EUROPHYSICS LETTERS
Europhys. Lett.
,
66
(4), pp. 572–578 (2004)
DOI:
10.1209/epl/i2003-10234-8
M. M. Fogler
Department of Physics, University of California San Diego
9500 Gilman Drive, La Jolla, CA 02139, USA
(received 15 December 2003; accepted in final form 17 March 2004)
PACS.
73.43.-f
– Quantum Hall effects.
PACS.
73.43.Lp
– Collective excitations.
PACS.
73.43.Nq
– Quantum phase transitions.
15 May2004
Abstract.
– I propose an effective theory of zero-temperature phases of the quantum Hall stripes: a smectic phase where the stripes are static and a novel quantum nematic phase where the positional order is destroyed by quantum fluctuations. The nematic is viewed as a Bose condensate of dislocations whose interactions are mediated by a
U
(1) gauge field. Collective mode spectrum and the dynamical structure factor in the two phases are calculated.
Motivation. – Stripe phases are extremelycommon in nature. Once studied mainlyin the context of pattern formation and soft condensed-matter systems (convection rolls, ferrofluids, diblock co-polymers, etc. [1]), they are now recognized to be important in venues ranging from neutron stars [2] to neural networks of the brain [3]. The “hard” condensed-matter communitywas alerted to the relevance of stripes after their discoveryin transitional metal oxides, especially, highT c cuprates [4]. The subject of this letter is the stripe phase in another interesting correlated system: a two-dimensional (2D) electron liquid in a transverse magnetic field. This phase forms when the Landau level occupation factor ν is close to a half-integer and is larger than some critical value
ν c
[5]. Unlike the still debated case of high
T c
, the theory of the quantum Hall stripes rests on a much more solid foundation [5, 6]. The quantity1
/ν plays the role of a small parameter controlling the strength of quantum fluctuations beyond mean field. Thus, the mean-field stripe solution is stable and adequatelycaptures the main properties of the ground state at
ν ν c
. Numerical calculations [7, 8] indicate that
ν c
∼
4 for the physically relevant case of Coulomb interaction, and so the magnetotransport anisotropy observed in GaAs heterostructures [9] at
ν ≥ 9
2 is naturallyexplained bythe stripe formation.
Still, the theoryof the stripe phases at modest
ν cannot be perceived as complete because at such
ν the fluctuations about the mean-field Hartree-Fock solution are large and several nearlydegenerate states are in competition. It is then possible that a slight modification of the microscopic parameters bymeans of, e.g.
, an in-plane magnetic field [9], clever sample geometry, etc., may bring to life novel ground states. Particularly intriguing are proposed quantum smectic and quantum nematic phases [10–15]. In the smectic phase the system lacks translational symmetry(is periodicallymodulated) along one of the spatial directions, say ˆ ,

M. M. Fogler
:
Theory of incompressible quantum Hall liquid crystals
573
Fig. 1 – Left: pictorial representation of stripes in (a) smectic, (b) nematic. One dislocation is encircled. Right: worldlines of dislocations in (c) smectic, (d) nematic.
but remains liquid-like along ˆ . An example of the smectic phase is a unidirectional chargedensity-wave Hartree-Fock state [5]; however, the true quantum smectic (see fig. 1(a)) must possess a certain amount of fluctuations beyond Hartree-Fock to prevent another periodic modulation, along ˆ [12, 16, 17]. The nematic state is fullytranslationallyinvariant but lacks the rotational invariance. It can be visualized as stripes that fluctuate stronglyand are riddled with dislocation defects yet retain a preferential alignment in the ˆ -direction, see fig. 1(b).
An interesting trial wave function for the nematic was proposed byMusaelian and Joint
(MJ) [11]:
Ψ = i<j
( z i
− z j
) ( z i
− z j
)
2 − a 2 exp
− | z k
| 2 /
4 l 2 ,
(1) k where z j
= x j
+ iy j is the complex coordinate of the j
-th particle, l is the magnetic length, and the complex parameter a of dimension of length sets the characteristic width of the stripes and their preferential orientation. The MJ wave function (1) corresponds to the filling fraction 1
/
3 and is intimatelyrelated to the 1
/
3 Laughlin liquid. (The Laughlin liquid is obtained from the MJ state bysetting a to zero.) Just as in the Laughlin state, the compact charge excitations in the MJ nematic are expected to have the charge e/
3 and an energy gap. Thus, the MJ state is incompressible , and so should exhibit the fractional quantum
Hall effect (FQHE). In contrast, other nematic trial states recentlysuggested for the 1 / 2filling [13, 15] should have gapless charge excitations and no FQHE. Unfortunately, none of the proposed trial states have a clear connection to the Hartree-Fock ones [5], and therefore, to the great deal of the correct physics that it presumably captures. While the search for other trial wave functions continues, it maybe worthwhile to approach the theoryof quantum nematics byother means, in particular, bytrying to construct their effective long-wavelength low-energytheory. This line of work was initiated byBalents [10] for the case of incompressible nematics, and byRadzihovskyand Dorsey[14] for the compressible case. Here I focus on the former (incompressible) class of states, in the conviction that it is more amenable to controlled analytical and numerical [11, 15] investigations. I discuss the zero-temperature properties and possible quantum phase transition between such states [18]. At present, the motivation for this studyis mainlytheoretical, but if the states with a FQHE and a broken rotational symmetry are ever discovered experimentally, it may prove to be relevant. The experimental implications of the proposed theoryare discussed at the end of this letter.
Smectic phase. – The smectic emerges from a uniform state when a pair of collective modes with wave vectors
Re { Ψ( r , t ) e − iq
0 x } , where q n
= ± q
0
ˆ goes soft and condenses [19]. Hydrodynamic fluctuations of the electron densityprojected onto the topmost Landau level become of the form n ( r , t ) + is a smooth densitycomponent (with q q
0
), Ψ = Ψ
0 e iq
0 u is the smectic order parameter, and u is the deviation of the stripes (densitymaxima) from

574
EUROPHYSICS LETTERS uniformity, see fig. 1(a). Since we are interested in the incompressible states, the Goldstone mode associated with u is the onlylow-energydegree of freedom of the system. In this case, the effective Hamiltonian for u and n , containing onlythe most relevant terms consistent with the symmetry [20–22], is
H =
Y
2
( ∂ x u )
2
+
K
2
∂ 2 y u 2
+
1
2
δn U δn + Cδn∂ x u .
(2)
Here
Y and
K are phenomenological compression and bending moduli,
δn
= n − n
0 deviation of local densityfrom the equilibrium value n
0
, and
U is the is the integral operator with kernel
U
( r
), the electron-electron interaction potential.
U is Coulombic at large r but is modified bymany-bodyscreening, exchange and correlation effects at short distances (see more details in [5]). The last term in eq. (2) accounts for the dependence of the smectic period on n
0
, with
C
=
Y ∂ ln q
0
/∂n
0
. It vanishes at the half-filling due to electron-hole symmetry but is nonzero otherwise. For the Hartree-Fock smectics,
Y and
K can be calculated microscopically[16, 23, 24] but, as explained above, at
ν ∼
4 a considerable softening of these parameters byquantum fluctuations maybe expected. Strictlyspeaking, because of such fluctuations and anharmonic couplings neglected in eq. (2), Y and K are not constants but some functions of q . However, in the present (2 + 1)D quantum theory, unlike the finiteT case [21], these are weak, most likely, logarithmic effects, which are neglected here.
One maywonder whyto include “incompressible” background n in our low-energyHamiltonian (2). The reason is the important role n plays in the dynamics of u
. Indeed, u determines the densityfluctuations near the star of the soft mode, the densityoperators n q e.g.
, n q
0 + k
= (Ψ
0 e projected onto a single Landau level are dynamically linked by a well-known commutation relation [25], which at small q takes the form
1
2
− iq
0 u
) k
, while n q
, n q
0 + k
= il 2
ˆ [ q × ( q
0
+ k )] n q
+ q
0 + k
.
(3)
Since we are dealing with the smectic condensate, we can use the standard Bogoliubov approximation, according to which on the right-hand side of eq. (3) we can neglect all operators n q + q
0 + k whose expectation value is zero and retain those of them whose expectation value is a macroscopicallylarge quantityapproximatelyequal to Ψ
0
L x
L y
/
2, where
L x
L y is the area of the system. In this manner we obtain the commutation relation
[ n q
, u k
] = (2
π
)
2 l 2 q y
δ
( q
+ k
)
.
(4)
Introducing a canonical momentum p by δn = − ∂ y effective imaginary-time action [26], p , eqs. (2) and (4) can be unified into the
A sm
=
0
β d
τ d
2 r − l i
2 p∂
τ u
+
Y
2
(
∂ x u
)
2
+
K
2
∂ u 2
+
1
2
(
∂ y p
)
U
(
∂ y p
)
− C∂ x u∂ y p .
(5)
This general form passes two important tests. First, after a change of variables it reproduces the effective actions derived for the Hartree-Fock smectics [17, 22, 24]. Second, the density structure factor, easilycalculated for the Gaussian theory(5),
S
( q , ω
) m
Im q Q ( q )
Q ( q ) ω 2 ( q y
/q ) 2 − ω 2 ω 2 − iωδ
,
(6) coincides up to the Bose factor and dissipative terms with the finite-temperature result of ref. [21]. The notations used here are Q ( q ) = ( ˜ + Kq ) /mn
0
, ˜ = Y − C 2 /U , and

M. M. Fogler
:
Theory of incompressible quantum Hall liquid crystals
575
ω p
( q ) = [ n
0
U ( q ) q 2 /m ] 1 / 2 . On the ω > 0 side, frequencyof the magnetophonon mode [21]
S ( q , ω ) consists of a single δ -function at the
ω
( q
) = ( q y
/q
)(
ω p
/ω c
)
Q
( q
)
.
(7)
This equation applies for q much smaller than the inverse stripe width, i.e.
, q
1
/l
, and predicts that
ω ∝ q 3
/
2 , unless q is nearlyparallel to the stripes. The latter agrees with the results of manyother authors [16, 17, 22], with the exception of Aoyama et al.
[27].
Dislocations and duality. – 2D smectics can exist onlyat zero temperature. At
T >
0, thermal fluctuations of the stripes restore the translational symmetry, so that the highest possible degree of ordering is that of the nematic [20]. The actual crossover of the Hamiltonian from the short-distance smectic (2) to the long-distance nematic form is quite nontrivial. It is driven bythermallyexcited dislocations, which have an abilityto screen the compressional stress [28]. It is natural to assume then that the smectic-nematic quantum phase transition is also driven bytopological defects. Pictorially, the difference between the smectic and nematic can be represented as follows. The dislocations are viewed as lines in the (2+1)D space. In the smectic phase, theyform small closed loops, see fig. 1(c) that depicts virtual pair creationannihilation events. In the nematic phase, arbitrarilylong dislocation worldlines exist and mayentangle (fig. 1(d)), similar to worldlines of particles in a Bose superfluid [29].
Now I present a mathematical formalism supporting these qualitative ideas. It is analogous to the dualitytransformation employed byFisher and Lee [30].
The first step is to incorporate the dislocations into the effective action (5).
This is accomplished byfactorizing the smectic order parameter, Ψ = Ψ single-valued function and Ψ
D is the phase factor due to the dislocations. The derivatives of u in eq. (5) should now be replaced by
∂
µ u −
( i/q
0
)Ψ
∗
D
∂
µ Ψ
D
≡
(
∂
µ
0 u
) tot e the quadratic part of the action with a Hubbard-Stratanovich field
σ iq
0 u ×
Ψ
D
, where u is a
,
µ
=
τ, x, y
. Decoupling
µ , one obtains
β
A
=
A
D
+ d
τ d
2 r
[
− i
(
∂
µ u
) tot
σ
µ
− H a ]
,
0
(8)
H a
= − i
C
Y l 2 σ x
∂ y
σ
τ
+
σ
2
Y
+ σ y
− 2 K∂ 2 y
−
1 σ y
+ l 4
2
∂ y
σ
τ
U −
C
Y
2
∂ y
σ
τ
, (9) where = 1,
σ
τ
≡ p/l 2
, and
A
D contains terms describing dislocation cores (see below).
Integration over u gives the constraint
∂
µ
σ
µ = 0, which can be implemented bymeans of an auxillary
U
(1) gauge field a
µ ,
σ
µ
=
&
µνλ
∂
ν a
λ
≡
[
∂ × a
]
µ
.
(10)
This brings the action to the form
A dual
=
A
D
+
0
β d
τ d
2 r
(
− i
Λ a
µ j
µ
+
H a
)
,
(11) where Λ = 2 π/q
0 and j
µ
= (2 πi )
−
1 &
µνλ
∂
ν
(Ψ
∗
D
∂
λ
Ψ
D
) is the dislocation 3-current. The latter can be expressed in terms of the second-quantized bosonic [31] dislocation field Φ, j
µ = t
µ Φ
∗
(
− i∂
µ + Λ a
µ )Φ
,
(12) which leads to the action
A dual
=
0
β d
τ d
2 r t
µ
2
|
(
− i∂
µ
−
Λ a
µ )Φ
| 2
+
V
(Φ) +
H a [ a
]
.
(13)

576
EUROPHYSICS LETTERS
Equations (9), (10), and (13) define the desired dual theory. The fundamental objects in theoryare Φ bosons interacting with each other and with the U (1) gauge field a
µ phenomenological parameters introduced above are as follows. Parameter t
τ
E c is the dislocation core energy. It was estimated within the Hartree-Fock approximation in refs. [5] and [23]. At ν ∼ 4, where the quantum fluctuations overlooked bythe Hartree-
Fock approximation are significant,
E challenging. Parameter t x of dimension of energy
×
(length) x c
-direction, can be greatlyreduced, but anyprecise estimate is i.e.
, dislocation glide tunneling and is exponentiallysmall unless Λ l
. Parameter t
2
∼ 2 /E c
. The
, where
. Such a glide requires quantum y is the hopping matrix element governs the dislocation climb, which also originates from the dynamics on the microscopic length scales. One may recall that the climb requires mobile point defects. Although those are not among fundamental low-energyexcitations of the theory e.g.
, short stripe segments or dislocation pairs, do exist, and so mayassist the climb. In general, I am not aware of any theorem that would protect r
Φ
|
Yet another phenomenological variable in eq. (13) is the potential
V
(Φ) = m
Φ
| 4 t y = 0 value, but it seems reasonable that t y t x in our case.
Φ
|
Φ
| 2
+
+
· · ·
, which accounts for a self-energyand a short-range interaction between the dislocations. Note that the dualitytransformations in continuum models are known to overlook terms like
V
(Φ), and so
V was added byhand (cf. [30]). The scales of m
Φ and r
Φ are set by
E c and E c
Λ 2 , respectively.
Another few comments are in order. The derived theoryis meant to capture onlythe dynamics of neutral (dipolar) excitations of the system. The underlying incompressible state has its own dynamics characterized, most importantly, by the quantized Hall conductivity.
Finally, if one decides to integrate out the gauge field a
µ from
A dual
, one can obtain a model of dislocation lines interacting via an effective Biot-Savart potential (which in our case is short-range for Coulombic U ). One maythen argue [32] that a proliferation of the dislocation loops occurs when the energycost for creating a large loop of length L is compensated bythe
“entropic gain” L/l
0
, where the persistence length l
0 is determined byshort-range physics.
Despite the physical appeal of this argument, the gauge theory (13) is presumably better suited for a quantitative analysis (perhaps, along the lines of ref. [33]).
Phases and their collective modes. – Let us now see how the smectic and nematic states are reproduced in the dual theory. As discussed above, the smectic phase corresponds to
Φ = 0. In this case, A dual reduces to H a
, which is quadratic. The low-energydynamics is that of a gas of noninteracting Goldstone bosons, which are the aforementioned magnetophons.
It is a simple exercise to verifythat their dispersion relation is given byeq. (7).
In the nematic phase dislocations have condensed, Φ = Φ
0
= 0. In conventional local
U (1) gauge theories, the appearance of such an order parameter triggers the Anderson-Higgs mechanism, eliminating the gapless Goldstone modes. This cannot be the case here because the nematic state does break the continuous symmetry with respect to spatial rotations.
The seeming paradox is resolved due to the peculiar feature of the present gauge theory: the nonlocalityof the gauge-field strength term
H a
, see eq. (9). Byvirtue of that, the condensation of Φ merelystiffens the collective mode, leaving it gapless. Neglecting terms proportional to
C
, I find that
1
/
2
ω
1
( q
) = m x m
τ q
+ m x
Kq , m
µ
≡ t
µ
Λ
2 |
Φ
0
| 2 .
(14)
Thus, the Goldstone mode dispersion is acoustic, in agreement with the earlier result of
Balents [10]. In contrast, in a recent work of Radzihovskyand Dorsey[14], the collective mode dispersion in the nematic was predicted to be superlinear, i.e.
, softer than in a smectic. It

M. M. Fogler
:
Theory of incompressible quantum Hall liquid crystals
577 seems that this discrepancyoriginates from different assumptions made about the dislocations dynamics in the present work and in ref. [14].
at q
The magnetophonon mode of the smectic (7), in the nematic acquires a small gap m y
= 0. It anti-crosses with the acoustic branch (14) near the point ω 2
1
( q ) ∼ m larger q becomes the lowest frequencycollective mode with the dispersion relation y
Y
Y , and at
ω
2
( q
) = q 2 q 2 m 2 ω 2
Y U
( q
) + m y
Y
1 / 2
(15) onlyslightlydifferent from (7). At such q
, the structure factor of the nematic has two sets of
δ
-functional peaks,
S
(
ω, q
) =
π q mω 2
Kq mn
0
δ ω 2 − ω 2
1
+
Y q mn
0
δ ω 2 − ω 2
2
, which split between themselves the spectral weight of the single collective mode of the smectic.
The presence of the two modes can be explained bythe existence of two order parameters: a unit vector (more precisely, director)
N normal to the local stripe orientation and the complex wave function Φ
0 of the dislocation condensate. Classical 2D nematics have two (overdamped) modes virtuallyfor the same reason [28].
Finally, I wish to address measurable properties of the novel quantum Hall states considered above. At low temperature, both the parent incompressible state and its liquid-crystal descendants will show FQHE. If T is not too small, formation of stripe superstructures can be deduced from the anisotropic magnetoresistance [9]. On the other hand, the microwave absorption will be anisotropic even at low T and would enable one to further distinguish between the smectic and nematic phases: the nematic will show two dispersing collective modes while the smectic will produce a single one. To circumvent disorder pinning effects, such measurements should be done at high enough q
. As mentioned above, our predictions regarding the functional form of the collective mode dispersion differ from that of ref. [14]. Further work is needed to resolve this controversy.
∗ ∗ ∗
This work is supported byMIT Pappalardo Fellowship in Physics and byC. & W. Hellman
Scholarship at UCSD. I wish to thank
X.-G. Wen for useful discussions, and also
A. Dorsey,
L. Radzihovsky, and
C. Wexler for valuable comments on the manuscript.
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