Supplemental Material for:
Bosonic Analogue of Dirac Composite Fermi Liquid
COUPLED-WIRE CONSTRUCTION OF
BOSONIC INTEGER QUANTUM HALL
STATE
In the main text, we used strips of bosons at
ν
= 2
as
a building block for our network model of a quantum Hall
plateau transition. It is useful to view each such strip in
Fig. 2(a) of the main text as itself composed of quantum wires
[1] labeled by
j
= 1
,...,N
. Each wire hosts charge-
e
bosons
∼
e
iφ
j
described by
L
wire
=
∂
x
θ
j
π
(
∂
t
φ
j
−
A
0
) +
v
2
π
[(
∂
x
φ
j
−
A
1
)
2
+ (
∂
x
θ
j
)
2
]
where
∂
x
θ
j
π
is conjugate to
φ
j
. When a boson hops between
neighboring wires at non-zero magnetic field—conveniently
taken in the gauge
A
2
=
Bx
—it acquires an Aharonov-
Bohm phase
exp[
i
2
πedB
hc
x
]
that prevents condensate forma-
tion. These oscillating phases can be compensated, however,
when a phase slip
∼
e
2
iθ
+
i
2
πdρ
0
x
accompanies boson hop-
ping. For
ν
= 2
this occurs for second-neighbor hopping
described by
L
boson IQH
=
g
IQH
∑
j
cos (
φ
j
+1
−
φ
j
−
1
+ 2
θ
j
)
.
(1)
A bosonic IQH state with
σ
xy
= 2
e
2
/h
[2–4] emerges when
g
IQH
flows to strong coupling. The
ν
= 2
strip then hosts edge
states with two flavors
α
=
±
of charge-
e
bosons
b
y
=1(2)
,α
∼
e
iφ
y
=1(2)
,α
at the lower (upper) edge, where
φ
y
=1
,
+
≡
φ
1
,
φ
y
=1
,
−
≡
φ
2
+ 2
θ
1
,
φ
y
=2
,
+
≡
φ
N
−
1
−
2
θ
N
, φ
y
=2
,
−
≡
φ
N
.
(2)
The Lagrangian density for the lower and upper edges is suc-
cinctly written as
L
edge
=
(
−
1)
y
K
αα
′
4
π
∂
x
φ
y,α
∂
t
φ
y,α
′
+
u
4
π
(
∂
x
φ
y,α
)
2
,
(3)
where
K
=
σ
x
, and
α,α
′
are implicitly summed over. This
is the edge theory for the boson IQH strips used in the main
text and generalizes straightforwardly to the full 2D system in
Fig. 2(a) when edges are enumerated by integers
y
. The more
microscopic picture provided here allows us in particular to
consider microscopic inversion symmetry and verify its action
on the low-energy edge fields quoted in the main text.
GENERAL COMPOSITE-FERMION BAND
STRUCTURE
In the absence of any symmetry,
the most gen-
eral momentum-independent bilinear perturbations to the
composite-fermion Hamiltonian
H
0
= Ψ
†
4
(
σ
x
k
1
+
σ
y
k
2
)Ψ
4
(4)
are given by
δH
=
∑
α,β
=0
,x,y,z
c
αβ
ˆ
h
αβ
,
(5)
ˆ
h
αβ
≡
Ψ
†
4
σ
α
τ
β
Ψ
4
,
(6)
where
c
αβ
are real constants. We now discuss these terms
according to their symmetries:
1.
Particle-hole symmetry broken.
The six terms
ˆ
h
i
0
and
ˆ
h
0
i
with
i
=
x,y,z
, are odd under particle-hole
symmetry. Among these,
ˆ
h
x/y,
0
and
ˆ
h
0
,x/y
break ro-
tation symmetry; the former moves the Dirac cones in
momentum while the latter splits them in energy. The
terms
ˆ
h
z
0
and
ˆ
h
0
z
are rotation symmetric; the former
opens a gap while the latter is a chemical potential with
opposite sign for the two cones.
2.
Particle-hole and rotation symmetries.
The combina-
tion of particle-hole and (fourfold) rotation symmetry
allows only the terms
ˆ
h
zz
,
ˆ
h
xx
+
ˆ
h
yy
, and
ˆ
h
xy
−
ˆ
h
yx
.
The latter two can be turned into each other using a uni-
tary transformation
e
iθτ
z
which commutes with
C
and
R
(Φ)
. One may therfore without loss of generality re-
strict the analysis to
ˆ
h
zz
and
ˆ
h
xx
+
ˆ
h
yy
only, as we did
in the main text. Depending on the relative magnitude
of
ˆ
h
zz
and
ˆ
h
xx
+
ˆ
h
yy
one either finds a quadratic band
touching (CFL
2
π
), or a gapped spectrum (CFL
0
).
3.
Particle-hole and inversion symmetries.
When the
rotation symmetry is broken down to twofold rota-
tions (i.e., inversion), additional terms
ˆ
h
xx
−
ˆ
h
yy
and
ˆ
h
xy
+
ˆ
h
yx
are allowed. In the CFL
2
π
regime, their effect
is to split the quadratic band touching into two Dirac
cones at different momenta. Inversion and particle-hole
symmetries map these cones onto one another and pro-
tect them from opening a gap.
4.
Particle-hole symmetry only.
When particle-hole is
the only symmetry present,
ˆ
h
zx
,
ˆ
h
zy
,
ˆ
h
xz
and
ˆ
h
yz
are
also allowed and generically give rise to a gapped spec-
trum.
2
DIAGONALIZATION OF
C
- AND
R
(
π/
2)
-SYMMETRIC HAMILTONIAN
We consider the Hamiltonian
H
R
= Ψ
†
4
h
R
Ψ
4
with
h
R
(
~
k
) =
−
∆
ke
−
iφ
k
0
0
ke
iφ
k
∆
2
g
0
0
2
g
∆
ke
−
iφ
k
0
0
ke
iφ
k
−
∆
,
(7)
where
ke
iφ
k
=
k
1
+
ik
2
. The eigenvalues of
h
R
(
~
k
)
are
E
1
(
k
) =
√
k
2
+
g
2
+
+
g,
E
2
(
k
) =
√
k
2
+
g
2
−
−
g,
E
3
(
k
) =
−
√
k
2
+
g
2
+
+
g,
E
4
(
k
) =
−
√
k
2
+
g
2
−
−
g,
where
g
±
≡
g
±
∆
and
E
1
> E
2
> E
3
> E
4
for
g >
0
. The
corresponding normalized eigenvectors are given by
u
1
/
3
(
~
k
) =
1
√
2 + 2
(
g
+
±
√
k
2
+
g
2
+
)
2
k
2
e
−
iφ
k
g
+
±
√
k
2
+
g
2
+
k
g
+
±
√
k
2
+
g
2
+
k
e
iφ
k
,
(8)
u
2
/
4
(
~
k
) =
1
√
2 + 2
(
g
−
∓
√
k
2
+
g
2
−
)
2
k
2
−
e
−
iφ
k
g
−
∓
√
k
2
+
g
2
−
k
−
g
−
∓
√
k
2
+
g
2
−
k
e
iφ
k
.
(9)
EVOLUTION OF BANDS FROM CFL
2
π
TO
CFL
0
Figure 1 shows the composite-fermion band structure for
g
= cos
α
,
∆ = sin
α
over a range of
α
. (i)-(iii) As
α
in-
creases from zero, the curvatures of the positive and negative
energy bands that meet at the quadratic band touching become
unequal. (iv) The transition between CFL
2
π
and CFL
0
occurs
at
|
∆
|
=
|
g
|
where three bands meet at one point. (v-viii) For
|
∆
|
>
|
g
|
the spectrum is gapped, and the positive (negative)
energy bands become degenerate at
g
= 0
.
PSEUDOSPIN WINDING NUMBER
In the main text, we provided an argument why Berry
phases of
γ
Berry
= 0
and
γ
Berry
= 2
π
should be viewed as
(ii)
훼훼
=
휋휋
10
(i )
훼훼
=
0
(iii)
훼훼
=
2휋휋
10
(iv)
훼훼
=
휋휋
4
(vi)
훼훼
=
4휋휋
10
(v)
훼훼
=
3
휋휋
10
(vii)
훼훼
=
0
.
475휋휋
10
(viii)
훼훼
=
휋휋
2
퐸퐸
퐸퐸
μ
CF
μ
CF
FIG. 1. Composite-fermion band structure shown for parameters
g
=
cos
α
,
∆ = sin
α
.
distinct. Alternatively, one may sharply distinguish between
CFL
2
π
and CFL
0
band structures via a
pseudospin
ˆ
n
with
ˆ
n
=
(
ˆ
h
xx
−
ˆ
h
yy
ˆ
h
xy
+
ˆ
h
yx
)
,
(10)
where
ˆ
h
αβ
are defined in Eq. (6). Evaluating
ˆ
n
for the eigen-
states
|
u
j
(
~
k
)
〉
of the
C
- and
R
(
π/
2)
-invariant Hamiltonian,
Eq. (7), we find
〈
u
j
(
~
k
)
|
ˆ
n
|
u
j
(
~
k
)
〉
=
f
j
(
~
k
)
(
cos 2
φ
k
sin 2
φ
k
)
.
(11)
In the CFL
2
π
regime,
g >
|
∆
|
,
lim
~
k
→
0
(
f
1
,f
2
,f
3
,f
4
) = (0
,
−
1
,
1
,
0)
,
(12)
while in the CFL
0
regime,
∆
> g >
0
,
lim
~
k
→
0
(
f
1
,f
2
,f
3
,f
4
) = (0
,
0
,
1
,
−
1)
.
(13)
Consequently, the partially filled band,
j
= 2
, features a wind-
ing of the pseudospin in CFL
2
π
that is absent in CFL
0
.
PROJECTION ONTO QUADRATICALLY
TOUCHING BANDS
To make the nature of CFL
2
π
more transparent, it is con-
venient to focus on the two bands that touch quadratically.
Writing
Ψ
T
4
= (
ψ
1
,ψ
2
,ψ
3
,ψ
4
)
, the states created by
ψ
1
and
ψ
4
form a degenerate subspace at
~
k
= 0
. The corresponding
3
states at small but non-zero
|
~
k
| |
g
|−|
∆
|
are created by
ψ
+
=
ψ
1
+
ke
iφ
k
2
g
+
g
−
(∆
ψ
2
−
gψ
4
)
(14)
ψ
−
=
−
ψ
4
+
ke
iφ
k
2
g
+
g
−
(
gψ
2
−
∆
ψ
4
)
.
(15)
Projecting the Hamiltonian of Eq. (7) onto the two-
dimensional subspace spanned by
ψ
±
yields
Ph
R
(
~
k
)
P
=
1
2
g
+
g
−
(
k
2
∆
e
−
2
iφ
k
gk
2
e
2
iφ
k
gk
2
k
2
∆
)
.
(16)
The PH-breaking mass term
σ
z
projects as
Pσ
z
P
=
(
1 0
0
−
1
)
+
O
(
k
2
)
.
(17)
The fields
ψ
+
,ψ
−
transform under PH symmetry and rota-
tions as
C
ψ
±
C
−
1
=
−
ψ
∓
,
(18)
R
(Φ)
ψ
±
R
−
1
(Φ) =
e
±
i
Φ
ψ
±
,
(19)
which for
Ψ
T
2
= (
ψ
+
,ψ
−
)
gives Eq. (16).
BERRY CURVATURE INDUCED IN CFL
0
BY
PH SYMMETRY BREAKING
To estimate the Berry curvature induced in CFL
0
by weak
breaking of PH symmetry, we consider the limit
∆
g >
0
and focus on the two positive energy bands with wave func-
tions
u
1
,
2
(
~
k
)
specified in Eqs. (8) and (9). Weak breaking of
particle-hole symmetry
∼
mσ
z
τ
0
with
m
g,
∆
modifies
the wave-functions as
̃
u
1
/
2
(
~
k
) =
u
1
/
2
(
~
k
)
±
m
4
g
u
2
/
1
(
~
k
)
.
(20)
Using these to compute the Berry-flux enclosed in a Fermi-
surface of radius
K
F
∆
one finds
γ
Berry
=
−
2
π
m
g
K
2
F
8∆
2
(21)
where we expanded to leading order in
K
F
[cf. Eq. (17) from
the main text].
[1] J. C. Y. Teo and C. L. Kane, Phys. Rev. B
89
, 085101 (2014).
[2] T. Senthil and M. Levin, Phys. Rev. Lett.
110
, 046801 (2013).
[3] Y.-M. Lu and A. Vishwanath, Phys. Rev. B
86
, 125119 (2012).
[4] S. D. Geraedts and O. I. Motrunich, Annals of Physics
334
, 288
(2013).