Towards kinetic theory with anomalies

Piotr SurówkaTheoretische Natuurkunde, Vrije Universiteit Brussel

and the International Solvay Institutes

P and CP odd effects in hot dense matterBNL June 26, 2012

(Based on PS and R. Loganayagam arXiv:1201.2812)

Outline

Motivation

Hydrodynamics and kinetic theory

Anomaly and transport in 2d Weyl gas

Generalization to arbitrary dimensions

Berry phase effects in non-relativistic systems

Summary and outlook

Hydrodynamics is a very universal effective field theory used to

describe heavy-ion collisions and condensed matter systems

Violation of P and CP symmetry in QGP

The right description of parity-odd hydrodynamics requires inclusion of gauge and gravitational anomalies

Kinetic theory is a complementary semi-classical description of weakly coupled hydrodynamics

Where do anomalies appear in kinetic theory?

Motivation

Nucleus-Nucleus collision and hydro

Chiral magnetic effect

Chiral vortical effectStrong magnetic field and vortices in QGP

Kinetic theoryKinetic theory treats the evolution of the one-particle distribution function, which can be associated with the number of on-shell particles per unit phase space

If collisions between particles can be neglected and there is no Berry phase effects, the evolution of follows from Liouville’s theorem

Given this interpreation the particle number density should be proportional to

Summing instead with a weight of particle energy, one expects a result proportional to the product of number density and energy, or energy density, which is a part of the energy-momentum tensor.

Hydrodynamics ⇔ Kinetic theoryWe can derive hydrodynamic quantities from kinetic theory e.g.

If we take the distribution function in equilibrium we recover energy- momentum tensor of a perfect fluid. One can derive the correspondence between kinetic theory out of equilibrium and viscous hydrodynamics by considering small departures from equilibrium where

This procedure allows one to study dissipative effects (first order in the derivatives of fields). Performing the integral one gets perfect fluid contribution plus shear tensor

Weyl fermion in 2dWhat is the hydrodynamic description of such an ideal gas? Naively we have

However, a free Weyl fermion is a holomorphic 2d CFT and hence only the holomorphic components of the currents can be non-zero. The above relations are in clear contradiction with holomorphy - the charge/entropy currents are time-like rather than null as would be predicted by holomorphy.

We need correct the above expressions to recover the required properties of 2d CFT. We do that by populating solutions of the Dirac equation by means of kinetic theory

Anomalous partSolving the Weyl equation we obtain

Populating these states leads to anomalous correction to hydrodynamics

Gibbs currentThe above anomalous quantities can be generated from

where and we used Hodge duals for simplicity.

We have to evaluate one thermal integral to get

Crucial observation : the anomalous contribution is completely proportional to the U(1) anomaly coefficient and the Lorentz anomaly coefficient

Anomaly polynomialsThe anomaly coefficients of a system are summarized by a polynomial in gauge field strength and spacetime curvature:

Using this we can write a rule to get from the anomaly polynomial to the anomaly induced Gibbs current

Motivated by this result we can generalize the Gibbs current to higher dimensions introducing concept of chiral spectral current, repeat the analysis and match to hydrodynamics

Chiral spectral currentTo determine chiral spectral current we will use adiabaticity in the position and energy space

The Current in the energy direction is the sum of the electric force and pseudo force

Using insight from adiabaticity in hydrodynamics and thermodynamic relations we can solve the above equation. The Hodge dual reads

Berry phaseConsider a physical system described by a Hamiltonian that depends on time through a set of parameters

Insering the above expression to the Schrodinger equation and multiplying by bra one finds that the phase factor can be expressed as a path integral in the parameter space

where we have defined the Berry connection. We see that in addition to a dynamical phase quantum state will acquire an additional phase during the adiabatic evolution along closed contour.

Gas of fermionsConsider a gas of non-relativistic fermions with a Berry curvature on the fermi surface in the presence of electromagnetic field. The lagrangian of such system is given by:

We can derive the EOMs

where we have defined the Berry curvature

We see the so-called anomalous Karplus-Luttinger contribution to velocity

Density of statesLiouvile theorem guarantees constatnt density of states in the classical systems. This is not the case in the presence of the Berry phase and magnetic field. Let us calculate the dynamics of the volume element

We can solve the above equation

The fact that the Berry curvature is generally momentum dependent and the magnetic field is position dependent implies that the phase-space volume changes during time evolution. We can introduce a modified density of states

such that

Summary

QFT anomalies lead to hydrodynamic transport

New transport coefficients calculated by linear responce theory

Anomalous transport leads to modified density of states and the emergence of chiral spectral current linked to a Dirac monopole in the momentum space. Possible relation with Berry phase proven for magnetic field part in Landau’s Fermi Liquid by Son and Yamamoto

Calculation of Gibbs free energy which was expressed in terms of anomaly polynomials

Future goals

Boltzmann operator with anomalies

Berry phase derivation of chiral vortical effect

Check of the prescribed conjecture for anomaly polynomials by means of AdS/CFT and Kubo formulae

Wigner formulation of QM with anomalies

Applictions to CMT systems such as Weyl-semi metals