We generalize our previous model to an O(N) symmetric two-dimensional model which possesses chiral symmetry breaking (͗ ͘ condensate͒ and superconducting ͑Cooper pair ͗͘ condensates͒ phases at large N. At zero temperature and density, the model can be solved analytically in the large-N limit. We perform the renormalization explicitly and obtain a closed form expression of the effective potential. There exists a renormalization group invariant parameter ␦ that determines which of the ͗ ͘ (␦Ͼ0) or ͗͘ (␦Ͻ0) condensates exist in the vacuum. At finite temperatures and densities, we map out the phase structure of the model by a detailed numerical analysis of the renormalized effective potential. For ␦ positive and sufficiently large, the phase diagram in the -T ͑chemical potential-temperature͒ plane exactly mimics the features expected for QCD with two light flavors of quarks. At low temperatures there exists low-chiral symmetry breaking and high-Cooper pair condensate regions which are separated by a first-order phase transition. At high , when the temperature is raised, the system undergoes a second-order phase transition from the superconducting phase to an unbroken phase in which both condensates vanish. For a range of values of ␦ the theory possesses a tricritical point ( tc and T tc ); for Ͼ tc (Ͻ tc ) the phase transition from the low temperature chiral symmetry breaking phase to unbroken phase is first order ͑second order͒. For the range of ␦ in which the system mimics QCD, we expect the model to be useful for the investigation of dynamical aspects of nonequilibrium phase transitions, and to provide information relevant to the study of relativistic heavy ion collisions and the dense interiors of neutron stars.

Using the Majorana fermion representation of spin-1/2 local moments, we show how it is possible to directly read off the dynamic spin correlation and susceptibility from the one-particle propagator of the Majorana fermion. We illustrate our method by applying it to the spin dynamics of a nonequilibrium quantum dot, computing the voltage-dependent spin relaxation rate and showing that, at weak coupling, the fluctuation-dissipation relation for the spin of a quantum dot is voltagedependent. We confirm the voltage-dependent Curie susceptibility recently found by Parcollet and Hooley [Phys. Rev. B 66, 085315 (2002)].PACS numbers: 03.65. Ca, 72.15.Qm, 73.63.Kv, 76.20.+q The mathematical difficulties of representing spins in many body physics have long been recognized. The essence of the problem is that spin operators are nonabelian: they do not obey Wick's theorem and an expectation value of the product of many spin operators cannot be decomposed into products of two-operator expectation values, even within a free theory.A conventional response to this difficulty is to represent spins as bilinears of fermions [1] or as bosons [2]. One of the disadvantages of these approaches is that the Hilbert space of the fermions or bosons needs to be restricted by the application of constraints [3,4,5]. Another difficulty is the "vertex problem", which arises in the context of spin dynamics and spin relaxation. Once the spins are represented as bilinears, the spin-spin correlation functions are represented by two-particle Green's functions. The calculation of these quantities requires a knowledge of both the four-leg vertex and the single-particle Green's function. Typically, the vertex is simply neglected, or treated in a very approximate fashion.An alternative approach is to take advantage of the anticommuting properties of Pauli matrices, writing the spin operator in terms of Majorana fermions [6,7,8,9,10],where η = (η 1 , η 2 , η 3 ) is a triplet of Majorana fermions which satisfy {η a , η b } = δ ab . This representation does not require the imposition of a constraint: the fact that S 2 = 3/4 follows directly from the operator properties of the Majorana fermions. In this letter, we show how this representation also solves the vertex problem. To demonstrate this, we employ an alternative derivation [11] of the Majorana spin representation. Consider a spin-1/2 operator S with dynamics described by a Hamiltonian H. Let us now introduce a single Majorana fermion Φ which lives in a completely different Hilbert space, commuting with S and H. It follows that Φ is a fermionic constant of motion, dΦ/dt = −i[H, Φ] = 0: an object of fixed magnitude Φ 2 = 1/2 which anticommutes with all other fermion operators. We may now identify η in (1) with the operator identityWe may confirm that {η a , η b } = δ ab using the anticommuting algebra of spin-1/2 operators {S a , S b } = 1 2 δ ab . Furthermore, using the SU(2) algebra of spins, η × η = 2 S × S = 2i S, from which (1) follows immediately. As a last step in the derivation, we note tha...

Superconductivity mediated by spin fluctuations in weak and nearly ferromagnetic metals is studied close to the zero-temperature magnetic transition. We solve analytically the Eliashberg equations for p-wave pairing and obtain the quasiparticle self-energy and the superconducting transition temperature T(c) as a function of the distance to the quantum critical point (QCP). We show that the reduction of quasiparticle coherence and lifetime due to scattering by quasistatic spin fluctuations is the dominant pair-breaking process, which leads to a rapid suppression of T(c) to a nonzero value near the QCP. We point out the differences and similarities of the problem to that of paramagnetic impurities in superconductors.

We develop a theory of quadratic quantum measurements by a mesoscopic detector. It is shown that the quadratic measurements should have nontrivial quantum information properties, providing, for instance, a simple way of entangling two noninteracting qubits. We also calculate the output spectrum of a detector with both linear and quadratic response, continuously monitoring two qubits.