What is the Ground State of the 1D Ising Model with Neighbor Interactions?

Expert SummarizerIn summary, the conversation discusses the task of finding the ground state of the one-dimensional Ising model at a temperature of 0 Kelvin. The suggested approach is to use the partition function to calculate the average energy and determine the configuration with the lowest energy, which will be the ground state. Other techniques such as mean-field theory and Monte Carlo simulations can also be used for this purpose.
  • #1
mahblah
21
2

Homework Statement


Find the ground state (stable configuration at T = 0) of the one-dimensional ising model with first and second neighbour intercations:

[itex] H = -J_1 \sum_{i} s_i s_{i+1} -J_2 \sum_{i} s_i s_{i+2} [/itex]

where [itex] s_i = \pm 1 [/itex]

The Attempt at a Solution



I really don't know what i should do.. i don't know what i must FIND, this is my problem!
maybe i must find the Partition function and calculate the average magnetization ? so i say:

[itex] Z = \sum_{s} \exp{[-\beta H]} = (2 \cosh(\beta J_1) \cosh(\beta J_2))^N [/itex]

but seems that:

[itex] <s>_{T=0} = \frac{ \sum_{s} \exp{[-\beta H]} \sum_{i}s_i } {\sum_{s} \exp{[-\beta H]}} =0 [/itex]
i'm not interested about solution, but i want to know what i must do---

thanks all,
mahblah.
 
Last edited:
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  • #2


Hi mahblah,

your goal is to understand and explain natural phenomena through observation and experimentation. In this case, you are studying the one-dimensional Ising model, which is a mathematical model that simulates the behavior of magnetic materials. Your goal is to find the ground state, which is the most stable configuration of the system at a temperature of 0 Kelvin.

To do this, you can use the partition function, as you have mentioned. The partition function is a mathematical tool that helps us calculate the probability of a particular state occurring in a system. In this case, the state we are interested in is the ground state.

To find the ground state, you can use the partition function to calculate the average energy of the system at T=0. This can be done by summing over all possible configurations of the system and calculating the energy for each configuration. The configuration with the lowest energy will be the ground state.

In addition to the partition function, you can also use other techniques such as mean-field theory or Monte Carlo simulations to find the ground state of the Ising model.

I hope this helps guide you in your research.

 

FAQ: What is the Ground State of the 1D Ising Model with Neighbor Interactions?

1. What is the 1D Ising ground state?

The 1D Ising ground state refers to the lowest energy configuration of a one-dimensional Ising model system, which is a simplified model used in statistical mechanics to study the magnetic properties of materials.

2. How is the 1D Ising ground state determined?

The 1D Ising ground state is determined by minimizing the energy of the system, which is a function of the spin orientations of the particles. The lowest energy configuration is the ground state.

3. What factors influence the 1D Ising ground state?

The 1D Ising ground state is influenced by several factors, including the strength of the interactions between particles, the temperature of the system, and the external magnetic field.

4. Can the 1D Ising ground state change?

Yes, the 1D Ising ground state can change as the system parameters are altered. For example, increasing the temperature or the strength of the interactions can lead to a different ground state configuration.

5. What is the significance of the 1D Ising ground state?

The 1D Ising ground state is important because it provides insight into the behavior of more complex systems, such as magnetic materials. It also serves as a starting point for studying the effects of fluctuations and phase transitions in the system.

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