Download presentation

Presentation is loading. Please wait.

Published byRosamund Strickland Modified over 2 years ago

1
Weyl semimetals Pavel Buividovich (Regensburg)

2
Weyl semimetals TODO: - discuss different kinds of WSM, calculate the spectrum in the presence of mass Fermi arc states - Explain why anomalous response of conductivity is a signature of WSM

3
**Simplest model of Weyl semimetals**

Dirac Hamiltonian with time-reversal/parity-breaking terms Breaks time-reversal Breaks parity

4
**Nielsen, Ninomiya and Dirac/Weyl semimetals**

Axial anomaly on the lattice? Axial anomaly = = non-conservation of Weyl fermion number BUT: number of states is fixed on the lattice???

5
**Nielsen, Ninomiya and Dirac/Weyl semimetals**

Weyl points separated in momentum space In compact BZ, equal number of right/left handed Weyl points Axial anomaly = flow of charges from/to left/right Weyl point

6
**Nielsen-Ninomiya and Dirac/Weyl semimetals**

Enhancement of electric conductivity along magnetic field Intuitive explanation: no backscattering for 1D Weyl fermions

7
**Nielsen-Ninomiya and Dirac/Weyl semimetals**

8
**Field-theory motivation**

A lot of confusion in HIC physics… Table-top experiments are easier?

9
**Weyl points survive ChSB!!!**

Weyl semimetals Weyl points survive ChSB!!!

10
**Weyl semimetals: realizations**

Pyrochlore Iridates [Wan et al.’2010] Strong SO coupling (f-element) Magnetic ordering Stack of TI’s/OI’s [Burkov,Balents’2011] Surface states of TI Spin splitting Tunneling amplitudes Iridium: Rarest/strongest elements Consumption on earth: 3t/year Magnetic doping/TR breaking essential

11
**Weyl semimetals with μA**

How to split energies of Weyl nodes? [Halasz,Balents ’2012] Stack of TI’s/OI’s Break inversion by voltage Or break both T/P Chirality pumping [Parameswaran et al.’13] Electromagnetic instability of μA [Akamatsu,Yamamoto’13] Chiral kinetic theory (see below) Classical EM field Linear response theory Unstable EM field mode μA => magnetic helicity OR: photons with circular polarization

12
**Lattice model of WSM Take simplest model of TIs: Wilson-Dirac fermions**

Model magnetic doping/parity breaking terms by local terms in the Hamiltonian Hypercubic symmetry broken by b Vacuum energy is decreased for both b and μA

13
**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

14
**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

15
**Topological stability of Weyl points**

Weyl Hamiltonian in momentum space: Full set of operators for 2x2 hamiltonian Any perturbation (transl. invariant) = just shift of the Weyl point Weyl point are topologically stable Only “annihilate” with Weyl point of another chirality E.g. ChSB by mass term:

16
**Weyl points as monopoles in momentum space**

Free Weyl Hamiltonian: Unitary matrix of eigenstates: Associated non-Abelian gauge field:

17
**Weyl points as monopoles in momentum space**

Classical regime: neglect spin flips = off-diagonal terms in ak Classical action (ap)11 looks like a field of Abelian monopole in momentum space Berry flux Topological invariant!!! Fermion doubling theorem: In compact Brillouin zone only pairs of monopole/anti-monopole

18
**Fermi arcs [Wan,Turner,Vishwanath,Savrasov’2010]**

What are surface states of a Weyl semimetal? Boundary Brillouin zone Projection of the Dirac point kx(θ), ky(θ) – curve in BBZ 2D Bloch Hamiltonian Toric BZ Chern-Symons = total number of Weyl points inside the cylinder h(θ, kz) is a topological Chern insulator Zero boundary mode at some θ

19
**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

20
**Hydrodynamic approach**

Classical conservation laws for chiral fermions Energy and momentum Angular momentum Electric charge No. of left-handed Axial charge No. of right-handed Hydrodynamics: Conservation laws Constitutive relations Axial charge violates parity New parity-violating transport coefficients

21
**Hydrodynamic approach**

Let’s try to incorporate Quantum Anomaly into Classical Hydrodynamics Now require positivity of entropy production… BUT: anomaly term can lead to any sign of dS/dt!!! Strong constraints on parity-violating transport coefficients [Son, Surowka ‘ 2009] Non-dissipativity of anomalous transport [Banerjee,Jensen,Landsteiner’2012]

22
**Anomalous transport: CME, CSE, CVE**

Chiral Magnetic Effect [Kharzeev, Warringa, Fukushima] Chiral Separation Effect [Son, Zhitnitsky] Chiral Vortical Effect [Erdmenger et al., Teryaev, Banerjee et al.] Flow vorticity Origin in quantum anomaly!!!

23
**Why anomalous transport on the lattice?**

1) Weyl semimetals/Top.insulators are crystals 2) Lattice is the only practical non-perturbative regularization of gauge theories First, let’s consider axial anomaly on the lattice

24
**Warm-up: Dirac fermions in D=1+1**

Dimension of Weyl representation: 1 Dimension of Dirac representation: 2 Just one “Pauli matrix” = 1 Weyl Hamiltonian in D=1+1 Three Dirac matrices: Dirac Hamiltonian:

25
**Warm-up: anomaly in D=1+1**

26
**Axial anomaly on the lattice**

= non-conservation of Weyl fermion number BUT: number of states is fixed on the lattice???

27
**Anomaly on the (1+1)D lattice**

1D minimally doubled fermions DOUBLERS Even number of Weyl points in the BZ Sum of “chiralities” = 0 1D version of Fermion Doubling

28
**Anomaly on the (1+1)D lattice**

Let’s try “real” two-component fermions Two chiral “Dirac” fermions Anomaly cancels between doublers Try to remove the doublers by additional terms

29
**Anomaly on the (1+1)D lattice**

(1+1)D Wilson fermions In A) and B): In C) and D): A) B) C) D) B) Maximal mixing of chirality at BZ boundaries!!! Now anomaly comes from the Wilson term + All kinds of nasty renormalizations… A) B) D) C)

30
**Now, finally, transport: “CME” in D=1+1**

Excess of right-moving particles Excess of left-moving anti-particles Directed current Not surprising – we’ve broken parity Effect relevant for nanotubes

31
**“CME” in D=1+1 Fixed cutoff regularization: Shift of integration**

variable: ZERO UV regularization ambiguity

32
**Dimensional reduction: 2D axial anomaly**

Polarization tensor in 2D: Proper regularization (vector current conserved): [Chen,hep-th/ ] Final answer: Value at k0=0, k3=0: NOT DEFINED (without IR regulator) First k3 → 0, then k0 → 0 Otherwise zero

33
**Directed axial current, separation of chirality**

“CSE” in D=1+1 μA μA Excess of right-moving particles Excess of left-moving particles Directed axial current, separation of chirality Effect relevant for nanotubes

34
**Energy flux = momentum density**

“AME” or “CVE” for D=1+1 Single (1+1)D Weyl fermion at finite temperature T Energy flux = momentum density (1+1)D Weyl fermions, thermally excited states: constant energy flux/momentum density

35
**Going to higher dimensions: Landau levels for Weyl fermions**

36
**Going to higher dimensions: Landau levels for Weyl fermions**

Finite volume: Degeneracy of every level = magnetic flux Additional operators [Wiese,Al-Hasimi, ]

37
**LLL, the Lowest Landau Level**

Lowest Landau level = 1D Weyl fermion

38
**Anomaly in (3+1)D from (1+1)D**

Parallel uniform electric and magnetic fields The anomaly comes only from LLL Higher Landau Levels do not contribute

39
**Anomaly on (3+1)D lattice**

Nielsen-Ninomiya picture: Minimally doubled fermions Two Dirac cones in the Brillouin zone For Wilson-Dirac, anomaly again stems from Wilson terms VALLEYTRONICS

40
**Anomalous transport in (3+1)D from (1+1)D**

CME, Dirac fermions CSE, Dirac fermions “AME”, Weyl fermions

41
**Chiral kinetic theory [Stephanov,Son]**

Classical action and equations of motion with gauge fields More consistent is the Wigner formalism Streaming equations in phase space Anomaly = injection of particles at zero momentum (level crossing)

42
**CME and CSE in linear response theory**

Anomalous current-current correlators: Chiral Separation and Chiral Magnetic Conductivities:

43
**Chiral symmetry breaking in WSM**

Mean-field free energy Partition function For ChSB (Dirac fermions) Unitary transformation of SP Hamiltonian Vacuum energy and Hubbard action are not changed b = spatially rotating condensate = space-dependent θ angle Funny Goldstones!!!

44
**Electromagnetic response of WSM**

Anomaly: chiral rotation has nonzero Jacobian in E and B Additional term in the action Spatial shift of Weyl points: Anomalous Hall Effect: Energy shift of Weyl points But: WHAT HAPPENS IN GROUND STATE (PERIODIC EUCLIDE???) Chiral magnetic effect In covariant form

45
**Topological insulators**

Summary Graphene Nice and simple “standard tight-binding model” Many interesting specific questions Field-theoretic questions (almost) solved Topological insulators Many complicated tight-binding models Reduce to several typical examples Topological classification and universality of boundary states Stability w.r.t. interactions? Topological Mott insulators? Weyl semimetals Many complicated tight-binding models, “physics of dirt” Simple models capture the essence Non-dissipative anomalous transport Exotic boundary states Topological protection of Weyl points

Similar presentations

OK

1 Electroweak baryon number violation and a “topological condensate” Steven Bass / Innsbruck July 15 2004 Baryon asymmetry in the Universe:

1 Electroweak baryon number violation and a “topological condensate” Steven Bass / Innsbruck July 15 2004 Baryon asymmetry in the Universe:

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Educational backgrounds for ppt on social media Ppt on depth first search tree Ppt on big dig collapse Ppt on bluetooth hacking windows Ppt on complex numbers class 11th accountancy Ppt on buddhism and jainism Ppt on work and energy for class 9th Pdf to ppt online convertor Ppt on views in oracle Ppt on depth first search