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We consider the quantum-control task of obtaining a target unitary operation via control fields that couple to the quantum system and are chosen to best mitigate errors resulting from time-dependent noise. We allow for two sources of noise: fluctuations in the control fields and those arising from the environment. We address the issue of error mitigation by means of a formulation rooted in the Martin-Siggia-Rose (MSR) approach to noisy, classical statistical-mechanical systems. We express the noisy control problem in terms of a path integral, and integrate out the noise to arrive at an effective, noise-free description. We characterize the degree of success in error mitigation via a fidelity, which characterizes the proximity of the sought-after evolution to ones achievable in the presence of noise. Error mitigation is then accomplished by applying the optimal control fields, i.e., those that maximize the fidelity subject to any constraints obeyed by the control fields. To make connection with MSR, we reformulate the fidelity in terms of a Schwinger-Keldysh (SK) path integral, with the added twist that the `forward' and `backward' branches of the time-contour are inequivalent with respect to the noise. The present approach naturally allows the incorporation of constraints on the control fields; a useful feature in practice, given that they feature in real experiments. We illustrate this MSR-SK approach by considering a system consisting of a single spin $s$ freedom (with $s$ arbitrary), focusing on the case of $1/f$ noise. We discover that optimal error-mitigation is accomplished via a universal control field protocol that is valid for all $s$, from the qubit (i.e., $s=1/2$) case to the classical (i.e., $s \to \infty$) limit. In principle, this MSR-SK approach provides a framework for addressing quantum control in the presence of noise for systems of arbitrary complexity., 25 pages, 4 figures. To be published in Physical Review A

Low-energy quasiparticle states, arising from scattering by single-particle potentials in d-wave superconductors, are addressed. Via a natural extension of the Andreev approximation, the idea that sign-variations in the superconducting pair-potential lead to such states is extended beyond its original setting of boundary scattering to the broader context of scattering by general single-particle potentials, such as those due to impurities. The index-theoretic origin of these states is exhibited via a simple connection with Witten's supersymmetric quantum-mechanical model., Comment: 5 pages

Reflection of charge excitations at the step in the interaction strength in a Luttinger liquid can be of the Andreev type, even if the interactions are purely repulsive. The region with stronger repulsion plays the role of a normal metal in a normal-metal /superconductor junction, whereas the region with weaker repulsion plays the role of a superconductor. It is shown that this quasi-Andreev reflection leads to a number of proximity-like effects, including the local enhancement (suppression) of superconducting fluctuations on the quasi-normal (quasi-superconducting) side of the step, significant modification of the local density of states, as well as others. The observable consequences of these proximity effects are analyzed for the case of single- and two-particle tunneling from a normal-metal or superconducting tip into an inhomogeneous Luttinger-liquid wire., Comment: 5 pages, 2 figures (eps)

It is shown, within the framework of the Ginzburg-Landau theory for a superconductor with d_{x^2-y^2} symmetry, that the passing of a supercurrent through the sample results, in general, in the induction of order-parameter components of distinct symmetry. The induction of s-wave and d_{xy(x^2-y^2)-wave components are considered in detail. It is shown that in both cases the order parameter remains gapless; however, the structure of the lines of nodes and the lobes of the order parameter are modified in distinct ways, and the magnitudes of these modifications differ in their dependence on the (a-b plane) current direction. The magnitude of the induced s-wave component is estimated using the results of the calculations of Ren et al. [Phys. Rev. Lett. 74, 3680 (1995)], which are based on a microscopic approach., 15 pages, includes 2 figures. To appear in Phys. Rev. B

The impact of polymer-polymer interactions of various types on the thermodynamics, structure, and accommodation of topological constraints is addressed for systems comprising many directed polymers in two spatial dimensions. The approach is predicated on the well-known equivalence between the classical equilibrium statistical mechanics of directed polymers in two spatial dimensions and the imaginary-time quantum dynamics of particles in one spatial dimension, originally exploited by P.-G. de Gennes [P.-G. de Gennes, J. Chem. Phys. 48, 2257 (1968)]. Known results concerning two exactly solvable microscopic models of quantum particles moving in one spatial dimension---the Lieb-Liniger model of contact interactions and the Calogero-Sutherland model of long-range interactions---are used to shed light on the behavior of the corresponding polymeric systems. In addition, the technique of bosonization is used to reveal how generic polymer interactions give rise to an emergent polymer fluid that has universal collective excitations. Additionally, the response of the system to topological constraints such as pins though which polymers cannot pass is explored. Immediately on the compressed side of a pin there is a divergent pile-up in polymer density, while on the other side there is a gap of finite area in which polymer density is negligible. Comparison of this response to that of a system of simply noncrossing (i.e., noncrossing but otherwise noninteracting) directed polymers, explored in a companion paper, reveals that generic interactions leave the structure quantitatively unchanged on the line transverse to the pin, and leave it qualitatively unchanged throughout the two dimensions of the system's extent. Furthermore, the free-energy cost associated with a pin that partitions a system having generic interactions is found to be proportional to the pin-partitioning cost for a system of simply noncrossing polymers.

Cryogenic scanning tunneling microscopy has been used as a local probe of the superconducting proximity effect across a normal metal--superconductor interface of a short coherence length superconductor. Both the topography and the local electronic density of states were measured on a superconducting NbS${\mathrm{e}}_{2}$ crystal decorated with nanometer-size Au islands. The presence of a quasiparticle bound state could be inferred even when the probe was located directly on the bare NbS${\mathrm{e}}_{2}$ surface near an Au island, indicating a severe depression of the pair potential inside the superconductor due to the proximity effect.

The phase ordering process following a quench to both the uniaxial and biaxial nematic phases of a quasi-two-dimensional nematic liquid crystal is investigated numerically. The time dependences of the correlation function, structure factor, energy density, and number densities of topological defects are computed. It is found that the correlation function and the structure factor apparently collapse on to scaling curves over a wide range of times. The correlation length ${\mathit{L}}_{\mathrm{cor}}$(t) is found to grow as a power law in the time since the quench t, with a growth exponent of ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cor}}$=0.407\ifmmode\pm\else\textpm\fi{}0.005. The growth exponents of the characteristic length scales obtained from the energy length (${\mathrm{\ensuremath{\varphi}}}_{\mathrm{en}}$) and the defect number densities (${\mathrm{\ensuremath{\varphi}}}_{\mathrm{def}}$), however, are found to differ from ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cor}}$. The discrepancy between ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{cor}}$ and ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{def}}$ indicates a violation of dynamical scaling, a violation that is not apparent from our data for the correlation function alone. The observation that all the measured growth exponents are smaller than 0.5 (i.e., the value expected from dimensional analysis) is addressed in terms of properties of point defects in two dimensions, and the annihilation process of an isolated defect pair in a uniaxial nematic phase is investigated. Following the quench to the biaxial nematic phase, there are four topologically distinct defect species present in the system, the populations of which are studied in detail. It is found that only two types of defects are observed in large numbers at late times, and a mechansim for the selection of the prevailing defect species is proposed.

The rate at which dissipative fluctuations occur in narrow superconducting rings is considered within the Little-Langer-Ambegaokar-McCumber-Halperin framework. The mesoscopic regime, in which the ring circumference [ital L] does not greatly exceed the correlation length [xi], is specifically addressed. In this regime, significant differences arise between models that mimic voltage and current sources, the former (latter) exhibiting corrections to the fluctuation rate algebraic in [xi]/[ital L] (exponentially small in [ital L]/[xi]). The theory employs Forman's elaboration of the Gel'fand-Yaglom technique for computing fluctuation determinants.

A Bose gas subject to a light-induced Rashba spin-orbit coupling possesses a dispersion minimum on a circle in momentum space; we show that kinematic constraints due to this dispersion cause interactions to renormalize to universal, angle-dependent values that govern the phase structure in the dilute-gas limit. We find that, regardless of microscopic interactions, (a) the ground state involves condensation at two opposite momenta, and is, in finite systems, a fragmented condensate; and (b) there is a nonzero-temperature instability toward the condensation of pairs of bosons. We discuss how our results can be reconciled with the qualitatively different mean-field phase diagram, which is appropriate for dense gases., Comment: 5 pages, 2 figures

Low-temperature transport in disordered conductors exhibits a variety of fascinating quantum-mechanical interference effects associated with the phenomenon of weak localization. Such effects are typically isolated and probed by virtue of their sensitivity to applied homogeneous magnetic fields, which introduce Aharonov-Bohm phase factors into quantum-mechanical amplitudes. Analogous interference effects have been proposed in the context of the quantum transport of (possibly electrically neutral) particles with spin in the presence of inhomogeneous magnetic fields, which have the effect of introducing Berry phases. Thus, the possibility is raised of isolating and probing quantum interference effects through their sensitivity to the inhomogeneity of applied magnetic fields. In this paper we develop an approach to the study of quantum transport in disordered conductors in the presence of almost arbitrarily inhomogeneous magnetic fields, which is based on diagrammatic and semiclassical path-integral techniques and a subsequent adiabatic approximation. We illustrate these ideas with applications to three examples: anomalous weak-field magnetoconductance, conductance oscillations in mesoscopic multiply connected structures, and sample-dependent mesoscopic conductance fluctuations. Among other things, we find that while in the context of the disorder-averaged conductance it is accurate to regard systems as being composed of two independent subsystems (having spins aligned or antialigned with the local external magnetic field) a more interesting and refined structure emerges in the context of conductance fluctuations.

We present a microscopic theory of the transition to and properties of the amorphous solid state of a system of vulcanized (i.e., randomly crosslinked) macromolecules. Our approach invokes a variational hypothesis for the random static density fluctuations characterizing this solid state. The variational parameter is the inverse monomer localization length, which is zero in the liquid state and increases continuously as the mean number of crosslinks exceeds a critical value. The emergent solid is a homogeneous isotropic elastic medium, whose elastic moduli we compute near the transition

Superconducting fluctuations induce remnants of the Josephson effect above ${\mathit{T}}_{\mathit{c}}$. To explore this, we consider a paraconducting cylinder, composed of two thin films connected by tunnel junctions, and threaded by magnetic flux. The electrical conductance includes a component that is an oscillatory function of the flux, having an amplitude proportional to the square of the Josephson coupling, and decaying exponentially with the circumference over a length scale \ensuremath{\xi}(T). We estimate the typical magnitude of this effect, and contrast it with related mesoscopic effects in normal-metal systems.

Josephson tunneling between superconducting electrodes in the vortex-glass state is considered. The Josephson coupling energy and phase offset are stochastic, the typical value of the former scaling as the Eddwards-Anderson order parameter, Larkin-Ovchinikov length, and square root of the junction area. The influence of magnetic flux through the junction is explored, and magnetofingerprints, reminiscent of universal conductance fluctuations, are predicted. Properties of SQUIDs are also examined, and it is concluded that Josephson phenomena may provide a useful probe of the vortex-glass state

The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity. In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms. The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity. We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory. At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition. We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry. In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.

The gel point is a continuous transition between fluid and solid states, and therefore exhibits a variety of scaling laws for the linear viscoelastic behavior. These are derived, together with a scaling law for the nonlinear shear relaxation modulus and a sum rule, valid in both fluid and solid states, which has a counterpart in superconductivity. Our results are generic consequences of both dynamic critical phenomena and spontaneous symmetry breaking, and therefore are not equivalent to some previous predictions, based on an analogy with percolation.

Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the \emph{structure} of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their \emph{elastic properties}. Vulcanization theory studies semi-microscopic models of random-solid-forming systems, and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids, in terms of residual stress and Lam\'e-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long-ranged and governed by a universal parameter that also gives the mean shear modulus., Comment: 40 pages, 7 figures

Consideration de la variation des courants persistants avec le flux, dans les anneaux mesoscopiques traverses par le flux magnetique, et extension des arguments bien connus pour inclure les particules de spin 1/2. On trouve plusieurs consequences interessantes du spin telles que: la division de la periode et de l'amplitude par deux dans un seul anneau sous l'inclusion des interactions electron-electron, des canaux transverses, ou du desordre. Ces consequences dependent sensiblement du nombre fixe (modulo 4) des particules sur un anneau donne, et conduisent a de fortes fluctuations entre les echantillons contenant un petit nombre d'anneaux

We study the stochastic dynamics of superconductive-resistive switching in hysteretic current-biased superconducting nanowires undergoing phase-slip fluctuations. We evaluate the mean switching time using the master-equation formalism, and hence obtain the distribution of switching currents. We find that as the temperature is reduced this distribution initially broadens; only at lower temperatures does it show the narrowing with cooling naively expected for phase slips that are thermally activated. We also find that although several phase-slip events are generally necessary to induce switching, there is an experimentally accessible regime of temperatures and currents for which just one single phase-slip event is sufficient to induce switching, via the local heating it causes.

As a function of the magnetic flux threading the object, the Little-Parks oscillation in the critical temperature of a large-radius, thin-walled superconducting ring or hollow cylinder has a period given by $h/2e$, due to the binding of electrons into Cooper pairs. On the other hand, the single-electron Aharonov-Bohm oscillation in the resistance or persistent current for a clean (i.e., ballistic) normal-state system, having the same topological structure, has a period given by $h/e$. A basic question is whether the Little-Parks oscillation changes its character, as the radius of the superconducting structure becomes smaller, and if it is even comparable to the zero-temperature coherence length. We supplement a physical argument that the $h/e$ oscillations should also be exhibited with a microscopic analysis of this regime, formulated in terms of the Gor'kov approach to BCS theory. We see that, as the radius of the ring is made smaller, an oscillation in the critical temperature of period $h/e$ emerges in addition to the usual Little-Parks $h/2e$-period oscillation. We argue that, in the clean limit, there is a superconductor-normal transition at nonzero flux as the ring radius becomes sufficiently small and that the transition can be either continuous or discontinuous, depending on the radius and the external flux. In the dirty limit, we argue that the transition is rendered continuous, which results in continuous quantum phase transitions tuned by flux and radius.

A Landau theory is constructed for the gelation transition in cross-linked polymer systems possessing spontaneous nematic ordering, based on symmetry principles and the concept of an order parameter for the amorphous solid state. This theory is substantiated with help of a simple microscopic model of cross-linked dimers. Minimization of the Landau free energy in the presence of nematic order yields the neoclassical theory of the elasticity of nematic elastomers and, in the isotropic limit, the classical theory of isotropic elasticity. These phenomenological theories of elasticity are thereby derived from a microscopic model, and it is furthermore demonstrated that they are universal mean-field descriptions of the elasticity for all chemical gels and vulcanized media.

A nematic liquid-crystal cylinder may exhibit metastable cholesteric-type twisted states in which the director uniformly rotates along the sample in the plane orthogonal to the cylinder axis. If a constant torque is maintained by strongly anchoring end plates, thermal fluctuations can produce rotation, leading to torsionally viscous behavior. We present a theory for the fluctuation-induced intrinsic torsional viscosity of narrow cylinders of nematic, based on the Frank free energy and constructed by analogy with the work of Langer and Ambegaokar on the resistance of superconducting channels.

We have developed a superconducting phase gradiometer consisting of two parallel DNA-templated nanowires connecting two thin-film leads. We have ramped the cross current flowing perpendicular to the nanowires, and observed oscillations in the lead-to-lead resistance due to cross-current-induced phase differences. By using this gradiometer we have measured the temperature and magnetic field dependence of the superfluid density and observed an amplification of phase gradients caused by elastic vortex displacements. We examine our data in light of Miller-Bardeen theory of dirty superconductors and a microscale version of Campbell's model of field penetration., Comment: 5 pages, 6 figures

In response to recent experiments by the Berkeley group, we construct a model of superflow through an array of nanosize apertures that incorporates two basic ingredients: (1) disorder associated with each aperture having its own random critical velocity, and (2) effective interaperture coupling, mediated through the bulk superfluid. As the disorder becomes weak there is a transition from a regime where phase slips are largely independent to a regime where interactions lead to system-wide avalanches of phase slips. We explore the flow dynamics in both regimes, and make connections to the experiments.

The analytical zero-temperature phase diagram of the double exchange model for classical background spins as a function of the carrier density and Hund's coupling in the entire range of these parameters is presented. By constructing a continuum field theory we explore the possibility of a continuous phase transition from ferromagnetic state to a gently varying textured state. We find such a transition in and below two dimensions and show that the emerging stable state is a spin-spiral which survives the tendency towards phase separation into commonly considered phases, and is also energetically favored to the canted state, for low carrier density.

Entanglement in quantum XY spin chains of arbitrary length is investigated via a recently-developed global measure suitable for generic quantum many-body systems. The entanglement surface is determined over the phase diagram, and found to exhibit structure richer than expected. Near the critical line, the entanglement is peaked (albeit analytically), consistent with the notion that entanglement--the non-factorization of wave functions--reflects quantum correlations. Singularity does, however, accompany the critical line, as revealed by the divergence of the field-derivative of the entanglement along the line. The form of this singularity is dictated by the universality class controlling the quantum phase transition., Comment: 4 pages, 2 figures

The scaling of the shear modulus near the gelation/vulcanization transition is explored heuristically and analytically. It is found that in a dense melt the effective chains of the infinite cluster have sizes that scale sub-linearly with their contour length. Consequently, each contributes k_B T to the rigidity, which leads to a shear modulus exponent d\nu. In contrast, in phantom elastic networks the scaling is linear in the contour length, yielding an exponent identical to that of the random resistor network conductivity, as predicted by de Gennes'. For non-dense systems, the exponent should cross over to d\nu when the percolation length becomes much larger than the density-fluctuation length., Comment: 4 pages, 2 eps figures

In sufficiently high spatial dimensions, the formation of the amorphous (i.e. random) solid state of matter, e.g., upon sufficent crosslinking of a macromolecular fluid, involves particle localization and, concommitantly, the spontaneous breakdown of the (global, continuous) symmetry of translations. Correspondingly, the state supports Goldstone-type low energy, long wave-length fluctuations, the structure and implications of which are identified and explored from the perspective of an appropriate replica field theory. In terms of this replica perspective, the lost symmetry is that of relative translations of the replicas; common translations remain as intact symmetries, reflecting the statistical homogeneity of the amorphous solid state. What emerges is a picture of the Goldstone-type fluctuations of the amorphous solid state as shear deformations of an elastic medium, along with a derivation of the shear modulus and the elastic free energy of the state. The consequences of these fluctuations -- which dominate deep inside the amorphous solid state -- for the order parameter of the amorphous solid state are ascertained and interpreted in terms of their impact on the statistical distribution of localization lengths, a central diagnostic of the the state. The correlations of these order parameter fluctuations are also determined, and are shown to contain information concerning further diagnostics of the amorphous solid state, such as spatial correlations in the statistics of the localization characteristics. Special attention is paid to the properties of the amorphous solid state in two spatial dimensions, for which it is shown that Goldstone-type fluctuations destroy particle localization, the order parameter is driven to zero, and power-law order-parameter correlations hold., 20 pages, 3 figures

The low-energy quasiparticle states of a disordered d-wave superconductor are investigated theoretically. A class of such states, formed via tunneling between the Andreev bound states that are localized around extended impurities (and result from scattering between pair-potential lobes that differ in sign) is identified. Its (divergent) contribution to the total density of states is determined by taking advantage of connections with certain one-dimensional random tight-binding models. The states under discussion should be distinguished from those associated with nodes in the pair potential., Comment: 5 pages, 1 figure

The issue of probing the pseudogap regime of the cuprate superconductors, specifically with regard to the existence and nature of superconducting pairing correlations of d-wave symmetry, is explored theoretically. It is shown that if the d-wave correlations believed to describe the superconducting state persist into the pseudogap regime, but with pair-potential phase-fluctuations that destroy their long-range nature, then the low-energy quasiparticle states observed near extended impurities in the truly superconducting state should also persist as resonances in the pseudogap regime. The scattering of quasiparticles by these phase-fluctuations broadens what was (in the superconducting state) a sharp peak in the single-particle spectral function at low energy, as we demonstrate within the context of a simple model. This peak and its broadening is, in principle, accessible via scanning tunneling spectroscopy near extended impurities in the pseudogap regime. If so, such experiments would provide a probe of the extent to which d-wave superconducting correlations persist upon entering the pseudogap regime, thus providing a stringent diagnostic of the phase-fluctuation scenario., Comment: 8 pages, 2 figures

The low-temperature Hall resistivity \rho_{xy} of La_{2/3}A_{1/3}MnO_3 single crystals (where A stands for Ca, Pb and Ca, or Sr) can be separated into Ordinary and Anomalous contributions, giving rise to Ordinary and Anomalous Hall effects, respectively. However, no such decomposition is possible near the Curie temperature which, in these systems, is close to metal-to-insulator transition. Rather, for all of these compounds and to a good approximation, the \rho_{xy} data at various temperatures and magnetic fields collapse (up to an overall scale), on to a single function of the reduced magnetization m=M/M_{sat}, the extremum of this function lying at m~0.4. A new mechanism for the Anomalous Hall Effect in the inelastic hopping regime, which reproduces these scaling curves, is identified. This mechanism, which is an extension of Holstein's model for the Ordinary Hall effect in the hopping regime, arises from the combined effects of the double-exchange-induced quantal phase in triads of Mn ions and spin-orbit interactions. We identify processes that lead to the Anomalous Hall Effect for localized carriers and, along the way, analyze issues of quantum interference in the presence of phonon-assisted hopping. Our results suggest that, near the ferromagnet-to-paramagnet transition, it is appropriate to describe transport in manganites in terms of carrier hopping between states that are localized due to combined effect of magnetic and non-magnetic disorder. We attribute the qualitative variations in resistivity characteristics across manganite compounds to the differing strengths of their carrier self-trapping, and conclude that both disorder-induced localization and self-trapping effects are important for transport., Comment: 29 pages, 20 figures

Spin-glass models with orthogonal and unitary bond interactions are investigated using replica mean-field theory. The orthogonal model leads to a low-temperature phase described by a Parisi replica symmetry broken order parameter. The analogous solution for the unitary model is unstable to fluctuations which render the order parameter complex. In this case a simple parametrization of the Hermitian order parameter suggests a first-order transition.

Les systemes de polymere reticule (gels et vulcanisats), sont en fait des solides amorphes a l'equilibre

The replica mean-field theory of the isotropic glass phase of the $\mathrm{XY}$ spin glass is reconsidered. We notice that the conventional order parameter, ${q}_{\ensuremath{\mu}\ensuremath{\nu}}^{\ensuremath{\alpha}\ensuremath{\beta}}={s}^{\ensuremath{\alpha}\ensuremath{\beta}}{\ensuremath{\delta}}_{\ensuremath{\mu}\ensuremath{\nu}}$, should be augmented by an additional term, ${a}^{\ensuremath{\alpha}\ensuremath{\beta}}{\ensuremath{\varepsilon}}_{\ensuremath{\mu}\ensuremath{\nu}}$, which is antisymmetric in both replica and spin space. We show that, in this wider space of order parameters, the conventional phase is not an extremum of the free energy.