Lattice gauge theory treatment ofelectron correlation in Dirac semimetals
ディラック半金属における電子相関と格子ゲージ理論

Yasufumi Araki 1,2
荒木 康史

Kentaro Nomura 1
野村 健太郎1 Institute for Materials Research, Tohoku Univ.

東北大学 金属材料研究所
2 Frontier Research Institute for Interdisciplinary Sciences, Tohoku Univ.

東北大学 学際科学フロンティア研究所

Paper in preparation
基研研究会「熱場の量子論とその応用」
京都大学 基礎物理学研究所 - Aug. 31, 2015

IMR and FRIS

金属材料研究所 (IMR)

学際科学フロンティア研究所 (FRIS)Dirac semimetals
Dirac band structure realized in electronic spectrum:- gapless, band touching at “Dirac points”- protected by topology, crystalline symmetry, ...

Ekz

2Dgraphene (sheet of carbon atoms)
surface states of topological insulators (gapped in the bulk)
3D counterpart...?

k x, ky(“3D analogue of graphene”)

Theoretical prediction: BiO2 Young et al. PRL 108, 140405 (2012)
Experimentally realized recently.
Na3Bi

Liu et al. Science 343, 864 (2014)

Cd3As2

Neupane et al. Nat. Commun. 5, 3786 (2014)

Many-body effects?
Electron-electron interaction
electron correlation can change the Dirac spctrum drastically...
Dynamical mass generation (excitonic instability, chiSB)
Renormalization of Fermi velocity (absence of Lorentz invariance)
Dynamical screeninge.g.) graphene: vF renormalized at charge neutrality.

Yu et al. PNAS 110, 3282 (2013)

Many-body effects in 3D Dirac semimetals?
Change in the spectrum
turn into various phases...
e.g.) normal insulator, topological insulator,
Weyl semimetal, axionic insulator, ...

Important as a platform for many novel states. Liu et al. Science 343, 864 (2014)

Analysis on many-body effects
Choice of interaction:
Lorentz non-invariant QED
phenomenological potentialRG, self-consistent eqn.,
Lattice Monte Carlo, ... etc.
In 3D Dirac semimetals...

coupling strength g

How to investigate many-body effects?

number of d.o.f.s N■ RPA(large-N) analysis Hofmann et al. PRB 92, 045104 (2015)■ From weak coupling... self-consistent eqn. Gonzalez, arXiv:1502.07640
Throckmorton et al., arXiv:1505.05154
RG analysis■ From strong coupling... strong coupling expansion of lattice QED
Improve

Sekine & Nomura, JPSJ 83, 094710 (2014); PRB 90, 075137 (2014)
w/ hypothetical lattice models.

This work: Lattice QED analysis with realistic d.o.f.s/symmetries.
connect the analysis to continuum limit.

Lattice action for fermions
Lattice Hamiltonian:background charge(μbg→∞)
tight-binding Hamiltonian

Eigenvalue:

Path integral formalism:
Split the imaginary time β into Nτ steps: aτLeave aτ dependence (aτ → 0 is required.)around Dirac points kD

with forward differencecf.) A similar form appears in Astrakhantsev et al., arXiv:1506.00026

Symmetries of the Hamiltonian [e.g. spin SU(2)] is preserved.
e.g.) spin SU(2) is broken to U(1) in staggered fermion formalism.

Coupling to gauge fieldContinuum:
Scalar potential A0: mediates 1/r Coulomb interaction.
vF≪c: neglect the retardation (A1,2,3) effect.

e.g.) Na3Bi: vF/c ~ 10-3

: local charge density

Lattice:
Define the gauge variable

on each site.
(not on the links)

The total lattice action becomes

Take the “renormalized” couplingas an expansion parameter.

Strong coupling expansion(i) Expand around the strong coupling limit gR-2=0(ii) Integrate out the gauge degrees of freedom U0
Derive the effective action for fermions: Seff

Strong coupling limit [gR-2=0]:
Propagation of “photon” is suppressed.
Correlates charge density at the same site.

NLO correction [O(gR-2)]:
SG carries “photon” to a neighboring site.
Correlates charge densities between nearest neighboring sites.

Strong coupling limit
On-site repulsion is generated in the strong coupling limit:

Going back to Hamiltonian formalism:

Repulsive Hubbard interaction: U = 1/4aτ → ∞ in the continuum limit.
requires local charge neutrality:
i.e. “one electron per one site”

The system becomes Mott insulator in the continuum limit (aτ→0).

Mean-field analysis
On-site repulsion:
Introduce bosonic field(local spin polarization)(Hubbard-Stratonovich)

Mean field: antiferromagnetic (Neel) ordere.g.) On the cubic lattice:

T=0(t: hopping amplitude)

Integrate out the fermionic d.o.f.s:
Continuousaτ→0: Mott insulator (Dirac mass generation)

DiscreteDimensionality dependence
Introduce anisotropy in the hopping amplitude: tz/t⊥ = νe.g.) Na3Bi: ν ~ 0.25

ν = 1: 3D cubic symmetryν = 0: stacked 2D (e.g. graphene)
Observe the 3D-2D crossover at finite aτ ~ O(a).
2aτt⊥ = 0.3, T = 0t⊥
tz

2D3D

ν = tz/t⊥
Consistent with previous calculation with hypothetical staggered fermion.
Sekine & Nomura, PRB 90, 075137 (2014)

“Phase diagram”
Discrete timeContinuous time3D(T=0)
Diracsemimetal

ν = tz/t⊥Mott insulator(antiferromagnet)

2D2aτt⊥Finite aτ ~ O(a): consistent with the “staggered fermion” results.

aτ → 0 limit: local charge neutrality (Mott insulator) is required.
- connected by aτ-dependent formalism.

Away from strong coupling limit
Nearest-neighbor repulsion is generated in the next-to-leading order:(assume n ~ 0 : around local charge neutrality)

Contributes to
new type of order: “charge density wave”
not favorable around local charge neutrality...

renormalization of NN hopping → Fermi velocity
Mean-field description:
Introduce a mean field (Hubbard-Stratonovich) z.
NN hopping: tR = t (1 + z)
Fermi velocity: vFR = vF (1 + z)
- renormalization from the strong coupling limit.
Effective action:i.e. the behavior of φ can be obtainedby replacing t in the strong coupling limit with tR = t (1+z).

Velocity renormalization and RG flow
Once the renormalized tR at finite coupling gR is chosen...
(at strong coupling limit)

Coefficient ζ(tR) is obtained from the self-consistent eqn.:

ζ > 0i.e.
The effect of EM field on carriers
becomes smaller.

AFM order gets suppressed.
RG flow of effective coupling strength:

weak coupling expansion
Throckmorton et al., arXiv:1505.05154

strong coupling expansion
This work
fixed point?

Summary
Defined the lattice action for 3D Dirac semimetals,
includingimaginary time discretization aτ“background charge” field b

In the strong coupling limit (gR-2 = 0) of Coulomb interaction,
infinitely large on-site (Hubbard) repulsion for aτ→0Local charge neutrality is required. = Mott insulator.
Necessary to connect finite aτ analysis to aτ→0 limit.
At the next-leading-order [O(gR-2)] in strong coupling expansion:
renormalization of Fermi velocity (NN hopping amplitude)
Opposite to the RG flow from weak coupling expansion.
Include other novel topological phases...
Weyl semimetal, topological insulator, axionic insulator, ...