Motivated by recent experiments with Bechgaard salts, we investigate the competition between antiferromagnetism and triplet superconductivity in quasi one-dimensional electron systems. We unify the two orders in an SO(4) symmetric framework, and demonstrate the existence of such symmetry in one-dimensional Luttinger liquids. SO(4) symmetry, which strongly constrains the phase diagram, can explain coexistence regions between antiferromagnetic, superconducting, and normal phases, as observed in (TMTSF)2PF6. We predict a sharp neutron scattering resonance in superconducting samples.A common feature of many strongly correlated electron systems is proximity of a superconducting state to some kind of magnetically ordered insulating state. Examples include organic materials [1,2], heavy fermion superconductors [3,4], and high T c cuprates [5]. Several theoretical analyses suggest that a strong repulsion between the two orders plays an important role in determining the phase diagram and low energy properties of these materials [6]. The idea of competing orders was developed into the SO(5) theory of high T c superconductivity of S.C. Zhang [7]. SO(5) symmetry has also been applied to study competition of ferromagnetism and p-wave superconductivity in Sr 2 RuO 4 [8], and antiferromagnetism and d-wave superconductivity in κ-BEDT-TTF salts [9].In this paper we consider the interplay of antiferromagnetism (AF) and triplet superconductivity (TSC) in quasi one-dimensional (Q1D) electron systems. Our study is motivated by Q1D Bechgaard salts (TMTSF) 2 X. The most well studied material from this family, (TMTSF) 2 PF 6 , is an antiferromagnetic insulator at ambient pressure and a superconductor at high pressures [10,11,12,13]. The symmetry of the superconducting order parameter in (TMTSF) 2 PF 6 is not yet fully established, but there is strong evidence that electron pairing is spin triplet: the superconducting T c is strongly suppressed by disorder [14]; critical magnetic field H c2 in the interchain direction exceeds the paramagnetic limit [15]; the electron spin susceptibility, obtained from Knight shift measurements, does not decrease below T c [16]. In another material from this family, (TMTSF) 2 ClO 4 , superconductivity is stable at ambient pressure and also shows signatures of triplet pairing [17,18,19,20]. Insulator to superconductor transition as a function of pressure has also been found in(TMTSF) 2 AsF 6 [21].The phase diagram of interacting electrons in one dimension was obtained in Ref. 22 using bosonization and renormalization group (RG) analyses. At incommensurate filling, this system has a phase boundary between spin density wave (SDW) and TSC phases when K ρ = 1 and g 1 > 0 (K ρ is the Luttinger parameter in the charge sector, g 1 the backward scattering amplitude). The starting point of our discussion is the observation that, in the absence of umklapp, 1D Luttinger liquids have an "isospin" SO(4) iso symmetry [23] at the boundary between SDW and TSC phases. To define this symmetry, we introduce the charge of le...

The probabilities of clusters spanning a hypercube of dimensions two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen--Kopelman algorithm combined with Grassberger's "go with the winner" strategy for the site percolation. We carried out a finite-size analysis of the data and found that the probabilities confirm Aizenman's proposal of the multiplicity exponent for dimensions three to five. A crossover to the mean-field behavior around the upper critical dimension is also discussed.Comment: 5 pages, 4 figures, 4 table

We discuss collective spin-wave excitations in triplet superconductors with an easy axis anisotropy for the order parameter. Using a microscopic model for interacting electrons, we estimate the frequency of such excitations in Bechgaard salts and ruthenate superconductors to be 1 and 20 GHz, respectively. We introduce an effective bosonic model to describe spin-wave excitations and calculate their contribution to the nuclear spin-lattice relaxation rate. We find that, in the experimentally relevant regime of temperatures, this mechanism leads to the power law scaling of 1=T 1 with temperature. For two-and threedimensional systems, the scaling exponents are 3 and 5, respectively. We discuss experimental manifestations of the spin-wave mechanism of the nuclear spin-lattice relaxation.

We investigate the competition between antiferromagnetism and triplet superconductivity in quasi-onedimensional electron systems. We show that the two order parameters can be unified using a SO(4) symmetry and demonstrate the existence of such symmetry in one-dimensional Luttinger liquids of interacting electrons. We argue that approximate SO(4) symmetry remains valid even when interchain hopping is strong enough to turn the system into a strongly anisotropic Fermi liquid. For unitary triplet superconductors SO(4) symmetry requires a first order transition between antiferromagnetic and superconducting phases. Analysis of thermal fluctuations shows that the transition between the normal and the superconducting phases is weakly first order, and the normal to antiferromagnet phase boundary has a tricritical point, with the transition being first order in the vicinity of the superconducting phase. We propose that this phase diagram explains coexistence regions between the superconducting and the antiferromagnetic phases, and between the antiferromagnetic and the normal phases observed in ͑TMTSF͒ 2 PF 6 . For nonunitary triplet superconductors the SO(4) symmetry predicts the existence of a mixed phase of antiferromagnetism and superconductivity. We discuss experimental tests of the SO(4) symmetry in neutron scattering and tunneling experiments.

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