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) transport Momentum 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 anomaly Expand 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/2

ε – “momentum” of chiral chemical potential

Time-dependent chemical potential:

No ground state!!!

Anomalous correlators vs VVA correlator

Spatially modulated chiral chemical potential

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

Transverse form-factor

Not 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 interactions

Anomalous 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 semimetals

Time-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 study Lattice Dirac fermions with contact interactions

Lattice Dirac Hamiltonian V>0, like charges repel

Suzuki-Trotter decomposition

Hubbard-Stratonovich transformation

A simple mean-field study Taking 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]

μA0=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/(µA2 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 CME

In 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

Conclusions

In 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 to

observe in experiment

Parity-breaking Weyl semimetals can be

simulated using Rational HMC algorithm

Outlook

Dynamical 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!!!