We discuss shape (Pomeranchuk) instabilities of the Fermi surface of a two-dimensional Fermi system using bosonization. We consider in detail the quantum critical behavior of the transition of a two dimensional Fermi fluid to a nematic state which breaks spontaneously the rotational invariance of the Fermi liquid. We show that higher dimensional bosonization reproduces the quantum critical behavior expected from the Hertz-Millis analysis, and verify that this theory has dynamic critical exponent z = 3. Going beyond this framework, we study the behavior of the fermion degrees of freedom directly, and show that at quantum criticality as well as in the the quantum nematic phase (except along a set of measure zero of symmetry-dictated directions) the quasi-particles of the normal Fermi liquid are generally wiped out. Instead, they exhibit short ranged spatial correlations that decay faster than any power-law, with the law |x| −1 exp(−const. |x| 1/3 ) and we verify explicitely the vanishing of the fermion residue utilizing this expression. In contrast, the fermion auto-correlation function has the behavior |t| −1 exp(−const. |t| −2/3 ). In this regime we also find that, at low frequency, the single-particle fermion density-of-states behaves as N * (ω) = N * (0) + B ω 2/3 log ω + . . ., where N * (0) is larger than the free Fermi value, N (0), and B is a constant. These results confirm the non-Fermi liquid nature of both the quantum critical theory and of the nematic phase.

A recently proposed path-integral bosonization scheme for massive fermions in 3 dimensions is extended by keeping the full momentum-dependence of the one-loop vacuum polarization tensor. This makes it possible to discuss both the massive and massless fermion cases on an equal footing, and moreover the results it yields for massless fermions are consistent with the ones of another, seemingly different, canonical quantization approach to the problem of bosonization for a massless fermionic field in 3 dimensions.

Barci and Stariolo Reply: The focus of our work [1] was to identify conditions for the presence of an isotropicnematic phase transition in the context of a generic system with isotropic competing interactions. By taking into account nontrivial angular momentum contributions from the interaction, we found a second order isotropic-nematic phase transition at mean field level, which becomes a Kosterlitz-Thouless one [2] when fluctuations are taken into account.In his Comment [3], Levin criticizes our results by showing that the low temperature fluctuations of a stripe phase in 2d diverge linearly in the thermodynamic limit. His analysis is restricted to the stripe phase and, contrary to what is suggested in the Comment, does not apply to the central result of our Letter which is the existence of an isotropic-nematic phase transition. In fact, as clearly anticipated by us in the Letter [1], the corresponding analysis of the fluctuations of the nematic order parameter displays a logarithmic divergence leading to a low temperature phase with quasi-long-range order.In our model, despite the involved calculations, it is straightforward to understand this fact. Introducing the nematic order parameterQ ij n inj ÿ 1 2 i;j [wherê n i cos; sin is the director field] through a Hubbard-Stratonovich transformation, it is possible to decouple the quartic terms. Integrating out the field, we obtain the following long wavelength effective free energy for the nematic order parameter: FQ a 2 =2Tr Q 2 a 4 =4Tr Q 4 =4Tr QDQ . . . , where the symmetric derivative tensor D ij r i r j and a 2 , a 4 , and are temperature dependent coefficients given in terms of the parameters of the original model. At mean field, the last term is zero, and we find ÿa 2 =a 4 p for a 2 < 0, going continuously to 0 for a 2 > 0. Note that any global rotation of the order parameter costs no energy. Therefore, parametrizing the order parameter by a modulus and an angle, the long wavelength angle fluctuations x dominate the low energy physics. Computing the free energy at lowest order in the derivatives of the angle fluctuations, we find F 2 R d 2 xjrj 2 , where F is the excess of free energy relative to the saddle point value. Therefore, the free energy of fluctuations corresponds to that of the XY model. The only difference with the usual vector orientational order is that the system should have the symmetry ! modifying the vorticity of the topological defects. Thus, one finds for the angle fluctuations hxx 0 i lnk 0 x ÿ x 0 , which in turn lead to an algebraic decay of the order parameter correlations. In an extended paper we will show the explicit dependence of the Frank constant KT 2 with the parameters of our

We develop an effective low-energy theory of the quantum Hall ͑QH͒ smectic or stripe phase of a twodimensional electron gas in a large magnetic field in terms of its Goldstone modes and of the charge fluctuations on each stripe. This liquid-crystal phase corresponds to a fixed point that is explicitly demonstrated to be stable against quantum fluctuations at long wavelengths. This fixed-point theory also allows an unambiguous reconstruction of the electron operator. We find that quantum fluctuations are so severe that the electron Green function decays faster than any power law, although slower than exponentially, and that consequently there is a deep pseudo-gap in the quasiparticle spectrum. We discuss, but do not resolve, the stability of the quantum Hall smectic to crystallization. Finally, the role of Coulomb interactions and the low-temperature thermodynamics of the QH smectic state are analyzed.