Does a temperature dependent Hamiltonian make sense?
At the fundamental level, we think of a Hamiltonian as independent of temperature. It is describing the energy of all possible states of the system in the absence of any environment.
However, when one does mean-field theory (e.g. for an Ising model or BCS theory) the Hamiltonian involves temperature-dependent parameters that are determined self consistently.
I have been thinking about this because one of the proposed effective minimal Hamiltonians for spin crossover compounds is an Ising model with a temperature dependent field.
My immediate reaction was that this must be some sort of mean-field theory.
However, I now realise that is not the case.
Effective Hamiltonians can be temperature dependent without invoking any approximations. Temperature-dependent interactions can arise when one integrates out some degrees of freedom.
One can see this by simply considering the case of a system with two degrees of freedom x and q. The partition function can be written as a path integral where there is an action which involves the integral of the Lagrangian in imaginary time from 0 to 1/T where T is the temperature.
Integrating out x one obtains an effective action for q that will depend on temperature.
Here are three cases where this can be done explicitly.
1. The spin boson model. One integrates out the harmonic oscillators, leading to a ``Feynman-Vernon influence functional'' that is temperature dependent.
2. A two-state system in which each state has a series of sub-states (e.g. spin states or vibrational states). Consider the simple Hamiltonian.
This corresponds to the case of spin-crossover systems and one sees how one can end up with an Ising type model with a "field" that is related to the free energy difference between the two spin states.
3. A one-dimensional chain of spin-crossover molecules which have an elastic interaction that depends on the spin state. This is treated in
Elastic interaction among transition metals in one-dimensional spin-crossover solids
K. Boukheddaden, S. Miyashita, and M. Nishino
The classical phonons are integrated out and one is left with an Ising chain of pseudo-spins in an external ``field'' where the "exchange" interaction and field depend on temperature.
[See equation (13) in the paper].
However, when one does mean-field theory (e.g. for an Ising model or BCS theory) the Hamiltonian involves temperature-dependent parameters that are determined self consistently.
I have been thinking about this because one of the proposed effective minimal Hamiltonians for spin crossover compounds is an Ising model with a temperature dependent field.
However, I now realise that is not the case.
Effective Hamiltonians can be temperature dependent without invoking any approximations. Temperature-dependent interactions can arise when one integrates out some degrees of freedom.
One can see this by simply considering the case of a system with two degrees of freedom x and q. The partition function can be written as a path integral where there is an action which involves the integral of the Lagrangian in imaginary time from 0 to 1/T where T is the temperature.
Integrating out x one obtains an effective action for q that will depend on temperature.
Here are three cases where this can be done explicitly.
1. The spin boson model. One integrates out the harmonic oscillators, leading to a ``Feynman-Vernon influence functional'' that is temperature dependent.
2. A two-state system in which each state has a series of sub-states (e.g. spin states or vibrational states). Consider the simple Hamiltonian.
This corresponds to the case of spin-crossover systems and one sees how one can end up with an Ising type model with a "field" that is related to the free energy difference between the two spin states.
3. A one-dimensional chain of spin-crossover molecules which have an elastic interaction that depends on the spin state. This is treated in
Elastic interaction among transition metals in one-dimensional spin-crossover solids
K. Boukheddaden, S. Miyashita, and M. Nishino
The classical phonons are integrated out and one is left with an Ising chain of pseudo-spins in an external ``field'' where the "exchange" interaction and field depend on temperature.
[See equation (13) in the paper].
What is the proper way to deal with decoherence in the systems? Once temperature is invoked the coherence properties of the density matrix and the entanglement should show it
ReplyDeleteFor the classical Ising model, you get a temperature dependent Hamiltonian (without any approximation), when going from the Hamiltonian of the spins to the Hamiltonian of the continuous variable \phi. The mean-field approximation then gets you the standard Landau-Ginzburg picture.
ReplyDeleteAdam,
DeleteThanks for pointing this out. I had forgotten this example. Many years ago I found the corresponding derivation in the text by Negele and Orland illuminating.
An other example of a temperature dependent Hamiltonian comes from dimensional reduction. In this case one integrates out nonzero Matsubara modes. For example in QED this leads to an effective Lagrangian with massless magnetic fields and massive electric fields, with the mass proprtional to eT coming from the interaction with the now integrated out fermions. This reduction can only be done perturbatively, but the exact effective Hamiltonian would of course still be temperature dependent in this case.
ReplyDelete