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.

Multiplicative white-noise stochastic processes continue to attract attention in a wide area of scientific research. The variety of prescriptions available for defining them makes the development of general tools for their characterization difficult. In this work, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we define the time reversal transformation for such processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. Representing the stochastic process in a functional Grassmann formalism, we avoid the necessity of fixing a particular prescription. In this framework, we analyze equilibrium properties and study hidden symmetries of the process. We show that, using a careful definition of the equilibrium distribution and taking into account the appropriate time reversal transformation, usual equilibrium properties are satisfied for any prescription. Finally, we present a detailed deduction of a covariant supersymmetric formulation of a multiplicative Markovian white-noise process and study some of the constraints that it imposes on correlation functions using Ward-Takahashi identities.

We consistently quantize a class of relativistic non-local field equations characterized by a non-local kinetic term in the lagrangian. We solve the classical non-local equations of motion for a scalar field and evaluate the on-shell hamiltonian. The quantization is realized by imposing Heisenberg's equation which leads to the commutator algebra obeyed by the Fourier components of the field. We show that the field operator carries, in general, a reducible representation of the Poincare group.We also consider the Gupta-Bleuler quantization of a non-local gauge field and analyze the propagators and the physical states of the theory.

We construct a path-integral representation of the generating functional for the dissipative dynamics of a classical magnetic moment as described by the stochastic generalization of the Landau-Lifshitz-Gilbert equation proposed by Brown [1], with the possible addition of spin-torque terms. In the process of constructing this functional in the Cartesian coordinate system, we critically revisit this stochastic equation. We present it in a form that accommodates for any discretization scheme thanks to the inclusion of a drift term. The generalized equation ensures the conservation of the magnetization modulus and the approach to the Gibbs-Boltzmann equilibrium in the absence of nonpotential and time-dependent forces. The drift term vanishes only if the mid-point Stratonovich prescription is used. We next reset the problem in the more natural spherical coordinate system. We show that the noise transforms non-trivially to spherical coordinates acquiring a non-vanishing mean value in this coordinate system, a fact that has been often overlooked in the literature. We next construct the generating functional formalism in this system of coordinates for any discretization prescription. The functional formalism in Cartesian or spherical coordinates should serve as a starting point to study different aspects of the out-of-equilibrium dynamics of magnets. Extensions to colored noise, micro-magnetism and disordered problems are straightforward.

We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the LandauLifshitz-Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.