Frustration meets topology: from C>1 fractional Chern insulators - - PowerPoint PPT Presentation
Frustration meets topology: from C>1 fractional Chern insulators to tilted Weyl semimetals Emil J. Bergholtz Mathematical Physics Seminar Maynooth University, Ireland February 2016 Today, I will Briefly introduce two frontiers of
Emil J. Bergholtz
Mathematical Physics Seminar Maynooth University, Ireland February 2016
1) Fractional Chern insulators
Briefly introduce two frontiers of condensed matter physics
2) Weyl semimetals
?
(b)
Key ingredient: Geometrical frustration + interactions and spin-orbit coupling
Report on related progress on both topics
New phenomena … and intriguing first experiments (by others)
Jan Budich, Innsbruck Eliot Kapit, Oxford/New York Dmitry Kovrizhin, Cambridge Andreas Läuchli, Innsbruck Roderich Moessner, Dresden Masaaki Nakamura, Tokyo Masafumi Udagawa, Tokyo
External In Berlin
Jörg Behrmann
Piet Brouwer
Jens Eisert
Irina Gancheva Kevin Madsen Gregor Pohl Björn Sbierski
Maximilian Trescher
Flore Kunst
Zhao Liu, Princeton -> Berlin
4
Reviews:
Topological Flat Band Models and Fractional Chern Insulators
Fractional Quantum Hall Physics in Topological Flat Bands
Fractional quantum Hall states in a strong magnetic field are truly amazing!
B
fractional charge and statistics
Extremely low temperatures
∆E ∼ e2/`B ∝
√ B
But no “topological quantum computer” in service, no Nobel prize for non-Abelian anyons,…
Very strong magnetic fields
|B| ∼ 10 − 30 Tesla
T . 1 Kelvin
Robust experiments? Topological quantum computation?
Fractional Chern insulators!?
Lattice scale realizations?
Integer Chern insulators recently realized!
How about strongly interacting versions?
easy to find
t1, λ1
connected to corresponding FQH states!
Theory: FQH/FCI states survive can despite strong lattice effects
2) Are there topologically ordered states qualitatively different from the FQH states?
Questions:
1) Where are FCIs likely to form?
P . Hosur and X.-L. Qi, Recent developments in transport phenomena in Weyl semimetals, arXiv:1309:4464 A.M. Turner and A. Vishwanath, Beyond Band Insulators: Topology of Semi-metals and Interacting Phases, arXiv:1301.0330
Reviews:
Topological gapless phase in three dimensions
C = 1 C = −1 C = sign(det(vij)) = ±1 Topological stability of a Weyl node
C = 1 4π
d · ∂ˆ d ∂kx × ∂ˆ d ∂ky
Broken symmetry
(= d(k) · σ)
i,j
Robust nodal points
E = ± s X
m,n,l
vmlvnlknkm + E0(k)
C = 1 C = 0 C = 0 C = 1 C = 0 C = 1
kx ky z
Zero total Chern flux in any periodic band structure
The topology is manifested through exotic surface states, “Fermi arcs”
Vishwanath, and S. Y. Savrasov,
Theory first
Discovery of Weyl semimetal TaAs
Experimental observation of Weyl points
Lu et. al. arXiv:1502.03438 (photonic crystals @ MIT) Xu et. al. arXiv:1502.03807 (TaAs @ Princeton) Lv et. al. arXiv:1502.04684 (TaAs @ Beijing)
Experimental realization of a Weyl semimetal phase with Fermi arc surface states in TaAs
Now with an avalanche of experiments!
transitions, …
. W. Brouwer
… and many others
Diffusive metal Pseudoballistic semimetal
K
Kc
Questions: 1) How about interaction effects? 2) Is the correspondence between bulk and surface
3) Breaking of Lorentz invariance?
References:
Flat bands with higher Chern number in pyrochlore slabs
Fractional Chern Insulators in Topological Flat bands with Higher Chern Number
E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to C > 1 Fractional Chern Insulators
?
(b)
Perovskite materials, ABO3, routinely grown in sandwich structures in the [100] direction
Nature Commun. 2, 596 (2011).
2nd order SOC B O B’ B AO3 AB’O3 ABO3 ABO3 AB’O3 x y z X Y B a ~ a0 A a b c d e eg t2g j=1/2 j=3/2 10Dq a1g eg’ λ λ&∆ ∆
C=-1 C=0 C=1 C=0 C=1 C=0 C=-1 C=0
Γ Γ Γ K M
b
good for topological physics (relatively flat C=1 bands).
Epitaxial growth of (111)-oriented LaAlO3/LaNiO3 ultra-thin superlattices
The epitaxial stabilization of a single layer or superlattice structures composed of complex oxide materials on
arXiv:1212.0590v1 [cond-mat.mtrl-sci] 4 Dec 2012
Our suggestion: Consider (111) slabs of pyrochlore transition metal oxides, in particular A2Ir2O7 iridate thin films
E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa,
Why did nobody report on fractional Chern insulators in C>1 bands?
13
to the lattice setting: Landau levels always have C=1!
Is it possible to make N C=1 bands hybridize so that one band absorbs all the topology (C=N) while the others become trivial (C=0)?
Frustrated lattices are especially promising
?
(b)
t1, λ1
t2, λ2
t⊥
Consider frustrated systems with a layered structure!
For N kagome layers we find an almost flat band with C=N!
(d)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
(e)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
(f)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
(g)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
(h)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
(i)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
C=8 C=100 C=3 C=12
(a)
K
Γ
M
(b)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
Dispersion for one layer
C=1
(c)
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
C=2
Dispersion with two layers
νb = 1/(C + 1) Bosonic FCIs at
νf = 1/(2C + 1) Fermionic FCIs at
but absent at higher filling fractions!
Strong evidence also for C>1 generalizations of non-Abelian FQH states found in this model!
E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Phys. Rev. Lett. 114, 016806 (2015)
B.A. Bernevig, and N. Regnault, Phys. Rev. B 87, 205137 (2013)
Different also from conventional multi-layer FQH systems
Ek/t1
“Graphene + a flat band”
H = t1 X
hi,ji
c†
icj
Example: nearest neighbor hopping on a kagome lattice But these states are neither topological nor Wannier functions!
Localized modes explain the flat band
|ψi = 1 p 6 X (1)n|ni
n ∈
A brief interlude: Flat bands and localized modes on frustrated lattices
Start by considering a single chain Stack N identical chains
m = 1
m = 2
m = N
. . .
H(kx) = d(kx) · σ E±(kx) = ±|d(kx)| A suitable gauge choice making the hopping to the intermediate (green sites) real always exists.
in preparation
spin-orbit coupling and in presence of magnetic fields
|ψ±(kx)i = X
m
m|φ±(kx)im
amplitude to the green sites
r±(kx) = − φ1
±(kx) + φ2 ±(kx)
φ1
±(kx) + eikxφ2 ±(kx)
With spin-orbit coupling there are two cases:
|r(kx)| = 1
(no edge state!)
Cylinder spectra and edge localization |r(kx)| = 1
(no edge state!)
No spin-orbit coupling or magnetic fields
unit cells
|r(kx)| 6= 1 !
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
a1
a2
r(k) = − φi
1(k) + φi 2(k) + φi 3(k)
e−ik2φi
1(k) + ei(k1−k2)φi 2(k) + φi 3(k)
N
X
m=1
⇣ r(k) ⌘m |φi(k)im
Surface bands localized to the kagome layers iff the total hopping amplitude to the intermediate triangular layer vanish.
components of the single-layer Bloch spinor
|r(k)|
Ψ(k) = N(k) r2(k)φ1(k) r2(k)φ2(k) r2(k)φ3(k) r(k)φ1(k) r(k)φ2(k) r(k)φ3(k) φ1(k) φ2(k) φ3(k)
state localized to the bottom
state localized to the top state delocalized!
top view
Non-trivial due to the twisted layer structure
r(k)
r(k) = − φi
1(k) + φi 2(k) + φi 3(k)
e−ik2φi
1(k) + ei(k1−k2)φi 2(k) + φi 3(k)
frustrated lattices!
N
X
m=1
⇣ r(k) ⌘m |φi(k)im
Ψ(k) = N(k) r2(k)φ1(k) r2(k)φ2(k) r2(k)φ3(k) r(k)φ1(k) r(k)φ2(k) r(k)φ3(k) φ1(k) φ2(k) φ3(k)
N bands, each with C=1, hybridize so that the surface band absorbs all the topology (C=N) while the others become trivial (C=0)
etc.
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
Increase the interlayer tunnelling —> bulk phase transition with surface band unchanged!
t⊥ = 2.0
t2 = 0.3
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
t2 = −0.3
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 4 E(k)
t2 = 0.1
Change the nearest neighbor hopping (no change in topology)
E.J. Bergholtz, Z. Liu, M. Trescher,
Γ K M Γ −6 −5 −4 −3 −2 −1 1 2 3 E(k)
t⊥ = 1.3
Another look at the bulk spectrum... Band touching described by a tilted Weyl Hamiltonian
i,j
Constant energy lines, “Fermi circles”, are split into Fermi arcs
localized to top layer localized to bottom layer delocalized
Here we have an exact solutions for the Fermi arcs, and seen as a family, they carry a huge Chern number. The Fermi arcs also exist in absence of Weyl nodes in the bulk!
E.J. Bergholtz, Z. Liu, M. Trescher,
Projections of the Weyl points for
(chemical potential at the Weyl point)
Vishwanath, and S. Y. Savrasov,
(Tilted) Weyl semimetal or layered Chern insulator in the large C limit
Generic absence of FCIs in the 3D limit
E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa,
# layers C=1 FCI C=2 FCI C~10 FCI
Intriguing dimensional crossover New type of fractionalization in the C>1 FCIs?
Very clean (111) slabs of Eu2Ir2O7 recently grown!
Fujita et. al., arXiv:1508.01318
Hall effect at zero B-field!
(a) (b)
Suggestion: probe the controversial disorder induced phase transition by tilting the Weyl cones Tilts of the Weyl cones are forbidden by Lorentz invariance
Tipping the Weyl cone
anisotropic Dirac and Weyl cones
Bernevig, arXiv:1507.01603
Recommended with a commentary by Carlo Beenakker, Leiden University
Diffusive metal Pseudoballistic semimetal
(disorder strength)
tilt
See also
.W. Brouwer,
Frustration & topology combine well
1 + 1 + 1 → 3 + 0 + 0
etc.
Less symmetry gives richer physics!
surface bands of thin Weyl semimetal slabs
N
X
m=1
⇣ r(k) ⌘m |φi(k)im
Tilted Weyl cones: Higher Chern number generalizations of Weyl cones: transport, defects, … Frustrated layer construction in other dimensions and symmetry classes “Second generation” of fractionalization in C>1 FCIs — phenomenology essentially unexplored — how about proximity effects?
Dislocations as non-Abelian wormholes? Microscopic picture? Experiments!