1
Loops of Dirac points in three dimensions
Line of Dirac nodes in hyperhoneycomb lattices
Kieran Mullen, Bruno Uchoa and Daniel T. Glatzhofer, Phys. Rev. Lett. 115, 026403
(2015)
Recommended with a commentary by Rahul Nandkishore, CU Boulder
Introductory courses in solid state physics typically begin with a discussion of (Fermi
liquid) metals and band insulators. At zero temperature in d spatial dimensions, the former
have a Fermi surface of dimension d−1 (and co-dimension [1] 1) where the chemical potential
intersects a partially filled band, while the latter have no Fermi surface, with the chemical
potential sitting in the band gap. The discovery of Dirac semimetals (of which graphene is
probably the earliest and certainly the most famous example) has introduced a qualitatively
new possibility. In Dirac semimetals, the conduction and valence bands touch at a discrete
set of points, and when the chemical potential is tuned to that point then there arises a zero
dimensional Fermi surface with co-dimension d, which in spatial dimensions two and three is
clearly intermediate between a metal and a band insulator. This fact, combined with the π
Berry phase associated with Dirac points, is responsible for much of the rich phenomenology
that has captivated condensed matter physicists over the past decade.
Very recently, it has been realized that in three spatial dimensions there can arise Dirac
line semimetals, where the conduction and valence bands touch on a line. To my knowledge,
this notion was first advanced in [2], albeit in a context where having the line contact at
constant energy required considerable fine tuning. The focus of [3] is on a situation where
this line contact is naturally non-dispersive (i.e. at constant energy), such that the Fermi
surface at perfect compensation consists of a line of Dirac nodes with co-dimension two a case intermediate between three dimensional Dirac point semimetals and ordinary three
dimensional metals. Such ‘Dirac line semimetals’ provide a qualitatively new playground
for solid state physics, of (one may hope) comparable richness to Dirac point semimetals.
Specifically, Mullen, Uchoa and Glatzhofer show that in a certain class of three dimensional
hyperhoneycomb lattices (Fig.1), a simple tight binding model (without spin orbit interaction) gives rise to a closed loop of Dirac points at zero energy. As pointed out in [2],
such a bulk Dirac line node also results in a flat band of surface states. The simplest such
Ener
veloc
arXiv:1408.5522v3 [cond-mat.mes-
1
dicted
in topological
superconductors
[3] and
the
Fig.
1(a)].to exist
Allonly
three coplanar
bondsbyspaced
separated
the vector ~↵, connecting an atom of the
e curve
defined
by
katoms
0 form
and
z =
3D Dirac semimetals
[4] in which the parameters such as
kind ↵ with its nearest neighbor of the kind . The sum
lar
by 120°.
Theandtight
binding
basistuned
is [2].
of⌘the form
ψ overðrÞ
¼
interactions
magnetic
finely
is carried α;k
all nearest neighbor vectors ~↵, among
⇣p
⌘ field are
⇣p
ik·r
Theoretically,
graphene is not the
only possible latno
ϕα4ðkÞe
, 3k
withwith
α¼
1;
2; 3; 4 connected
labeling
the components(2)
of a
tice realization
planar
trigonally 3k
cos
cos
x a/2
y a/2atoms= 1.
ere
It is therefore
to ask
if there are variations
four [5].
vector
Φk, natural
which
describes
the amplitudes of the
0
on the honeycomb geometry that might produce exotic
0
0.5
[3]
electronic
wavewith
function
on the four
atoms in thea) unit cell.
Fermi surfaces
Dirac-like excitations
and topologi1
q.as (2) defines
a zerostates.
energy
linewekpropose
cally non-trivial
In this Letter,
fam0 =a (k
x ( ), ky ( ), 0)
φ/π
x
ily of trigonally connected 3D lattices that admit sim-
0
-2
1
2
Γ
M’
R
Γ
2
4
own in theplesolid
white
lineshaving
in Fig.
2a,withwhere is the
3e)
tight-binding
Hamiltonians
Dirac loops,
out requiring any tuning or spin-orbit coupling. Some
lindrical polar
angle liewith
respect to the center of the z
ice
(a) structures
of these
in the family of harmonic honeyy
comb
lattices,
which
have
been
studied in the
context
t is
rillouin
zone
(BZ) at the point.
The
reciprocal
latb)
1
p
of the Kitaev model [7–11], and experimentally
realized
2
q^ z
theis generated
in honeycomb
[12]. The simplest
is the 3a, 0, ⇡/3a),3
ce
the
vectors
b1 example
= (2⇡/
x lattice,
1
p by iridates
Γ
hyper-honeycomb
shown in Fig. 1a.
mi
x
3a, the⇡/3a)
and b23of this
=3family
(0,4of0, 2⇡/3a),
as 4 5 6 7
low energy Hamiltonian
2 = (0, 2⇡/We derive
systems, and analyze the quantization of the conductivity
on8
own in Fig.
2b, and
has Even
fourthough
high
points,
and possible
surface states.
these symmetry
systems
of
3D semimetals, their Fermi surface is multiply conq^ ρ
R, X, andare
Z.
The
3D
BZ
has
four-fold
rotational
symz
q^ φ
y
nected, with the shape of a torus,
z and highly anisotropic.
y
htWhen a(a)
magnetic field with toroidal geometry is applied,
(c)
etry aroundwe the
[001] direction. The energy spectrum of
find
uir(b)that the Hall conductivity is quantized in 3D at FIG. 1. (Color online) Simple lattice structures where all
sufficiently
large
field.
Additional
spin-orbit
coupling
efamiltonianfects
(1)canhas
four
bands, shown in Fig.atoms
2c,are where
connected by three co-planar
spaced by 120
.
FIG. bonds
2. (Color
online)
a) BZ in the kz = 0 plane showing the
ese
create 2
topologically protected surface states in
3 in the presence of spin-orbit a) The hyper-honeycomb lattice (H-0), with a four atom unit
theseenergy
crystals.
We
claim
that
ees,two lowest
bands
are
particle
hole-symmetric
FIG.
1.
(a)
and
(b)
show
the
unit
cells
two
simplest
hyperhoneycomb
lattices
that boundary
realize of the BZ,
Dirac
loop
(solid
white).
Black line:
cell. Atoms
1, 2of
andthe
3 (xy plane);
atoms
2,
3 and 4lines
(yz
plane).
1
coupling, these structures conceptually correspond to a
Atoms 1 and 2 form a vertical chain (black links); atoms 3 and
nd
cross
along
the
nodal
lines,
in
the
k
=
0
plane.
The
centered
at
the
point.
b)
3D
Brillouin
zone.
Black arrows:
new
family
of
strong
3D
topological
insulators
[13,
14].
z
4
6
4 form horizontal chain (blue links). The chains are connected
del
Wexfinally
discussloops
the experimental
feasibility
of realizing
5 tight
7 by links
(red) in the z-direction.
b) An eight
atom
unit cell the Fermi surface at small but
Dirac
in
their
binding
dispersion.
Subfigure
(c)
shows
directions
of
the
reciprocal
lattice
vectors
b
,
i
= 1, 2, 3. c)
ands
in Fig
2callotropic
follow
path shown
in 1-4
the
i
tes displayed
(H-1). Atoms
create a vertical chain of hexagons along
those structures
as new
formsthe
of carbon.
the x-direction. Atoms 5-7 create
a horizontal
chain whenof the four bands of Eq.
Tight-binding lattice. Our discussion starts with the
Energy
spectra
(1)
in
8repeated
iangular
line
ofnon-zero
panels a,hyper-honeycomb
b, with
point
R
located
in in momentum space which shrinks to a circle as the units of t
ce,
y-direction.
- thethe
Fermi
surface
isinathetorus
simplest
structure, thedoping
lattice
(see Fig.
plotted along the path shown in the red line of panels a) and
e middle of the flattened corners of the BZ.
b). The low energy bands cross along the Dirac loop. d)
doping
is
taken
to
zero.
Figures
are
taken
from
[3].
z
of
Projected
Hamiltonian. Expanding
the
eigenvecy
Velocity of the quasiparticles at the Dirac line in units of ta,
ity around the nodal line and projecting the Hamiltors
as a function of the cylindrical polar angle with respect to
are
supportingsubspace
a loop ofthat
Dirac
nodes is a hyperhoneycomb
lattice withred:
a four
unit v⇢ ( ); blue:
an (1) in the lattice
two component
accounts
. In cylindrical coordinates,
vz ( site
); black:
FIG. 1 (color online). Simple lattice structures where all atoms
ed,
r the lowest
energy
bands,
projected
Hamiltonian
v ( ). Theallotrope
orange and
curves
describe
vx ( ) and vy ( )
are connected
three
coplanar
bonds spaced
byshould
120°. (a)beThe
cell,bywhich
thetheauthors
argue
a metastable
ofviolet
carbon.
Such
a three
na
n
in the Dirac-like
hyperhoneycomb
lattice (H-0), form
with a four atom unit cell. Atoms respectively. e) Red arrows: cylindrical moving basis around
ndbe written
dimensional
allotrope
if realized,
a paradigmatic
example
of a for energies
the then
line ofprovide
Dirac nodes.
Toroidal Fermi
surfaces
1,
2,
and
3
(xy
plane);
atoms
2,
3,
and 4of(yzcarbon,
plane). Atoms
1 and 2 may
tly
E/t
=
0.1,
0.2,
0.3
and
0.4
around
the
Dirac
loop.
chain
(black
links);
atoms
3
and
4
form
a
Hp (q)form
= a [vvertical
(
)q
+
v
(
)
q
]
+
v
(
)q
,
(3)
x
y
x
z
z z
ate
Diracxline ysemimetal.
Alternative
material realizations have also been discussed elsewhere.
horizontal chain (blue links). The chains are connected by links
We
zkdirection.
(b) however,
An
eight atom
cell (H-1).
Atoms
here q (red)
⌘ k(in the
) My
) is the
momentum
measured
away
wasunit
piqued
more
by the universal physics that should be associated
0 (interest,
ese
1–4
create
a
vertical
chain
of
hexagons
along
the
x
direction.
om
the
nodal
line,
,
are
2
⇥
2
Pauli
matrices
(we
oscillates between 0 and 0.19. Away from half-filling,
x
z
of
Atoms 5–7
create
a horizontal
when
repeated
in the y of the material realization.
with
such
a loop ofchain
Dirac
points,
regardless
tuss~ ! direction.
1) and
the Fermi surfaces are toroids containing the nodal line
[3] takes
thesmall
form energies, the crossk0 ( ), asdiscussed
shown ininFig.
2e. For
q The low energy k · p Hamiltonian for the material
2
2
section
is
nearly
circular,
and
the
energy varies linearly
± Ek = ± [vx ( )qx + ©
vy 2015
( ) qyAmerican
] + [vz ( Physical
)qz ] Society
(4)
X
026403-1
(vx (φ)δpx + vwith
(1)analysis can
H(p) =
the
from
y (φ)δp
y )σdistance
x + vz (φ)δp
z σz the loop. A similar
the low energy spectrum. The quasiparticlesp of Hamilbe done for the unit cell shown in Fig. 1b, which has
nian (3) are where
chiral σin and
thatσ there
is
a
Berry
phase
8 carbon atoms in the unit cell. In that case, the tight
x
z are Pauli matrices, the δpi denote the momentum offset from the closest
~
~
binding Hamiltonian is an 8⇥8 matrix with 8 di↵erent
h k |rk k i · dk = ⇡ [15, 16] associated with paths
point
on
the
Dirac
loop,
the
v
are
the
Fermi
velocities
in the
corresponding
andinto the low
i
bands. This
Hamiltonian
candirections,
be projected
momentum space that encircle the nodal line.
energy
states,
resulting
in a Much
Hamiltonian
with the same
The Fermi velocities
( )is(ian=azimuthal
x, y, z) are
plotted
in
0 ≤ φ <vi2π
angle
parametrizing
position
along
the loop.
of the key
form as Eq. (3).
g. 2d, and can be approximated by simple trigonophysics, however, can be exposed by a cylindrically symmetric ‘toy model’ where the Dirac
etric functions. The quasiparticles disperse linearly in
The above structures are merely two in a hierarchy
loop isto
a circle
of radius
in
the
x-y
plane.
This
has a low
energy
k·p
Hamiltonian
of the
e normal directions
the nodal
line p(Fig
2e)
and
are
of
possible
lattices
that
can
be made with
perpendicular
F
spersionless along the Dirac loop. In the cylindrical
zigzag chains of trigonally connected carbon atoms. We
form
oving basis shown in Fig. 2e, the velocities are X
given
denote these structures with two integers (nx , ny ), where
H(p)
= line
v⊥ (p⊥n−x p(n
+ the
vz pz σ
(2) complete
F )σ
z
y vz ( ), v⇢ ( ) and v ( ). Even though the
nodal
number
of vertical (horizontal)
y )x is
p
not a perfect circle, the ratio v ( )/vz ( ) is small and
honeycomb hexagons contained in the unit cell. In this
p 2
2
where p⊥ = px + py . The dispersion takes the form
p
E± (p) = ± (v⊥ (p⊥ − pF ))2 + (vz pz )2
(3)
This toy model contains a great deal of physics. At perfect compensation, the Fermi
surface is a circle with radius pF and codimension two, and there is a π Berry phase along any
3
loop that interlocks the Fermi surface. The low energy density of states vanishes linearly with
energy - much as in graphene, but now in three dimensions. This too is intermediate between
a metal (where the low energy density of states is constant) and a perfectly compensated
clean Dirac point semimetal (which in three dimensions has a quadratically vanishing low
energy density of states) Meanwhile, on doping away from perfect compensation the Fermi
surface turns into a torus.
Dirac loops in three dimensions can support new phenomena not conventionally associated with 3D systems. The authors of [3] highlight this by pointing out one particularly
striking new phenomenon: a three dimensional quantized Hall effect in the presence of an
constant azimuthal magnetic field. Intuitively, if we consider a single point on the Fermi
ring, the Hamiltonian v⊥ δp⊥ σx +vz pz σz looks just like the Hamiltonian for a two dimensional
Dirac fermion moving in the p⊥ − z plane, and the application of an azimuthal magnetic
field should give rise to a quantized Hall response just as the application of an out of plane
magnetic field gives rise to a quantized Hall response for the two dimensional Dirac fermion.
More formally, a magnetic field B ∝ Bφ φ̂ may be represented by a ‘Landau gauge’ vector
p
potential A = −Bφ ρẑ, where ρ = x2 + y 2 . Introducing the vector potential into the
Hamiltonian through the usual minimal coupling prescription, and squaring the Hamiltonian, Mullen et al find that the Hamiltonian can be diagonalized by introducing ladder
operators, and the spectrum takes the form
p
√
EN = sign(N )( 2v⊥ vz /lB ) |N |
(4)
√
where lB ∝ 1/ B is the usual magnetic length i.e. the spectrum breaks up into a (particlehole symmetric) tower of Landau levels indexed by integer N , with a ‘square root’ dependence of energy on magnetic field and on Landau level index, just as in graphene. When
the chemical potential lies in the gap between the N th and (N + 1)th level, the system
has σzρ = (2N + 1)pF e2 /h, allowing for a factor of two coming from spin degeneracy i.e.
a radial current induces a voltage bias along the ẑ axis, with a quantized proportionality
constant that depends linearly on the Fermi radius. Colloquially, an azimuthal magnetic
field produces a Hall response in the ẑ − ρ̂ plane that is like the response of pF copies of
graphene to an out of plane field. This behavior should survive even when the Fermi line
is deformed away from a perfect circle, as long as the bulk gaps do not close, and thus
may well be observable in Dirac loop materials. Of course, to produce a radially uniform
4
azimuthal magnetic field one would have to apply a time varying electric field ∝ 1/ρ along
the ẑ direction, which experimentally may not be a simple task.
The combination of a Berry phase, a linearly vanishing low energy density of states in
three dimensions, and a non-trivial Fermi surface topology provides a qualitatively new playground for condensed matter physics, and there remain many questions still to explore. For
example, what are the signatures of Dirac loops in transport, aside from the three dimensional quantum Hall effect discussed above? What is the effect of disorder? Of interactions?
Of a combination of the two? What new phases can we access starting from Dirac loop
systems and turning on interactions, disorder, and/or external fields? How does the phenomenology differ for Dirac loops and for Weyl loops (without spin degeneracy)? And where
and how will this new physics be seen experimentally? While some preliminary steps along
these directions have been taken [2, 4, 5], much surely remains to be done. A new chapter
has been opened in the study of Dirac materials. There are exciting times ahead.
[1] The codimension is the number of directions in which one can move away from the Fermi surface.
For a traditional Fermi liquid metal, the codimension is 1 - the direction normal to the Fermi
surface. A Dirac point semimetal, however, has co-dimension d, and a Dirac line semimetal has
codimension d − 1. The codimension dc is important in part because it controls the low energy
density of states, which scales as ν(E) ∼ E dc −1 for systems with linear dispersion, and this
in turn controls the response functions, the relevance/irrelevance of disorder and interactions,
and much besides.
[2] A.A. Burkov, M.D. Hook and L. Balents, Phys. Rev. B 84, 235126 (2011)
[3] K. Mullen, B. Uchoa and D. T. Glatzhofer, Phys. Rev. Lett. 115, 026403 (2015)
[4] Y. Huh, E.-G. Moon and Y. B. Kim, arXiv: 1506.05105
[5] R. Nandkishore, arXiv: 1510.00716