# Instabilities Toward Charge Density Wave and Paired Quantum Hall State of Half-Filled Landau Levels

###### Abstract

We study the stability of spin-resolved Landau levels at electron filling factor , where is a positive integer. Representing the half-filled topmost Landau level by fermions and the filled inner Landau levels by bosons, coupled to the Chern-Simons gauge fields, we show that the ground states exhibit charge density wave order for , where is the 2D carrier density and is the effective Bohr radius. We find that the pairing interaction mediated by the fluctuating gauge field is enhanced near the charge density wave instability such that p-wave pairing of the Chern-Simons fermions prevails for . The competition between charge density wave order and paired quantum Hall state is discussed in connection to recent experiments.

###### pacs:

PACS numbers: 73.20Dx,73.40Kp,73.50.Jt[

]

The physics of a sufficiently clean two dimensional electron system in strong perpendicular magnetic field has been highlighted by its correlated many-body ground state of profound origin known as the fractional quantum Hall liquid [1]. Recently, it has become evident that a different family of correlated ground states emerges when the Landau level (LL) filling factor is centered around , , corresponding to filled inner LLs and a half-filled valence LL. For , an anomalous “metal” state has been known in which is not quantized and shows a shallow minimum [2]. In sharp contrast, quantized Hall plateaus in and a low-temperature activation dominated has been observed at [3, 4], indicative of an incompressible quantum Hall state. Most recently, it has been discovered that beyond the second LL (), the low-temperature magnetotransport becomes highly anisotropic and nonlinear, the strong resistance peak turns into a deep minimum when the current direction is rotated by 90 degrees [5, 6]. The anomalous anisotropic transport can be interpreted as a manifestation of a spontaneously broken orientation symmetry in the ground states at [7, 8], such as in a unidirectional charge density wave (CDW) state, which has been predicted to occur in higher LLs by the Hartree-Fock theory [9, 10] and numerical diagonalizations [11]. Experiments have shown that the isotropic incompressible quantum Hall state at is destroyed when a large in-plane component of the magnetic field is present, taking its place is a state with highly anisotropic resistances similar to those observed in higher LLs [12, 13]. That these ground states are close in energy are further supported by numerical calculations using various interacting potentials [14, 15].

In this paper, we study at zero-temperature the ground state stability of the half-filled LLs by a generalization of the fermion Chern-Simons (CS) theory of Halperin, Lee and Read (HLR) for [16] to . In the theory of HLR, electrons in the lowest LL are represented by fermions bound to two flux quanta through the introduction of the statistical CS gauge field. Uniform smearing of the flux in the direction opposite to the external field leads to a meanfield metallic state in which the CS fermions experience zero effective magnetic field. Using this as the starting point, a CS Fermi liquid coupled to strong gauge field fluctuations has been formulated and shown to correctly describe the longitudinal responses at . Naturally, one would like to understand the ways by which such a Fermi liquid-like state could become unstable due to residual quasiparticle interactions mediated by strong gauge field fluctuations. For example, a p-wave BCS-like pairing instability [17] of the CS fermions would indicate the tendency towards the formation of incompressible paired quantum Hall state [18] of the type proposed by Moore and Read [19]. Moreover, a unidirectional CDW instability would give rise to a state with spontaneous rotational symmetry breaking and anisotropic transport.

While no evidence has been found for such breakdown of the CS Fermi liquid at (), we show in this paper that both CDW and p-wave pairing instabilities exist in higher LLs (finite ). The basic reason for the emergence of these instabilities is the mutual screening of interactions between electrons in the filled inner LLs and in the half-filled valence LL. It produces an effective interaction between the CS fermions that differs from the Coulomb interaction at finite and can become attractive at short distances. We calculate the density-density response and show that the CS Fermi liquid becomes unstable towards a CDW state for , i.e. , where is the effective Bohr radius, is the magnetic length at , and is the 2D carrier density. The onset of the CDW instability occurs at a wave vector . Close to the CDW instability, the pair interactions are enhanced by the strong fluctuations of the gauge field at finite . We find that the CS Fermi liquid becomes unstable towards a p-wave paired (quantum Hall) state for , i.e. . Taking Å for GaAs and /cm, these results predict a CDW instability for () and a p-wave paired quantum Hall state for () in remarkable agreement with experimental findings [3, 4, 5, 6].

We start by writing down an effective theory at , in which the electrons in the filled lower LLs are represented by bosons [20] and those in the half-filled valence LL by fermions [16], coupled to the statistical CS gauge fields. The physical picture is that the electrons in each of the filled LL form an independent incompressible state which will be described as a condensate of the bosons. Excitations above the condensate will have an energy gap . On the other hand, the electrons in the half-filled topmost LL form a compressible CS Fermi liquid supporting low-lying quasiparticle excitations and gauge fluctuations. The Lagrangian density is given by

(1) | |||||

(2) | |||||

(3) | |||||

(4) | |||||

Here and , are the CS fermion and boson fields respectively, is the band mass in GaAs, and is the vector potential for the external magnetic field . We work in the Coulomb gauge where the CS gauge fields and satisfy . Integration over and enforces the constraints on the 2D curl () of the gauge fields,

(5) | |||||

(6) |

which correspond to attaching (even) flux quanta to the fermions and (odd) flux quanta to each component of the bosons, turning them back to the original electrons. For and , the external magnetic field is canceled out exactly by the vacuum expectations of the statistical flux in Eqs (2) and (3). Thus at the mean-field level, both the CS fermions , having a filling fraction , and the CS bosons , having , see zero total magnetic field. The electron filling factor is given by .

The electron interaction is described by in Eq. (4), where is the Coulomb potential, . For GaAs, the dielectric constant . contains the fermion-fermion and the boson-boson interactions and, most importantly, the fermion-boson interaction that couples the topmost and the lower LLs. The latter leads to the mutual screening of the interacting potential between the CS fermions by the bosons and vice versa. A natural question arises as to whether one should treat the fermion or the boson sector of the problem first. We shall follow a close analogy to the classic electron-phonon problem in metals, where the answer is known due to Migdal [21], and first solve the boson problem taking into account the screening by the CS Fermi liquid that has no bosons included. Then we solve for the screened fermion-fermion interaction using the renormalized boson properties. In terms of the total symmetric boson density operator , , the bosonic part of the interaction becomes,

(7) |

with the screened Coulomb interaction,

(8) |

where is the fermion (Coulomb) irreducible density-density correlation function derived by HLR. In the limit and ,

(9) |

The screening of the bare Coulomb potential in the lower LLs by the compressible CS Fermi liquid is of the Thomas-Fermi type in Eq. (8). The screening length is, after making use of Eq. (9), with the effective Bohr radius . The fact that is short-ranged turns out to be crucial for modifying the effective fermion interactions at short distances in the CS Fermi liquid.

With the screened interaction, we now solve the boson part of the problem. Since the bosons see on average a zero total magnetic field, they Bose condense into incompressible states, . The excitations above the condensate can be studied by integrating out the CS gauge field in Eq. (3) and decompose the boson field into density and phase fluctuations, and . Notice that the fermions couple only to the totally symmetric boson density fluctuations in Eq. (4) which is the only mode affected by the interaction. We thus perform a unitary rotation on the boson fields and keep only the totally symmetric mode: . The resulting effective boson theory is given to quadratic order by,

(10) | |||||

with given in Eq. (7). The dynamical density-density correlation can be obtained from Eq. (10),

(11) |

where , and the cyclotron frequency . This mode corresponds to the cyclotron resonance in accordance with Kohn’s theorem. It is interesting to note that the mode disperses quadratically due to the screening of Coulomb interaction by the CS Fermi liquid and becomes overdamped as seen from Eqs (8) and (9).

It is now straightforward to integrate out the boson fluctuations in Eq. (1), using Eq. (11), to obtain an effective theory for the CS fermions in the topmost LL coupled to gauge field fluctuations and ,

(12) | |||||

Here , , is the transverse component of the gauge field, and the fermion-gauge field coupling vertices

(13) |

The renormalized theory differs from the lowest LL theory of HLR only in the effective fermion-fermion interaction, the last term in Eq. (12), where the dielectric function governing the mutual screening effects is given by, from Eqs (4) and (11),

(14) |

with given in Eq. (8). In the limit , the dielectric function behaves as ,

(15) | |||||

(16) |

As a result of the mutual screening, the effective interaction between the CS fermions is significantly modified at finite , and becomes attractive at large enough so long as . Notice that one recovers to leading order the dielectric function obtained by Aleiner and Glazman [22] from projecting out the lower LLs, which was used in the Hartree-Fock studies [9], in the limit , i.e. neglecting the Thomas-Fermi screening of the lower LL interaction by the compressible CS Fermi liquid in the topmost LL. We next demonstrate that the effective theory given in Eqs (12)-(16) produces CDW and pairing instabilities of the CS Fermi liquid outside the regime of the Hartree-Fock theory.

From Eq. (5), it follows that the fermion density-density correlation function is given by

(17) |

To determine the gauge field propagators, , we follow the RPA approach of HLR and integrate out the fermion fields in Eq. (12) to arrive at an effective action for the gauge field,

(18) |

where the inverse propagator matrix is given by

(19) |

for and . In Eq. (19) and . The density response is then, from Eqs (17) and (19),

(20) |

where . Substituting the dielectric functions in Eqs (15) and (16) into Eq. (20), one finds that may diverge at a finite , giving rise to CDW formation. Near , we obtain,

(21) |

where is a -dependent constant and

(22) |

The onset of the instability, i.e. the critical point for the zero temperature phase transition, is marked by , close to which . When , which happens for , the Fermi liquid-like ground state becomes unstable and is replaced by a CDW state with an initial ordering wave vector . Since the charge density is related to the statistical flux density through Eq. (5), the system becomes energetically more favorable to spontaneously condensing the statistical flux into a flux density wave rather than uniformly smearing the gauge flux which is the starting point for the CS Fermi liquid, making the latter unstable. One expects, based on energetic considerations of this effect when there is particle-hole symmetry in the half-filled topmost LL, that the ordering vector . As a result of the enhanced (divergent) static susceptibility near the instability, a unidirectional CDW state in the absence of an in-plane field can be induced by a weak sample anisotropy which may be born out of the underlying crystallography conditions. For , i.e. half-filled valence LL, using , we can rewrite the CDW instability condition as . Taking Å for GaAs and a 2D carrier density cm, the CDW order emerges for , i.e. for , consistent with experimental findings [5, 6].

We next turn to the pairing instability of the CS Fermi liquid towards the paired quantum Hall state. The proximity to the CDW formation enhances the pair interactions mediated by the gauge field fluctuations. The static pair interaction in the Cooper channel is given by

(23) |

where the coupling vertices are given in Eq. (13). In the long wavelength limit, the attractive interaction mediated by the current-density fluctuations () in the p-wave channel [18] is known to be overcome by the logarithmically singular pair-breaking interaction mediated by the transverse current-current fluctuations [23]. However, close to the CDW instability, gauge fluctuations in all channels are significantly enhances near ,

(24) |

where and are given in Eq. (22), and become singular as . These contributions dominate over those in the small limit and allows the attractive part of the interactions to compete, resulting in a pairing instability. Within the standard BCS framework, we determine the pairing instability by calculating the effective interaction averaged on the Fermi surface in the -angular momentum channel,

(25) |

where are the angles that and make with the -axis on the Fermi circle: . The sign convention is such that corresponds to attractive interactions in the -wave channel. For our spin-polarized case, pairing is only allowed for integer. In the p-wave channel, we calculate using Eqs (23)-(25) near the CDW instability, i.e. for ,

(26) |

where we have put . The condition for is thus given by or equivalently . As in the CDW case, we can solve these inequalities to obtain, as the condition for the p-wave pairing instability of the CS Fermi liquid and the emergence of the p-wave paired quantum Hall states. Using the for GaAs and cm, we find that the paired state is possible within the RPA for , i.e. in good agreement with experimental observations [3, 4].

We have concentrated in this paper on half-filling LLs which correspond to choosing . The same theory applies to cases when the electron filling factor in the topmost LL is at other even denominator fractions, e.g. when and . Eqs (17)-(22) show that the CDW instability also exists in these flanks of the LL where re-entrant integer quantum Hall states have been observed experimentally [12].

The author thanks D. Broido, J. Engelbrecht, Y. B. Kim, and N. Nagaosa for useful discussions. This work was supported in part by DOE Grant No. DE-FG02-99ER45747 and an award from Research Corporation.

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