Nodal line semimetals

László Oroszlány ELTE-KRFT

Outline

  • Topology in real and momentum space

  • Topological metals

  • Magnetic oscillations of nodal loop semimetals

Topology

in

real space

 Berezinskii–Kosterlitz–Thouless transition

Berezinskii, Sov. Phys. JETP, 32 493 (1971)

Kosterlitz,  Thouless,  J. Phys. C, 6  1181 (1973)

2D XY model has a phase transition despite the

Mermin-Wagner theorem!

Skyrmions in magnetic systems

 Phys. Rev. Lett. 87, 037203 (2001)

\vec{r}= \left(\begin{array}{c} r_x\\ r_y \end{array}\right)
\vec{d}(\vec{r})= \left(\begin{array}{c} d_x(\vec{r})\\ d_y(\vec{r})\\ d_z(\vec{r}) \end{array}\right)
\displaystyle C=\frac{1}{4\pi}\int \vec{d}\cdot \left(\partial_x\vec{d}\times\partial_y\vec{d}\right) \mathrm{d}S

Chern

number

Nuclear Physics 31, 556 (1962)

Topology

in

momentum space

Two band lattice models

U
T_x
T_y
H(\vec{k})=U+\displaystyle\sum_p \mathrm{e}^{\mathrm{i}\vec{k}\cdot \vec{a}_p}T_p+\mathrm{e}^{-\mathrm{i}\vec{k}\cdot \vec{a}_p}T^\dagger_p

two bands

H(\vec{k})=d_0(\vec{k})\sigma_0+\vec{d}(\vec{k})\cdot\vec{\sigma}

also parametrizes the eigenstates on the Bloch spehre

SSH:The mother of all topological insulators

H=\vec{d}(k)\cdot \vec{\sigma}
\vec{d}(k)=\left(\begin{array}{c} v+w\cos(k)\\ w\sin(k)\\ 0 \end{array}\right)

W. P. Su, J. R. Schrieffer, and A. J. Heeger Phys. Rev. Lett. 42, 1698 (1979)

Winding

number

chiral symmetry

Bulk boundary correspondence

Finite bulk winding number                          edge states

Chern insulators

\vec{k}= \left(\begin{array}{c} k_x\\ k_y \end{array}\right)
\vec{d}(\vec{k})= \left(\begin{array}{c} d_x(\vec{k})\\ d_y(\vec{k})\\ d_z(\vec{k}) \end{array}\right)

BZ

d_x
d_z
d_y
d_x
d_z
d_y
C=0
C=1

"sitting in the origin staring towards infinity"

\rightarrow C

v. Klitzing, Dorda, Pepper  Phys. Rev.  Lett. 45,  494 (1980)

Haldane Phys. Rev. Lett. 61,  2015 (1988)

Qi, Wu, Zhang Phys. Rev. B 74, 085308  (2006)

Quantum Hall effect & Chern insulators

TRI topological insulators in 2D & 3D

Hasan,  Kane Rev. Mod. Phys. 82 3045 (2010)

Topological metals

Weyl Semimetals

TaAs surface, Nat. Comm. 6, 7373 (2015)

Weyl points+symmetry =nodal lines

\displaystyle H=\left(5t-2t\sum_i \cos(k_i)\right)\sigma_x+2t\sin(k_y)\sigma_y+2t\sin(k_z)\sigma_z
E=0
H=\vec{d}(\vec{k})\cdot \vec{\sigma}
E_\pm=\pm|\vec{d}(\vec{k})|

chiral symmetry

Physical realizations and interesting models

Nodal knots

Phys. Rev. B 96, 201305(R) (2017)

Nodal links 

Phys. Rev. B 96, 081114(R) (2017)

A good summary: Adv. Phys.  X 3, 1414631 (2018)

PbTaSe

ZrSiS

Magnetic oscillations

in

nodal loop semimetals

SdH & dHvA in 3D: Extremal Fermi surfaces

|\psi_R\rangle
\mathcal{C}
\vec{B}

Topological content of magnetic oscillations

\frac{\mathcal{A}(E)\hbar}{eB}=2\pi(n+\gamma)

Onsager quantization condition:

Phil. Mag. 43, 1006 (1952)

Berry's phase:

 

 

 

 

Proc. R. Soc. Lond. A 392, 45 (1984)

\phi=\mathrm{i}\displaystyle\oint_C\mathrm{d}R\langle \psi_R|\partial_R|\psi_R\rangle
d_x
d_y
d_z
=\pi\left(1-2\gamma\right)
H=\vec{d}_R\cdot\vec{\sigma}

Two band models

Chiral symmetry quantizes Berry's phase!

\varrho(E)=\displaystyle{\sum_n \delta(E-E_n)}

DOS oscillatory in 1/B

nice pedagogical summary for 2D

JN Fuchs https://arxiv.org/pdf/1306.0380.pdf

Simplest case

H=\frac{p^2}{2m}
\vec{A}=By\vec{e_x}
H=\frac{(p_x-eBy)^2+p_y^2+p_z^2}{2m}
E=\frac{\hbar e B}{m}\left (n+\frac{1}{2} \right)+\frac{p_z^2}{2m}
\gamma=\frac{1}{2} !

No Berry's phase!

There is "nothing to wind" !

only trivial oscillations!

\vec{B}=-B\vec{e_z}

Weyl semimetals

H=v\vec{p}\cdot\vec{\sigma}
\vec{A}=By\vec{e_x}
H=v\left [(p_x-eBy)\sigma_x+p_y\sigma_y+p_z\sigma_z\right ]
\vec{B}=-B\vec{e_z}
H^2=v^2\left [(eBy)^2+p_y^2+p_z^2\right ]\sigma_0+v^2\hbar eB\sigma_z
H^2=\left [2v^2\hbar eB\left (n+\frac{1}{2}\right)+v^2p_z^2\right ]\sigma_0+v^2\hbar eB\sigma_z
E_n=\pm v\sqrt{2\hbar eB n+p_z^2}
\gamma=0!

non trivial Berry's phase!

"it always winds"

n=0 is special ...

Nature 438, 201 (2005)

Berry phase and the triumph of graphene

H=v\vec{p}\cdot\vec{\sigma}
E=\pm v\sqrt{2\hbar eBn}
\gamma=0,\,\phi=\pi

Magnetic oscillations in ZrSiS: experiments

Science Advances 2, e1601742  (2016)

Nature Physics 14, 178 (2018)

Frontiers of Physics 13, 137201 (2017)

" A transition like this, which is highly sensitive and depends only on a 10° or less change in the magnetic field angle, opens the door to creating new types of devices based on subtle details of the Fermi surface."

Effective model for nodal loops

\hat{H}=\left(\Delta-\frac{p_x^2+p_y^2}{2m}\right )\sigma_x+v\hat{p}_z\sigma_z=\mathbf{d}(\mathbf{p})\cdot\mathbf{\sigma}
E_\pm=\pm|\mathbf{d}(\mathbf{p})|

Extremal surfaces of nodal loops

\hat{H}=\left(\Delta-\frac{p_x^2+p_y^2}{2m}\right )\sigma_x+v\hat{p}_z\sigma_z

Including magnetic fields

\hat{H}=\left(\Delta-\frac{p_x^2+e^2B^2\left[p_y+z\sin\vartheta-x\cos\vartheta\right]^2}{2m}\right )\sigma_x+v\hat{p}_z\sigma_z
\vartheta=0
E=\pm\sqrt{\left(\Delta-\frac{eB}{m}\left[n+\frac{1}{2}\right]\right)^2+v^2p_z^2}

trivial oscillations

\hat{H}=\left(\Delta-\frac{p_x^2+p_y^2}{2m}\right )\sigma_x+v\hat{p}_z\sigma_z

Magnetic field

perpendicular  to the loop

\vec{A}=B(x\cos(\vartheta)-z\sin(\vartheta))\vec{e}_y
E
B

Including magnetic fields

\hat{H}=\left(\Delta-\frac{p_x^2+e^2B^2z^2}{2m}\right )\sigma_x+v\hat{p}_z\sigma_z
\vartheta=\frac{\pi}{2}
E_{n}\approx \left\{\begin{array}{cc}\pm 2\sqrt{\sqrt{\Delta}veBn/\sqrt{2m}}\\ \pm\left(\frac{veB}{\sqrt m}\left(n+\frac 12\right)\right)^{2/3} \end{array}\right.

semiclassics:

topological

trivial

in plane magnetic field

\hat{H}=\left(\alpha-\lambda^2\right )\sigma_x+\hat{p}_\lambda\sigma_z

inspiration: Montambaux et al. Eur. Phys. J. B  72 509 (2009)

Phase diagram

Oscillation spectra

Oscillation spectra

\vartheta=\frac{\pi}{2}

The Team

Balázs Dóra

József Cserti

Alberto Cortijo

Thanks:







​


details at:

Phys. Rev. B 97, 205107 (2018)

https://arxiv.org/abs/1801.04721

https://github.com/oroszl/nodalloopsemimetal

2017-1.2.1-NKP-2017-00001

A nice, pedagogical summary of topological insulators

:)

https://arxiv.org/abs/1509.02295

A 2D two band insulator with chiral symmetry...

  • A) Has always C=0 because of symmetric bands ensures a that the system remains a metal.
     
  • B) Has always C=0 because the torus can never contain the origin and remain an insulator.
     
  • C) Can have C=1 if the torus, the surface defined by d(k) in d-space, intersects the origin.
     
  • D) 2D systems can not have chiral symmetry and thus it is meaningless to define a Chern number for such a system.  

 

We should teach this!

A really good implementation

http://pybinding.site/

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