PHYSICAL REVIEW B 86, 035122 (2012)
Fractional quantum Hall states in two-dimensional electron systems with anisotropic interactions
Hao Wang,
1
Rajesh Narayanan,
1,2,3
Xin Wan,
4
and Fuchun Zhang
1,4
1
Department of Physics, The University of Hong Kong, Hong Kong SAR, China
2
Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India
3
Department of Physics, Hong Kong University of Science and Technology, Hong Kong SAR, China
4
Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China
(Received 26 March 2012; revised manuscript received 18 June 2012; published 16 July 2012)
We study the anisotropic effect of the Coulomb interaction on a 1/3-filling fractional quantum Hall system
by using an exact diagonalization method on small systems in torus geometry. For weak anisotropy the system
remains to be an incompressible quantum liquid, although anisotropy manifests itself in density correlation
functions and excitation spectra. When the strength of anisotropy increases, we find the system develops a Hall-
smectic-like phase with a one-dimensional charge density wave order and is unstable towards the one-dimensional
crystal in the strong anisotropy limit. In all three phases of the Laughlin liquid, Hall-smectic-like, and crystal
phases the ground state of the anisotropic Coulomb system can be well described by a family of model wave
functions generated by an anisotropic projection Hamiltonian. We discuss the relevance of the results to the
geometrical description of fractional quantum Hall states proposed by Haldane [Phys. Rev. Lett. 107, 116801
(2011)].
DOI: 10.1103/PhysRevB.86.035122 PACS number(s): 73.43.Cd, 73.43.Nq, 71.10.Pm
I. INTRODUCTION
The fractional quantum Hall (FQH) effect at an odd
denominator filling of ν has been understood as a property
of an incompressible quantum liquid in an interacting two-
dimensional (2D) electron system. Laughlin’s trial wave
function
1
is the first successful theory to describe this many-
body effect, where the interacting system is implicitly assumed
to be isotropic. Since then most theoretical works on the FQH
system have followed this simple assumption, and the FQH
states are considered to be isotropic with rotational symmetry.
However, the real FQH systems may be anisotropic. One
natural source for this is the anisotropic dielectric tensor,
which in turn leads to an anisotropic Coulomb interaction.
Other mechanisms for various anisotropic FQH systems have
also been discussed theoretically
2–6
and experimentally.
7,8
For example, an anisotropic FQH state in a ν = 7/3system
has been observed in experiment.
8
In these anisotropic FQH
systems, rotational symmetry of the consequent ground state
is expected to be broken. In the extreme anisotropic interaction
limit, where the Coulomb interaction can be effectively treated
to be one-dimensional (1D), the ground state of the system will
be a quasi-1D crystal.
9
Thus, the properties of the FQH state
in an anisotropic interaction may not always be associated
with the isotropic incompressible liquid. A comprehensive
investigation on the effect of the interaction anisotropy is called
for.
Motivated by the anisotropic transport properties experi-
mentally reported at the partially filled higher Landau level
(LL), trial wave functions
10–13
have been proposed to describe
the anisotropic FQH states. These variational wave functions
modify the original isotropic Laughlin wave function by
splitting the multiple-order zeros in the wave function. Very
recently, Haldane
2
has constructed a family of the Laughlin
states, which are the exact ground states of the corresponding
projected Hamiltonians and can be parameterized according
to the interaction anisotropy. This variational state can be
compared numerically with the anisotropic FQH state. An
effort to map the underlying wave function of this variational
state has been reported.
5
In this paper we study FQH states
in 2D electron systems with anisotropic Coulomb interaction
and discuss the relevance of our results with the geometric
description of the FQH states.
2
The paper is organized as follows. In Sec. II we introduce
our model with anisotropic Coulomb interaction and set
up the Hamiltonian on a torus geometry. In Sec. III we
discuss the properties of ν = 1/3 FQH states at different
regimes of the interaction anisotropy using energy spectra,
charge density, and correlation functions. We also compare the
anisotropic FQH state with variational Laughlin states using
wave-function overlap. Section IV summarizes the paper.
II. MODEL AND NUMERICAL SETUP
We study a 2D electron system under a perpendicular mag-
netic field B = B
ˆ
z. The electron-electron Coulomb interaction
with an in-plane biaxial dielectric tensor has the form
V
c
(r) =
e
2
4πε
A
c
x
2
+ y
2
/A
c
, (1)
where A
c
is the interaction anisotropy parameter, and the
directions of
ˆ
x and
ˆ
y are along the two principal axes of
the dielectric tensor. The effective mass tensor is considered
isotropic so that noninteracting electrons move in the cir-
cular cyclotron orbitals. However, equipotential lines of the
Coulomb interaction are generally elliptical with A
c
= 1. In
the following discussion, we choose A
c
1 such that
ˆ
x is the
hard axis. For A
c
< 1, one simply swaps the easy and hard
axes. At A
c
1, the Coulomb interaction is effectively a 1D
repulsion along the hard axis.
In our numerical calculations, we use Landau gauge
(0,Bx) for the magnetic vector potential. Periodic boundary
conditions for the magnetic translational operators are imposed
with a quantized flux N
φ
through the rectangular unit cell
L
x
× L
y
. The magnetic length is taken as the unit length
035122-1
1098-0121/2012/86(3)/035122(7) ©2012 American Physical Society
HAO WANG, RAJESH NARAYANAN, XIN WAN, AND FUCHUN ZHANG PHYSICAL REVIEW B 86, 035122 (2012)
and the energy is in units of e
2
/4πε. To reduce the size
of the Hilbert space, we carry out our calculation at every
pseudomomentum K = (K
x
,K
y
),
14
where K
x
(K
y
) is in units
of 2π/L
x
(2π/L
y
). The magnetic field is assumed to be strong
enough so that the spin degeneracy of the Landau levels is
lifted.
14,15
One can thus project the system Hamiltonian into
the valence Landau level.
14
For the lowest Landau level, the
projected Hamiltonian has the form
H
c
=
1
N
φ
q
V (q)e
−q
2
/2
i<j
e
iq·(R
i
−R
j
)
, (2)
where the momentum q = (q
x
,q
y
) takes discrete values suit-
able for the lattice of the unit cell, and R
i
is the guiding center
coordinate of the ith electron. V (q) = 1/
q
2
x
/A
c
+ A
c
q
2
y
is
the Fourier transform of the Coulomb interaction. From the
geometrical point of view, we generalize q
2
= q
2
x
+ q
2
y
to
q
2
g
= g
ab
q
a
q
b
, where
g =
1/A
c
0
0 A
c
(3)
is the inverse metric for the Coulomb interaction.
III. NUMERICAL RESULTS AND DISCUSSION
For a ν = 1/3 FQH system, we first study the low-lying
energy spectra using an exact diagonalization method. Here
and in following subsections, the default size of the ν = 1/3
system is N
e
= 10, and the default shape of the unit cell is
square unless otherwise specified. We find qualitatively similar
results in systems with other sizes and/or different shapes of
the unit cell.
Figure 1 plots the excitation energy gap as a function
of the Coulomb interaction anisotropy. In the range up to
A
c
= 100 the curve is nonmonotonic and develops several
distinct regimes. For small interaction anisotropy up to
1 10 100
0.00
0.02
0.04
0.06
0.0 0.2
0.00
0.06
E
gap
A
c
1
Liquid
Crystal
Intermediate
Phase
E
min
1/N
e
FIG. 1. (Color online) Excitation energy gap versus interaction
anisotropy for an N
e
= 10 and ν = 1/3 FQH system with a square
unit cell. The horizontal axis is plotted in the log scale to show the
transition at low anisotropy. The system is in a Laughlin-liquid-like
state for A
c
< 2.0 (Sec. III A), and becomes a quasi-1D crystal for
A
c
> 33.0 (Sec. III C), whose excitation gap scales as 1/
√
A
c
.The
intermediate regime is related to a Hall-smectic-like phase, which
is unstable towards the quasi-1D crystal and will be discussed in
Sec. III B. The inset shows a linear size-scaling for the minimum gap
in the intermediate regime.
A
c
= 2.0, the energy gap remains nearly constant, indicating
the incompressible liquid phase in the isotropic case (i.e.,
A
c
= 1) is robust against weak interaction anisotropy. When
the interaction anisotropy further increases, the energy gap
decreases to a minimum at around A
c
= 8.0. For larger A
c
the energy gap increases with the interaction anisotropy to a
maximum at around A
c
= 33.0. The finite-size scaling shown
in the inset reveals that the minimum gap can close in the
thermodynamic limit, suggesting there might exist a switch
between different order parameters ruling the system. Beyond
A
c
= 33.0 the energy gap decreases roughly as 1/
√
A
c
,
indicating the regime of the quasi-1D repulsion limit. We have
studied other system sizes and found that these boundaries are
size dependent. But, in general, the ground state of the system
maintains a threefold degeneracy. This adiabatic transition
with complex regimes typically occurs between distinct phases
with a competition in the intermediate region. In the following
subsections, we will focus on these different regimes in A
c
and
reveal an interesting competition between liquid and crystal
phases.
A. Anisotropic Laughlin liquid at small interaction anisotropy
The energy gap plot suggests that the ground state at small
interaction anisotropy is an incompressible liquid similar to
the isotropic Laughlin state. The anisotropy in interaction,
however, is expected to be imprinted, e.g., in the static structure
factor of the resulting incompressible liquid. The projected
static structure factor is defined as
14
S
0
(q) =
1
N
e
0|
i=j
e
iq·(r
i
−r
j
)
|0, (4)
where |0is the calculated ground state and r
i
is the coordinate
of the ith particle.
In Fig. 2(a), we draw the three-dimensional (3D) and
contour plots of the structure factor for a calculated FQH
state at A
c
= 1.8. It exhibits a craterlike feature, which is
similar to that of the isotropic liquid. However, the overall
shape of the crater is deformed, stretching along the hard
axis direction. Therefore, the elliptical symmetry replaces the
circular symmetry in the isotropic liquid case.
This anisotropic signature is more prominent in the 2D
cuts along the two principal axes as shown in Fig. 2(b).We
note that the structure factor behaves asymptotically as ∼ q
4
in the long wavelength limit. This agrees with the single
mode approximation
16
(SMA) for an incompressible liquid.
However, the prefactor of the quartic term is orientation
dependent, revealing the anisotropic nature of the structure
factor. According to Ref. 6, the ratio of prefactors at the q
x
and
q
y
axes is equal to (1/A
∗
L
)
4
, where the parameter A
∗
L
defines an
intrinsic metric, describing how the correlated quasiparticles
bind to each other in the anisotropic environment. The fitting
lines in the plot have revealed A
∗
L
∼ 1.45 for interaction
anisotropy A
c
= 1.8. The peaks in the orientation-dependent
plots represent the crater ridge in Fig. 2(a).
According to the SMA, the maximum in the structure factor
corresponds to a minimum gap in the excitation spectrum,
or the roton minimum, which corresponds to the excitonic
binding of the neutral quasiparticle-quasihole pairs.
16
This is
evident in Fig. 2(c), where we plot the orientation-dependent
035122-2
FRACTIONAL QUANTUM HALL STATES IN TWO- ... PHYSICAL REVIEW B 86, 035122 (2012)
-
6
-
4
-2
0
2
4
6
-
6
-
4
-
2
0
2
4
6
0.0
0.5
1.0
1.5
-6 -4 -2 0 2 4 6
q
y
q
x
0.000
0.3000
0.6000
0.9000
1.200
1.500
S
0
(q)
(b)
~0.7q
4
S
0
(q)
~0.16q
4
0.1
(c)
Δ
E
q
x
q
y
(a)
0.0
FIG. 2. (Color online) (a) 3D and contour plots of the structure
factor, (b) the structure factor along the q
x
and q
y
axes, and (c)
the excitation spectrum along the q
x
and q
y
axes for the N
e
= 10
system with Coulomb interaction anisotropy A
c
= 1.8. To overcome
the discrete momentum limitation, we use unit cells with different
aspect ratios R
a
= L
x
/L
y
in (b) to obtain more data points and a unit
cell with R
a
= 0.5in(c).
low-energy excitation spectra in the momentum space. The
location of the roton minimum is sensitive to the direction, but
the gap value is less sensitive. The ratio of the roton-minimum
locations along the q
x
and q
y
axes is found close to A
∗
L
as
expected, and these two locations match the peak locations of
the structure factor in Fig. 2(b).
The isotropic Laughlin wave function, with order-3 zeros
at the locations of other particles, triumphed in the explana-
tion of the isotropic incompressible liquids of the ν = 1/3
FQH system. Corresponding to the deformed electron-hole
correlation from the anisotropic interaction, order-3 zeros in
the wave function are expected to split. Several works have
suggested that the relative coordinate part of the anisotropic
wave function
5,10–13
has the form
w(z
i
) ∼
i<j
(z
i
− z
j
)
(z
i
− z
j
)
2
+ z
2
0
, (5)
where z
i
is the complex coordinate of the ith particle and z
0
is
a complex constant related to the splitting of the zeros due to
anisotropy. This zero-splitting effect in the wave function can
be detected using the pair correlation function defined as
17
g(r) =
L
x
L
y
N
e
(N
e
− 1)
0|
i=j
δ(r − (r
i
− r
j
))|0. (6)
In Fig. 3(a), we plot the pair correlation functions along the
ˆ
x
and
ˆ
y directions for the ν = 1/3 FQH state with the interaction
anisotropy A
c
= 1.8. The two curves are distinguishable from
their isotropic counterparts. We point out that the curves
behave asymptotically as αr
2
in the limit of r → 0, with the
0246
0.0
0.5
1.0
0.0 0.1 0.2 0.3
1.2 1.6 2.0
0.0
0.2
r
g(r)
Isotropic g(r)
g(x)
g(y)
(a)
α
r
2
(b)
A
c
FIG. 3. (Color online) (a) Pair correlation function along the
ˆ
x
and
ˆ
y axes for the N
e
= 10 system at interaction anisotropy A
c
= 1.8.
The dotted-dashed line represents the correlation function at A
c
= 1
for comparison. The inset reveals the αr
2
behavior at small r for
the anisotropic case. (b) A linear fit for the prefactor
√
α at small
anisotropy.
prefactor α ∝|z
0
|
4
. This is entirely different from the isotropic
Laughlin wave function, which exhibits an r
6
asymptotic
behavior in its pair correlation function. The nonmonotonic
behavior in g(y) at the small r region, which manifests
itself more clearly at a larger A
c
, is also consistent with the
zero-splitting scenarios.
5,13
We point out that for a suitable
deformed model wave function, there is also an additional
contribution to the Gaussian Landau level form factor,
5,18
which can be observable in the disk geometry with a boundary.
In Fig. 3(b), we plot the square root of the prefactor α
at several small anisotropy. The linear fit of
√
α to A
c
is
expected as (A
c
− 1) [or (
√
A
c
− 1)], which characterizes
the perturbation away from the isotropic point. However, the
resulting nonzero intercept at A
c
= 1 suggests that we may
have overestimated the prefactor, possibly due to the higher
order contributions at small A
c
.
B. Hall-Smectic-like phase in the intermediate interaction
anisotropy regime
The explicit construction
5
of the model wave function by
unimodular transformation on the disk geometry suggests that
the geometrical description of the quantum Hall system accepts
the following deformation of the isotropic Laughlin state (i.e.,
γ = 0):
=
i<j
z
ij
z
2
ij
+
12γ
∗
1 −|γ |
2
e
−
i
γz
2
i
/4
e
−
i
|z
i
|
2
/4
, (7)
where z
ij
= z
i
− z
j
and γ characterizes the amount of mixing
between the guiding center creation and annihilation operators
in the unimodular transformation. Note that the model wave
function is expected to be valid for small γ . In the present
parametrization γ =
A
∗
L
− 1 is real. We can postulate the
breakdown criterion for the anisotropic Laughlin liquid to be
π
12γ
∗
1 −|γ |
2
=
2π
ν
, (8)
i.e., the area occupied by a set of three splitting zeros is the
average area per particle. This suggests that at the breakdown
035122-3
HAO WANG, RAJESH NARAYANAN, XIN WAN, AND FUCHUN ZHANG PHYSICAL REVIEW B 86, 035122 (2012)
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6
0
6
0
2
4
-
6
0
6
-
6
0
6
0
1
2
-
6
0
6
-
6
0
6
0
.
0
0
.
6
1
.
2
1
.
8
-
6
0
6
110
0
8
A
c
=20.0
q
y
q
x
(c)
A
c
=11.0
A
c
=5.0
(a)
(b)
(d)
CDW peak
Crystal peak
S
0
(q
*
)
A
c
50
FIG. 4. (Color online) (a)–(c) Structure factors for the ground
states of the N
e
= 10 system at A
c
=5.0, 11.0, and 20.0, respectively.
Note that the locations of the twin peaks for A
c
= 5.0 (CDW-like
peaks) differ from those for A
c
= 20.0 (crystal peaks). Both types
of peaks coexist at A
c
= 11.0. (d) Peak values of the CDW-like and
crystal peaks are plotted as a function of the interaction anisotropy in
a range of 1 <A
c
< 50.
A
∗
L
≈ 2 (i.e., A
c
≈ 3 according to the estimation in the
Sec. III D), consistent with the onset of the rapid decrease
of the excitation gap.
In other words, the anisotropic Laughlin liquid is stable
when the long-distance (i.e., at average particle spacing)
behavior of the Jastrow factor is still as z
1/ν
ij
. The collective
excitation of the liquid is the neutral magnetoroton excitations,
which becomes anisotropic. When the liquid phase breaks
down, it cannot sustain further anisotropy by the spatial
deformation in the roton spectrum.
One possible outcome of the system after this breakdown
is that the mode at the roton minimum goes softer, developing
some charge-density-wave (CDW) order. Due to the orien-
tation effect of the anisotropy, this CDW is expected to be
unidirectional (stripelike), and the characterizing sharp peaks
in the structure factor are along the stretching direction. This is
clearly visible in the structure factor at A
c
= 5.0in Fig.4(a).
The background in the structure factor resembles that of an
anisotropic Laughlin liquid, but its peak value is significantly
smaller than the two sharp peaks along the q
x
axis. The CDW
twin peaks correspond to a period in real space, which can be
roughly anticipated as the splitting of zeros |z
0
|=
√
6atthe
critical A
∗
L
= 2. The peak value (subtracting background) in
the structure factor suffices as the order parameter. The plot in
Fig. 4(d) shows that this CDW-like order parameter rules the
system in the regime 2.0 <A
c
< 8.0.
We term this phase, which breaks one-dimensional trans-
lational symmetry, as a Hall-smectic-like phase since we
speculate that it is related to the Hall smectic discussed
earlier in the context of liquid crystal phases in the FQH
system.
10–12,19–21
The rise of the smectic phase softens the
magnetoroton mode and appears to be responsible for the
reduction of the excitation gap for 2.0 <A
c
< 8.0asshownin
Fig. 1.AsdiscussedinRef.11 the transition from the Laughlin
liquid to the Hall smectic can be second order and its critical
behavior is in the XY universality class.
Beyond A
c
= 8.0, the reverse trend in the excitation gap
as a function of A
c
indicates that the system is under the
influence of a distinct mechanism. This is evidently shown
in the structure factor plot of Fig. 4(b) at A
c
= 11.0. Two
additional peaks along the q
x
axis are clearly visible with the
different wave vectors from the CDW-like twin peaks. These
additional twin peaks are corresponding to the unidirectional
crystal order in the quasi-1D repulsion limit that we will
discuss in Sec. III C. The peak value of them is plotted as
the crystal order parameter in Fig. 4(d). There we can see that
the crystal order parameter is continuously increasing with
the interaction anisotropy. The crossover for the competition
with the CDW-like order occurs around A
c
= 13.0. For larger
anisotropy the crystal order dominates as illustrated in Fig. 4(c)
at A
c
= 20.0, where only the crystal peaks remain. Thus, the
Hall-smectic-like phase is found unstable towards a 1D crystal.
C. Quasi-1D crystal in the large anisotropy limit
The crystal phase at the large anisotropy limit can be
probed using the charge distribution. In Fig. 5(a),weplot
the average LL orbital occupations and the charge density
along the hard axis at A
c
= 40.0. The charge density appears
smoother as an integral from local Gaussian wave packets
over orbital occupations, and a 2% fluctuation above the
background can be observed in the exaggerated plot. Both
the charge occupation and density fluctuate along the hard axis
with the crystalline period λ
∗
= L
x
/N
e
. The maximum charge
occupation is close to unity as expected in the ultimate 1D
crystal limit. The 2D distribution of the charge density reveals
that the system is a unidirectional crystal with each electron
spreading into a stripe perpendicular to the hard axis. For basis
states in the torus geometry, the guiding-center coordinate
along the hard axis is coupled with the momentum along the
q
y
axis.
17
Thus, the calculated ground states are expected to
carry a period of 2π/λ
y
in the momentum space along the q
y
FIG. 5. (Color online) For ground states of the N
e
= 10 system
at A
c
= 40.0. (a) Average occupations (scattered squares) of the
Landau-level orbital at guiding centers and the charge density (solid
line) are plotted along the hard axis. A period of λ
∗
= L
x
/N
e
is
clearly visible. The density values have been shifted and exaggerated
to emphasize a ∼ 2% fluctuation over the background. (b) Structure
factor with sharp twin peaks in the
ˆ
q
x
direction and periodic
oscillation in the
ˆ
q
y
direction. (c) Pair correlation function is plotted
along two principal axes. The function g(y) shows an oscillation with
the period L
y
/3.
035122-4
FRACTIONAL QUANTUM HALL STATES IN TWO- ... PHYSICAL REVIEW B 86, 035122 (2012)
axis, where λ
y
= L
y
/3. The structure factor plot in Fig. 5(b)
demonstrates this characteristic order along q
y
axis. As its real
space counterpart, the pair correlation plot in Fig. 5(c) shows
oscillation in the
ˆ
y direction with a period λ
y
.
The above results support that at filling ν = 1/3 the ground
state of the system is a crystal in the large interaction anisotropy
limit and the system undergoes some transition from an
incompressible liquid to a solid as anisotropy increases. A
similar story has been discussed in an isotropic FQH system
with extreme geometry, such as in a thin torus or a cylinder
limit,
22–24
and in a recent work
25
on the graphene ribbon with
flat bands. They can be explained under the same principle
in Ref. 24 by a sorting Hamiltonian. When the interaction
anisotropy increases, the repulsion-related diagonal terms
dominate, which has the similar effect as geometry on the
isotropic FQH and as the local orbital expansion on the
flat-band graphene ribbon. The low-energy physics is governed
by the strong repulsion so that the system tends to form crystal.
At small anisotropy, the hopping-related off-diagonal terms are
comparable and screen the repulsion, resulting in the liquid
phase.
D. Generalized variational Laughlin state
In the discussion above, we have seen that the isotropic
Laughlin wave function is insufficient to fully capture the
features of an anisotropic FQH system. For such a system
at the lowest LL filling ν = 1/q, Haldane has suggested to
use a family of Laughlin states,
2
which is generally defined
as the densest zero-energy eigenstate of a projected two-body
anisotropic Hamiltonian:
H
v
(A
L
) =
m<q
P
m
(A
L
). (9)
For the fermion system with an odd denominator q, m are
limited to be odd. This Hamiltonian is a truncated summation
over anisotropic pair interactions:
P
m
(A
L
) =
1
N
φ
q
L
m
(Q
2
)e
−Q
2
/2
i<j
e
iq·(R
i
−R
j
)
(10)
for two particles with the relative angular momentum of
m¯h in the guiding-center coordinates. In the above expres-
sion, L
m
(x)aremth Laguerre polynomials and Q(A
L
) =
q
2
x
/A
L
+ A
L
q
2
y
, which, like in Eq. (3), defines a wave-
function metric parameterized by A
L
. The parameterized
Laughlin states (A
L
) satisfy
P
m
(A
L
)|(A
L
)=0,m < q. (11)
The isotropic Laughlin wave function corresponds to the
Laughlin state with A
L
= 1. With this family of parameterized
states, we are able to variationally approximate the ground
state of a FQH system with anisotropic interaction. According
to Haldane’s proposal, if the mass or orbital metric (in our
case, an identity matrix for isotropic mass) is different from
the interaction metric (parameterized by A
c
), the resulting
variational state (A
∗
L
) should be described by a metric
interpolating the mass metric and the interaction metric, i.e.,
1 <A
∗
L
<A
c
. This intrinsic metric describes how correlated
1.0 1.5 2.0
0.6
0.8
1.0
E
a
(arb. unit)
|<A
L
|A
c
>|
2
A
L
N
e
=6
N
e
=8
N
e
=10
N
e
=12
FIG. 6. (Color online) Wave-function square overlap (lines) and
expected value of Coulomb energy (symbols) as a function of the
variational parameter A
L
for the ν = 1/3 FQH system at interaction
anisotropy A
c
= 1.8. The energy values have been shifted and
enlarged to emphasize that the location of the minimum coincides
with that of the largest overlap. The comparison of different sizes of
N
e
= 6, 8, 10, and 12 shows a weak size dependence only.
quasiparticles effectively feel each other in such an anisotropic
FQH system.
In Fig. 6, we study the anisotropic ν = 1/3 FQH system
with the Coulomb anisotropy A
c
= 1.8. The optimal Laughlin
state (A
∗
L
) is obtained by tracing either the maximum of
the wave-function overlap or the minimum of the expected
Coulomb energy
E
a
(A
L
) =(A
L
)|H
c
(A
c
)|(A
L
). (12)
The optimal parameter is found at A
∗
L
∼ 1.43, which is weakly
size dependent. This parameter is indeed an intermediate value
between unity and the Coulomb anisotropy as expected.
2
It
also agrees with the intrinsic metrics through the analysis of
the anisotropic structure factor in the Sec. III A. The overlaps
between the optimal Laughlin state, and the exact ground
states are larger than 90% for various system sizes, which
supports the validity of the variational state. We also note
that the expected Coulomb energy quadratically approaches
its minimum, which suggests a linear approximation of the
anisotropic Laughlin state with A
L
in the liquid phase regime.
To gain a further understanding for the validity of this
variational approach, we approximately expand the Coulomb
interaction in the anisotropic pair interactions as
H
c
(A
c
) ≈
m
V
m
(A
c
,A
L
)P
m
(A
L
), (13)
where the average expansion coefficients V
m
define the
effective anisotropic pseudopotentials in a form of
V
m
=
2π
0
dθ
∞
0
dx
L
m
(x
2
)e
−F (θ,x)/2
2π
√
G(θ)
(14)
with
G(θ) = (A
c
/A
L
) cos
2
θ + (A
L
/A
c
)sin
2
θ (15)
and
F (θ,x) = x
2
(1 + A
L
cos
2
θ + sin
2
θ/A
L
). (16)
An example of V
m
is plotted in Fig. 7(a) with A
c
= 1.8
and A
L
= 1.47. These pseudopotential parameters are found
035122-5
