Anomalous transport on the lattice

Pavel Buividovich(Regensburg)

Why anomalous transport?Collective motion of chiral fermions• High-energy physics:

Quark-gluon plasma Hadronic matter Leptons/neutrinos in Early Universe

• Condensed matter physics: Weyl semimetals Topological insulators Liquid Helium [G. Volovik]

Weyl semimetals: “3D graphene”

No mass term for Weyl fermions

Weyl points survive ChSB!!!

[Pyrochlore iridate]

Anomalous (P/T-odd) transportMomentum shift of Weyl points: Anomalous Hall Effect

Energy shift of Weyl points:

Chiral Magnetic Effect[Experiment ZrTe5:

1412.6543]Also: Chiral Vortical Effect, Axial Magnetic

Effect…Chiral Magnetic Conductivity and Kubo relations

T-invariace Ground-state transport???

MEM

Bloch theorem

?

CME and axial anomalyExpand current-current correlators in μA:

VVA correlators in some special kinematics!!!

The only scale is µ

k3 >> µ !!!

CME and axial anomaly

Difference between the gauge-invariant and non-invariant results: “surface” Chern-Simons term

General decomposition of VVA correlator

• 4 independent form-factors • Only wL is constrained by axial WIs

[M. Knecht et al., hep-ph/0311100]

Anomalous correlators vs VVA correlator

CME: p = (0,0,0,k3), q=(0,0,0,-k3), µ=1, ν=2, ρ=0

IR SINGULARITY

Regularization: p = k + ε/2, q = -k+ε/2ε – “momentum” of chiral chemical potential

Time-dependent chemical potential:

No ground state!!!

Anomalous correlators vs VVA correlatorSpatially modulated chiral chemical potential

By virtue of Bose symmetry, only w(+)(k2,k2,0)

Transverse form-factorNot fixed by the anomaly[Buividovich 1312.1843]

CME and axial anomaly (continued)In addition to anomaly non-renormalization,new (perturbative!!!) non-renormalization theorems[M. Knecht et al., hep-ph/0311100] [A. Vainstein, hep-ph/0212231]:

Valid only for massless fermions!!

CME and axial anomaly (continued)Special limit: p2=q2

Six equations for four unknowns… Solution:

Might be subject to NP corrections due to ChSB!!!

Anomalous transport and interactionsAnomalous transport coefficients:• Related to axial anomaly• Do not receive corrections IF• Screening length finite

[Jensen,Banerjee,…]• Well-defined Fermi-surface [Son,

Stephanov…]• No Abelian gauge fields

[Jensen,Kovtun…]

In Weyl semimetals with μA/ induced mass:• Screening length is zero (Goldstones?)• Electric charges STRONGLY interact• Non-Fermi-liquid [Buividovich’13]

Interacting Weyl semimetalsTime-reversal breaking WSM:

• Axion strings [Wang, Zhang’13]• RG analysis: Spatially modulated chiral condensate [Maciejko, Nandkishore’13]• Spontaneous Parity Breaking [Sekine,

Nomura’13]

Parity-breaking WSM: not so clean and not well studied… Only PNJL/σ-model QCD studies

• Chiral chemical potential μA: • Dynamics!!!• Circularly polarized laser• … But also decays dynamically??? [Akamatsu,Yamamoto,…]

[Fukushima, Ruggieri, Gatto’11]

Interacting Weyl semimetals + μA

Dynamical equilibrium / Slow decay

A simple mean-field studyLattice Dirac fermions with contact interactions

Lattice Dirac HamiltonianV>0, like charges repelSuzuki-Trotter decomposition

Hubbard-Stratonovich transformation

A simple mean-field studyTaking everything together…

Partition function of free fermions with

one-particle hamiltonian

Action of the Hubbard field

Possible homogeneous

condensates (assume unbroken Lorentz symmetry)

Linear response and mean-field

External perturbation

change the condensate

CME and vector/pseudo-vector “mesons”

Vector meson propagator

CME response:

Meson mixing with μA (kz ≠ 0)

ρ-mesons

Pseudovector mesons

Effect of interactions on CME:Continuum Dirac fermions, cutoff

reg.[Buividovich, 1408.4573]μA

0=0

μA0=0.2

• μA shifts spontaneous chiral symmetry breaking to smaller V

• μA is enhanced by interactions• Miransky scaling of chiral condensate at

small V Meff ~ Exp[-A/(µA

2 V)]

CME response: explicit calculation

Green = μAk/(2 π2)“Conserved” currents!!!

Chiral magnetic conductivity vs. V

Chiral magnetic conductivity vs. V(rescaled by µA)

Effect of interactions on CME: Wilson-Dirac

• SChSB is replaced with spontaneous parity breaking

Axionic insulator or Aoki phase • Phase transitions are still lowered by µA

• µA is still enhanced by (repulsive) interactions• No more Miransky scaling, 2nd order phase trans.

[Buividovich, Puhr 1410.6704]

Effect of interactions on CME: Wilson-Dirac fermions

Effect of interactions on CME:Wilson-Dirac lattice fermions

Still strong enhancement of CMEIn the vicinity of phase transition

Weyl semimetals+μA : no sign problem!• One flavor of Wilson-Dirac fermions

• Instantaneous interactions (relevant for condmat)

• Time-reversal invariance: no magnetic interactions

Kramers degeneracy in spectrum: • Complex conjugate pairs• Paired real eigenvalues• External magnetic field causes sign

problem!

• Determinant is always positive!!!• Chiral chemical potential: still T-

invariance!!!• Simulations possible with Rational HMC

Weyl semimetals: no sign problem!Wilson-Dirac with chiral chemical potential:

• No chiral symmetry• No unique way to introduce μA

• Save as many symmetries as possible [Yamamoto‘10]

Counting Zitterbewegung,not worldline wrapping

ConclusionsIn many physically interesting situations,anomalous transport coefficients receive nontrivial corrections due to interactions

CME and chiral imbalance strongly enhanced if chiral symmetry or parity are spontaneously broken should be easier toobserve in experiment

Parity-breaking Weyl semimetals can be simulated using Rational HMC algorithm

OutlookDynamical stability of chirally imbalanced matter? “Chiral plasma instability” scenario?[Akamatsu, Yamamoto’12, Zamaklar’11]

Real-time dynamics of “chirality pumping”?Effect of boundaries?

Chirally symmetric lattice fermions with chiral chemical potential [See also the poster by Matthias Puhr]

Thank you for your attention!!!