Universal adiabatic dynamics across a quantum critical point Anatoli Polkovnikov, Boston University

Consider slow tuning of a system through a quantum critical point. tuning parameter tuning parameter gap    t,   0   t,   0 Gap vanishes at the transition. No true adiabatic limit! How does the number of excitations scale with  ? This question is valid for isolated systems with stable excitations: conserved quantities, topological excitations, integrable models.

Use a general many-body perturbation theory. Expand the wave-function in many-body basis. Substitute into Schrödinger equation.

Uniform system: can characterize excitations by momentum: Use scaling relations: Find:

Caveats: 1.Need to check convergence of integrals (no cutoff dependence) Scaling fails in high dimensions. 2.Implicit assumption in derivation: small density of excitations does not change much the matrix element to create other excitations. 3.The probabilities of isolated excitations: should be smaller than one. Otherwise need to solve Landau- Zener problem. The scaling argument gives that they are of the order of one. Thus the scaling is not affected.

Simple derivation of scaling (similar to Kibble-Zurek mechanism): Breakdown of adiabaticity: From   t and we get  In a non-uniform system we find in a similar manner:

Example: transverse field Ising model. There is a phase transition at g=1. This problem can be exactly solved using Jordan-Wigner transformation:

Spectrum: Critical exponents: z= =1  d /(z +1)=1/2. Correct result (J. Dziarmaga 2005): Other possible applications: quantum phase transitions in cold atoms, adiabatic quantum computations, etc.