Blogged by Piers Coleman (See below)
Michelle Fabrizio Break-down of ergodicity in quantum phase transitions
Blogged by Piers Coleman
Michelle discusses how, in a classical phase transition, broken symmetry means a break-down of ergodicity. Question is - what happens for quantum systems. He talks about Anderson localization and Many Body localization as examples of a kind of "quantum ergodicity breakdown".
Now we are turning to photo-induced phase transitions. Here light is exciting the system. Commonly, optical pumping is regarded as a non-equilibrium, heated state. One would like to use photo-excitation to induce phase transitions from insulator to metal. This would, he said, have important technological applications.
Q: How is this something to do with ergodicity.
A: But because of an ultra-fast laser, one can heat very rapidly. Pumping is not heating - but perhaps ergodicity is broken - so we can't think of it as heating. There isn't enough time to explore all phase space - so is it really true that one explores typical areas of phase space.
Now he defines a kind of Lanzcos procedure for the operation of H on a starting state. This generates a set of states |psi_n> with parameters t_n and E_n, this produces a tight-binding Hamiltonian, and the quantum evolution of the initial wavefunction is equivalent to a particle that starts at time t=0 from the state 0 of the chain. The underlying dimensionality is hidden in the correlation of the various parameters.
The quantum evolution of |Psi_0> is not ergodic if a particle that starts at the intial site of the Lanczos chain does not propagate to the end. (Becomes localized, stops defusing.)
Natan Andrei points out that you don't need full ergodicity -but what you need is "typicality" - splitting up into sections that are somehow typical of the entire Hilbert space.
So - next - a case study - the Bose Hubbard model, hopping of bosons in a 2D lattice. Experiments on cold atoms trapped in optical lattices offer an opportunity to explore the ergodicity issue experimentally. In a recent expt, Trotkzky et al used Rb atoms in a 1D optical lattice, and let them propagate from their intiial density cofiguration - does the system equilibrate to uniform denity? Indeed, they found that the density did rapidly equilibrate, and the result were consistent with a density functional renormalization calcn.
But! The Bose Hubbard odel in 1D has a Mott transition for integer filling and large enough U/J. Worth considering the case of n=1, rather than 1/2. So to examine this case, Fabrizzio and collaborators did a numerical simulation with 2-0-2-0 inital configuration. He says that they found that for large U, the denisty remains inhomogeneous for extremely long times - the system does not relax.. the density kind of oscillated between 2 and 0 without any clear relaxation. So they decided to define a relaxation time - n_odd(t_R)=1. They found a sharp dynamical crossover at Uf~ 4.5 - a genuine transition though? Above this value of interaction, the system is trapped into an inhomogenious state.
So to examine this using the protocol MF and co. used the Lanczos chain approach, with up to 1k sites. For the U=2K - the ti are smooth, as are the energies, but for large U there is an oscillating component to the "potential" that lasts out to 300 or so sites. So does the particle propagate along the chain? This depends on U/J. For small U, the state is able to move to the end and is reflected back and so on, but for U = 10J, he says, the R(t)/Rmax never exceeds 0.2 - the state is localized.
Natan Andrei comments that this is related to his talk yesterday - that in a Leib Linniger model, you get localization for large interaction strength.
MF says that the simplest explanation of this situation can be made in terms of an effective spin chain description - from a kind of Schrieffer Wolff transformation. up spin is doubly occupied, down spin is empty. The singly occupied states are much lower energy and can be integrated out.
H* = 2 J^2/U sum [-8 S_z(i)S_z(j)+ (S+(i)S-(j) + h.c)]
For large enough U, you stay in an AFM. (Blogger didn't understand).
Further evidence of glassy-like dynamical transition, Study the correlation function
C(t) = (1/L)\sum i=1^L <delta n_i delta n_j>
One defines tau_R as the first time where C(t_R) =0, then one finds a sharp increase in tau_R. To explore this further, MF and co have applied variational Monde Carlo and found evidence of a dynamical transition around U~ 4.5.
MF ends with a speculation: he argues that there has to be a mobility edge separating low energy eigenstates of H, which are translationally invariant, from high energy eigenstates that are not translationally invariant. He argues that the dynamical transition will occur when E^*~ N.
A very nice talk. What is not clear is the relation between this and many body localization. It would appear to be quite similar (bloggers state).
Yu Lu states that it appears that you are making a description of the Upper Hubbard band. You are providing a snap shot of this.