Aditi Mitra (New York University, NY)
Quantum quenches in one-dimension: A renormalization group approach
Blogged by Piers coleman
Aditi discusses the challenges of a "Quantum Quench". One of the most exciting areas for Quantum Quenches (QQ) is in cold atoms systems. These are highly tunable systems - one can realize model Hamiltonians and tune the interactions between the atoms. Another method is the use of ultra-fast optical pump probe methods. This allows one to probe time evolutions on femto-pico time scales, when the physics should be governed by quantum unitary time evolution. One applicaion is work by Cavalieri et al.
AM gives an outline:
Quenches in free theories - result - non equlibrium steady state.
What happens then if one adds a periodic potential. Will study case of "irrelevant operators" in eqn. SHow that they become relevant out of eqn.
Then consider interactions that are relevant in eqn. This leads to a new kind of dynamical phase transition. She considers a Luttinger liquid (spinless), and shows that in a quench that suddenly changes the Luttinger constant K0-> K, the power-laws for the density density and phase-phase correlation function acuquire a new exponent. Correlations always decay more rapidly than in equilibrium but with a new powerlaw. Interestingly, the non-equilibrium state is not a Luttinger liquid because the new powerlaws are not the inverse of one-another.
AM now explains the reasoning behind these results. One looks at the bosonization the initial and final Hamiltonian's are non-interacting bosons. The initial distribution is conserved, and the results can be understood in terms of a "Generalized Gibb's ensemble" rhoGGE.
Now what happens with some non-linearities in the system. Look at the effect of a periodic potential. The final partition function is given by
Z = Lim Tr[U(t)rhogge U\dg(t)]
When the periodic potential is there with interactions, dissipation arise. Interestingly enough, now the density density decays exponentially fast. This is because of the generated noise. More importantly, The conductivity acuires a finite value (from an infinite one).
Now AM considers a different quench protocole in which K is shifted and the lattice potential is simultaneously turned on. In this case, the slow and fast modes separate out. Now the correlation function depends on both the relative time and the "center of mass" time. At short times, power-aw in space with exponent K0, but at long times, plaw in space and time with exponent Kneq. Teh crossover between these two times is governed by ...., which means that the scaling dimensions of the lattice depend on time.
Aditi introduces the scaling equations, which he time-dependent scaling dimensions. Providing one is only interested in times tm < 1/eta. The RG reduces to just three equations. One defines a g-eff = g Sqrt[IK(Tm)] which leads us to a Kostlitz phas transition. Four cases
a. periodic pot irrelevant at all times
b. perioidic potential relevant at all times.
c. PP relevant at short, irrelevant at long times
d. PP irrelevant at short times, relevant at long.
Integrate the RG up to the point where g Cos(phi) = 1- gphi^2. There is an exactly solvable point that can be used to check the results.Case b such that geff = epsilon (Tm*) There is a non-analytic behvior in the RG, corresponding to a dynamical phase transition. Even to leading order in the potential, multi-particle scattering processed.
Now its the conclusions.
Quantum quenches in Free Theories can lead ot interesting non-equilibrium behavior.
In non-linearities, an analytic approach to study dynamics is presented that is valid in teh thermodynamic and long time limit where numerical studies are hard to do.
Even when the periodic potential is irrelevan, its effect is no-trivial as it generates dissipation and noise.
When the PP is relevant a new kind of non-equilbrium dynamical phase transition takes place - a kind of non-equiibrium Kosterlitz Thouless transition.
Q: what happens when you include a cubic term?
A: will give new results, but we have not done this yet.
Q: Is there a quantum version of the Komogorov theorem.
A: Now and you have to be very careful about this.
AM now explains the reasoning behind these results. One looks at the bosonization the initial and final Hamiltonian's are non-interacting bosons. The initial distribution is conserved, and the results can be understood in terms of a "Generalized Gibb's ensemble" rhoGGE.
Now what happens with some non-linearities in the system. Look at the effect of a periodic potential. The final partition function is given by
Z = Lim Tr[U(t)rhogge U\dg(t)]
When the periodic potential is there with interactions, dissipation arise. Interestingly enough, now the density density decays exponentially fast. This is because of the generated noise. More importantly, The conductivity acuires a finite value (from an infinite one).
Now AM considers a different quench protocole in which K is shifted and the lattice potential is simultaneously turned on. In this case, the slow and fast modes separate out. Now the correlation function depends on both the relative time and the "center of mass" time. At short times, power-aw in space with exponent K0, but at long times, plaw in space and time with exponent Kneq. Teh crossover between these two times is governed by ...., which means that the scaling dimensions of the lattice depend on time.
Aditi introduces the scaling equations, which he time-dependent scaling dimensions. Providing one is only interested in times tm < 1/eta. The RG reduces to just three equations. One defines a g-eff = g Sqrt[IK(Tm)] which leads us to a Kostlitz phas transition. Four cases
a. periodic pot irrelevant at all times
b. perioidic potential relevant at all times.
c. PP relevant at short, irrelevant at long times
d. PP irrelevant at short times, relevant at long.
Integrate the RG up to the point where g Cos(phi) = 1- gphi^2. There is an exactly solvable point that can be used to check the results.Case b such that geff = epsilon (Tm*) There is a non-analytic behvior in the RG, corresponding to a dynamical phase transition. Even to leading order in the potential, multi-particle scattering processed.
Now its the conclusions.
Quantum quenches in Free Theories can lead ot interesting non-equilibrium behavior.
In non-linearities, an analytic approach to study dynamics is presented that is valid in teh thermodynamic and long time limit where numerical studies are hard to do.
Even when the periodic potential is irrelevan, its effect is no-trivial as it generates dissipation and noise.
When the PP is relevant a new kind of non-equilbrium dynamical phase transition takes place - a kind of non-equiibrium Kosterlitz Thouless transition.
Q: what happens when you include a cubic term?
A: will give new results, but we have not done this yet.
Q: Is there a quantum version of the Komogorov theorem.
A: Now and you have to be very careful about this.
No comments:
Post a Comment