Abstract
In many quantum quench experiments involving cold atom systems the post-quench phase can be described by a quantum field theory of free scalars or fermions, typically in a box or in an external potential. We will study mass quench of free scalars in arbitrary spatial dimensions d with particular emphasis on the rate of relaxation to equilibrium. Local correlators expectedly equilibrate to GGE; for quench to zero mass, interestingly the rate of approach to equilibrium is exponential or power law depending on whether d is odd or even respectively. For quench to non-zero mass, the correlators relax to equilibrium by a cosine-modulated power law, for all spatial dimensions d, even or odd. We briefly discuss generalization to O(N ) models.
Highlights
Introduction and summaryThermalization in integrable systems has been an important area of study for more than a decade [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
We explicitly find that expectation values and local correlators in the most general generalized Calabrese-Cardy (gCC) state evolve at long times to their respective values in an appropriate generalized Gibbs ensemble (GGE)
We will come back to an analysis of the time-independent part in section 4 where we show that this part exactly matches the corresponding two-point function in a GGE
Summary
Thermalization in integrable systems has been an important area of study for more than a decade [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. The question of interest in the above studies is, whether local correlators in a time-dependent state relax, in the infinite time limit, to their values in a GGE We will call this “thermalization” in a generalized sense. The main result we find is that, for a quench to zero mass, correlators relax to their GGE values as an exponential (∼ e−γt) or as a power law ∼ 1/tp, depending on whether the number of space dimensions is odd or even, respectively; we call this result the odd-even effect. This constitutes a proof of thermalization of arbitrary local correlators (section 4.2).
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