Thank you for the nice lecture, Dr. Mitchell. I might be wrong, but I think at the 46:37 mark, DeltaE = Eex -Egs should equal -4J. Not 4J. Since J is negative and the excited state must be higher in energy than the ground state, DeltaE should not be a negative number.
Thank you -- yes you are correct. Thanks for pointing out the typo. The excitation energy above the ground state should be positive by definition! I guess I meant Delta E = 4|J|
The energy E=NJ/4 (at 10.49) where "N is number of sites". I believe the statement should be "where N is number of pairs". Please correct me if I am wrong.
We sum over all i, which gives N, each pair is identified by the position of the first spin in the pair uniquely, so call it whatever it's really the same.
Eigenstates of 1D Ising model are product states. "We can therefore expect this ising model will essentially give us classical physics" well, given a separable initial state (without entanglement), the state can still be entangled as time evolves, no? In what sense is "their physics, classical" ? besides the obvious fact that energy eigen states are separable? oh... I think I got it... the partition function of the quantum one would be the same as the classical one.
Here by "like classical" I just mean that the eigenstates are product states and therefore the constituents are not entangled. If you prepare an initial state that is not an eigenstate of H and is entangled, then it will stay entangled but its time evolution controlled by the Hamiltonian will be separable.
I feel like one needs to be very careful in drawing conclusions from quantum-to-classical correspondence. For instance, one can create a naive argument why 2D classical ising model is equivalent to 1D classical ising model (which is clearly false because 2D classical ising model has phase transition. Classical 1D, doesn't.). Here is the naive argument: 2D classical ising model is "in some sense equivalent" to some 1D quantum ising model.. and 1D quantum ising model is truly equivalent to 1D classical ising model because they have the same partition function due to separability of energy eigenstates. Therefore 2D classical ising model is "in some sense equivalent" to some 1D classical ising model. I wonder where dose the argument go wrong. The corresponding 2D classical system is finite, so there is no phase transition in such finite system? The temperatures are different so the 1D classical system only maps to the non-ferromagnet temperature region of the 2D classical system? Either way, like I said, I feel like one needs to be very careful in drawing conclusions from quantum-to-classical correspondence.
No not quite. The 2d classical Ising model is equivalent to the 1d quantum Ising model with a transverse field. This means that the eigenstates of the quantum model are not separable product states but entangled. The partition function of this model is not equivalent to the classical 1d model, and the classical 1d model is not equivalent To the classical 2d model! The intuition behind the mapping, as mentioned in the lecture, is that the quantum spin fluctuations in *time* caused by the transverse field can be interpreted as classical fluctuations in *space*.
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Hi professor, thank you very much for these valuable lectures!
Thank you for the nice lecture, Dr. Mitchell. I might be wrong, but I think at the 46:37 mark, DeltaE = Eex -Egs should equal -4J. Not 4J. Since J is negative and the excited state must be higher in energy than the ground state, DeltaE should not be a negative number.
Thank you -- yes you are correct. Thanks for pointing out the typo. The excitation energy above the ground state should be positive by definition! I guess I meant Delta E = 4|J|
very valuable lecture
The energy E=NJ/4 (at 10.49) where "N is number of sites". I believe the statement should be "where N is number of pairs". Please correct me if I am wrong.
We sum over all i, which gives N, each pair is identified by the position of the first spin in the pair uniquely, so call it whatever it's really the same.
Eigenstates of 1D Ising model are product states. "We can therefore expect this ising model will essentially give us classical physics"
well, given a separable initial state (without entanglement), the state can still be entangled as time evolves, no? In what sense is "their physics, classical" ? besides the obvious fact that energy eigen states are separable?
oh... I think I got it... the partition function of the quantum one would be the same as the classical one.
Here by "like classical" I just mean that the eigenstates are product states and therefore the constituents are not entangled. If you prepare an initial state that is not an eigenstate of H and is entangled, then it will stay entangled but its time evolution controlled by the Hamiltonian will be separable.
I feel like one needs to be very careful in drawing conclusions from quantum-to-classical correspondence. For instance, one can create a naive argument why 2D classical ising model is equivalent to 1D classical ising model (which is clearly false because 2D classical ising model has phase transition. Classical 1D, doesn't.).
Here is the naive argument:
2D classical ising model is "in some sense equivalent" to some 1D quantum ising model.. and 1D quantum ising model is truly equivalent to 1D classical ising model because they have the same partition function due to separability of energy eigenstates. Therefore 2D classical ising model is "in some sense equivalent" to some 1D classical ising model.
I wonder where dose the argument go wrong. The corresponding 2D classical system is finite, so there is no phase transition in such finite system? The temperatures are different so the 1D classical system only maps to the non-ferromagnet temperature region of the 2D classical system?
Either way, like I said, I feel like one needs to be very careful in drawing conclusions from quantum-to-classical correspondence.
No not quite. The 2d classical Ising model is equivalent to the 1d quantum Ising model with a transverse field. This means that the eigenstates of the quantum model are not separable product states but entangled. The partition function of this model is not equivalent to the classical 1d model, and the classical 1d model is not equivalent
To the classical 2d model! The intuition behind the mapping, as mentioned in the lecture, is that the quantum spin fluctuations in *time* caused by the transverse field can be interpreted as classical fluctuations in *space*.