed?

Yes, the ground state of the one-dimensional spin-1/2 Ising model can be changed by altering the external conditions such as the temperature or the external magnetic field. It can also be changed by introducing defects or impurities in the system.

5. How does the ground state of the one-dimensional spin-1/2 Ising model relate to real-world systems?

The one-dimensional spin-1/2 Ising model is a simplified version of more complex systems, such as magnetic materials. The ground state of this model can provide insights into the behavior of real-world systems and can be used to make predictions about their properties. However, it is important to note that the one-dimensional model does not capture all the complexities of real-world systems.

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