Dynamics and transport of the Z2 spin liquid: application to kappa-(ET)2Cu2(CN)3

We describe neutron scattering, NMR relaxation, and thermal transport properties of Z2 spin liquids in two dimensions. Comparison to recent experiments on the spin S=1/2 triangular lattice antiferromagnet in kappa-(ET)2Cu2(CN)3 shows that this compound may realize a Z2 spin liquid. We argue that the topological "vison" excitations dominate thermal transport, and that recent thermal conductivity experiments by M. Yamashita et al. have observed the vison gap.

Quantum criticality and black holes

Many condensed matter experiments explore the finite temperature dynamics of systems near quantum critical points. Often, there are no well-defined quasiparticle excitations, and so quantum kinetic equations do not describe the transport properties completely. The theory shows that the transport coefficients are not proportional to a mean free scattering time (as is the case in the Boltzmann theory of quasiparticles), but are completely determined by the absolute temperature and by equilibrium thermodynamic observables. Recently, explicit solutions of this quantum critical dynamics have become possible via the anti-de Sitter/conformal field theory duality discovered in string theory. This shows that the quantum critical theory provides a holographic description of the quantum theory of black holes in a negatively curved anti-de Sitter space, and relates its transport coefficients to properties of the Hawking radiation from the black hole. We review how insights from this connection have led to new results for experimental systems: (i) the vicinity of the superfluid-insulator transition in the presence of an applied magnetic field, and its possible application to measurements of the Nernst effect in the cuprates, (ii) the magnetohydrodynamics of the plasma of Dirac electrons in graphene and the prediction of a hydrodynamic cyclotron resonance.

Quantum Hall to insulator transition in the bilayer quantum Hall ferromagnet

We describe a phase transition of the bilayer quantum Hall ferromagnet at nu=1. In the presence of static disorder (modeled by a periodic potential), bosonic S=1/2 spinons undergo a superfluid-insulator transition while preserving ferromagnetism. The Mott insulating phase has an emergent U(1) photon, and the transition is between Higgs and Coulomb phases of this photon. Consequences for charge and counterflow conductivity and for interlayer tunneling conductance are discussed.

Imaging bond order near nonmagnetic impurities in square-lattice antiferromagnets

We study the textures of generalized "charge densities" (scalar objects invariant under time reversal), in the vicinity of nonmagnetic impurities in square-lattice quantum antiferromagnets, by order parameter field theories. Our central finding is the structure of the vortex in the generalized density wave order parameter centered at the nonmagnetic impurity. Using exact numerical data from quantum Monte Carlo simulations on an antiferromagnetic spin model, we are able to verify the results of our field theoretic study. We extend our phenomenological approach to the period-4 bond-centered density wave found in the underdoped cuprates.

Infinite randomness fixed point of the superconductor-metal quantum phase transition

We examine the influence of quenched disorder on the superconductor-metal transition, as described by a theory of overdamped Cooper pairs which repel each other. The self-consistent pairing eigenmodes of a quasi-one-dimensional wire are determined numerically. Our results support the recent proposal by Hoyos et al. [Phys. Rev. Lett. 99, 230601 (2007)10.1103/PhysRevLett.99.230601] that the transition is characterized by the same strong-disorder fixed point describing the onset of ferromagnetism in the random quantum Ising chain in a transverse field.

Nodal quasiparticles and the onset of spin-density-wave order in cuprate superconductors

We present a theory for the onset of spin-density-wave order in the superconducting ground state of the cuprates. We compute the scaling dimensions of allowed perturbations of a "relativistic" fixed point with O4 x O(3) symmetry, including those associated with the fermionic nodal Bogoliubov quasiparticles. Analyses of up to six loops show that all perturbations with square lattice symmetry are likely irrelevant. We demonstrate that the fermion spectral functions are primarily damped by the coupling to fluctuations of a composite field with Ising nematic order. A number of other experimental implications are also discussed.

Theory of Néel and valence-bond solid phases on the kagome lattice of Zn paratacamite

Recently, neutron scattering data on powder samples of Zn paratacamite, ZnxCu4-x(OH)6Cl2, with small Zn concentration has been interpreted as evidence for valence-bond solid and Néel ordering [S.-H. Lee, Nat. Mater. 6, 853 (2007)10.1038/nmat1986]. We study the classical and quantum Heisenberg models on the distorted kagome lattice appropriate for Zn paratacamite at low Zn doping. Our theory naturally leads to the emergence of the valence-bond solid and collinear magnetic order at zero temperature. Implications of our results to the existing experiments are discussed. We also suggest future inelastic neutron and x-ray scattering experiments that can test our predictions.

Square-lattice algebraic spin liquid with SO(5) symmetry

We propose a critical spin liquid ground state for S=1/2 antiferromagnets on the square lattice. In a renormalization group analysis of the "staggered flux" algebraic spin liquid, we examine perturbations, present in the antiferromagnet, which break its global SU(4) symmetry to SO(5). At physical parameter values, we find an instability towards a fixed point with SO(5) symmetry. We discuss the possibility that this fixed point describes a transition between the Néel and valence bond solid states, and the relationship to the SO(5) nonlinear sigma model of Tanaka and Hu.

Superfluid-insulator transitions of the fermi gas with near-unitary interactions in a periodic potential

We consider spin-1/2 fermions of mass m with interactions near the unitary limit. In an applied periodic potential of amplitude V and period a_{L}, and with a density of an even integer number of fermions per unit cell, there is a second-order quantum phase transition between superfluid and insulating ground states at a critical V=V_{c}. We compute the universal ratio V_{c}ma_{L};{2}/variant Planck's over 2pi;{2} at N=infinity in a model with Sp(2N) spin symmetry. The insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs. We discuss implications for recent experiments.

Impurity induced spin texture in quantum critical 2D antiferromagnets

We describe the uniform and staggered magnetization distributions around a vacancy in a quantum critical two-dimensional S=1/2 antiferromagnet. The distributions are delocalized across the entire sample with a universal functional form arising from an uncompensated Berry phase. The numerical results, obtained using quantum Monte Carlo simulations of the Heisenberg model on bilayer lattices with up to approximately equal to 10(5) spins, are in good agreement with the proposed scaling structure. We determine the exponent eta' = 0.40+/-0.02, which governs both the staggered and uniform magnetic structure away from the impurity and also controls the impurity spin dynamics.

Quantum criticality of a Fermi gas with a spherical dispersion minimum

We describe the quantum phase transition of a Fermi gas occurring when the quasiparticle excitation energy has a minimum in momentum space which crosses zero on a sphere of radius k(0) not equal 0. The quasiparticles have a universal interaction which controls the physical properties in the vicinity of the quantum-critical point. We discuss possible applications to fermionic superfluids formed by pairing two fermion species, near the point where the densities of the two species become unequal.

Magnon decay in gapped quantum spin systems

In the continuum O(3) sigma-model description of gapped spin systems, S = 1 magnons can only decay into three lower energy magnons. We argue that the symmetry of the quantum spin Hamiltonian often allows decay into two magnons, and compute this decay rate in model systems. Two-magnon decay is present in Haldane gap S = 1 spin chains, even though it cannot be induced by any allowed term written in powers and gradients of the sigma-model field. We compare our results with recent measurements of Stone et al. on a two-dimensional spin system.

Universal relaxational dynamics of gapped one-dimensional models in the quantum sine-Gordon universality class

A semiclassical approach to the low-temperature real-time dynamics of generic one-dimensional, gapped models in the sine-Gordon model universality class is developed. Asymptotically exact universal results for correlation functions are obtained in the temperature regime T << Delta, where Delta is the energy gap.

Low-temperature broken-symmetry phases of spiral antiferromagnets

We study Heisenberg antiferromagnets with nearest- (J1) and third- (J3) neighbor exchange on the square lattice. In the limit of spin S-->infinity, there is a zero temperature (T) Lifshitz point at J(3)=1/4J(1), with long-range spiral spin order at T=0 for J3>1/4J(1). We present classical Monte Carlo simulations and a theory for T>0 crossovers near the Lifshitz point: spin rotation symmetry is restored at any T>0, but there is a broken lattice reflection symmetry for 0< or =T<T(c) approximately (J3-1/4J(1))S2. The transition at T=T(c) is consistent with Ising universality. We also discuss the quantum phase diagram for finite S.

Quantum phase transition in an atomic bose gas with a feshbach resonance

We show that in an atomic Bose gas near a Feshbach resonance a quantum phase transition occurs between a phase with only a molecular Bose-Einstein condensate and a phase with both an atomic and a molecular Bose-Einstein condensate. We show that the transition is characterized by an Ising order parameter. We also determine the phase diagram of the gas as a function of magnetic field and temperature: the quantum critical point extends into a line of finite temperature Ising transitions.

Universal conductance of nanowires near the superconductor-metal quantum transition

We consider wires near a zero temperature transition between superconducting and metallic states. The critical theory obeys hyperscaling, which leads to a universal frequency, temperature, and length dependence of the conductance; quantum and thermal phase slips are contained within this critical theory. Normal, superconducting, and mixed (SN) leads on the wire determine distinct universality classes. For the SN case, wires near the critical point have a universal dc conductance which is independent of the length of the wire at low temperatures.

Deconfined Quantum Critical Points

The theory of second-order phase transitions is one of the foundations of modern statistical mechanics and condensed-matter theory. A central concept is the observable order parameter, whose nonzero average value characterizes one or more phases. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. We show that near second-order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm, and we present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional "confining" order parameters. Nevertheless, the critical theory contains an emergent gauge field and "deconfined" degrees of freedom associated with fractionalization of the order parameters. We propose that this paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems and offer a new perspective on the properties of complex materials.

Radiation-induced magnetoresistance oscillations in a 2D electron gas

Recent measurements of a 2D electron gas subjected to microwave radiation reveal a magnetoresistance with an oscillatory dependence on the ratio of radiation frequency to cyclotron frequency. We perform a diagrammatic calculation and find radiation-induced resistivity oscillations with the correct period and phase. Results are explained via a simple picture of current induced by photoexcited disorder-scattered electrons. The oscillations increase with radiation intensity, easily exceeding the dark resistivity and resulting in negative-resistivity minima. At high intensity, we identify additional features, likely due to multiphoton processes, which have yet to be observed experimentally.

Fractionalized fermi liquids

In spatial dimensions d>or=2, Kondo lattice models of conduction and local moment electrons can exhibit a fractionalized, nonmagnetic state (FL(*)) with a Fermi surface of sharp electronlike quasiparticles, enclosing a volume quantized by (rho(a)-1)(mod 2), with rho(a) the mean number of all electrons per unit cell of the ground state. Such states have fractionalized excitations linked to the deconfined phase of a gauge theory. Confinement leads to a conventional Fermi liquid state, with a Fermi volume quantized by rho(a)(mod 2), and an intermediate superconducting state for the Z2 gauge case. The FL(*) state permits a second order metamagnetic transition in an applied magnetic field.

PIASy, a nuclear matrix-associated SUMO E3 ligase, represses LEF1 activity by sequestration into nuclear bodies

The Wnt-responsive transcription factor LEF1 can activate transcription in association with beta-catenin and repress transcription in association with Groucho. In search of additional regulatory mechanisms of LEF1 function, we identified the protein inhibitor of activated STAT, PIASy, as a novel interaction partner of LEF1. Coexpression of PIASy with LEF1 results in potent repression of LEF1 activity and in covalent modification of LEF1 with SUMO. PIASy markedly stimulates the sumoylation of LEF1 and multiple other proteins in vivo and functions as a SUMO E3 ligase for LEF1 in a reconstituted system in vitro. Moreover, PIASy binds to nuclear matrix-associated DNA sequences and targets LEF1 to nuclear bodies, suggesting that PIASy-mediated subnuclear sequestration accounts for the repression of LEF1 activity.

Spin-ordering quantum transitions of superconductors in a magnetic field

We argue that recent neutron scattering measurements by Lake et al. [Science 291, 1759 (2001)] of the spin excitation spectrum of La(2-delta)Sr(delta)CuO4 in a magnetic field can be understood in terms of proximity to a phase with co-existing superconductivity and spin density wave order. We present a general theory for such quantum transitions, and argue that their low energy spin fluctuations are controlled by a singular correction from the superflow kinetic energy, acting in the region outside the vortex cores. We propose numerous experimental tests of our theory.

Spin orthogonality catastrophe in two-dimensional antiferromagnets and superconductors

We compute the spectral function of a spin S hole injected into a two-dimensional antiferromagnet or superconductor in the vicinity of a magnetic quantum critical point. We show that, near Van Hove singularities, the problem maps onto that of a static vacancy carrying excess spin S. The hole creation operator is characterized by a new boundary anomalous dimension and a vanishing quasiparticle residue at the critical point. We discuss possible relevance to photoemission spectra of cuprate superconductors near the antinodal points.

Impurity in a d-wave superconductor: Kondo effect and STM spectra

We present a theory for recent STM studies of Zn impurities in the superconductor Bi2Sr2CaCu2O8+delta, using insights from NMR experiments which show that there is a net S = 1/2 moment on the Cu ions near the Zn. We argue that the Kondo spin dynamics of this moment is the origin of the low bias peak in the differential conductance, rather than a resonance in a purely potential scattering model. The spatial and energy dependence of the STM spectra of our model can also fit the experiments.

Quantum phase transitions in d-wave superconductors

Motivated by the strong, low temperature damping of nodal quasiparticles observed in some cuprate superconductors, we study quantum phase transitions in d(x(2)-y(2)) superconductors with a spin-singlet, zero momentum, fermion bilinear order parameter. We present a complete, group-theoretic classification of such transitions into seven distinct cases (including cases with nematic order) and analyze fluctuations by the renormalization group. We find that only two, the transitions to d(x(2)-y(2))+is and d(x(2)-y(2))+id(xy) pairing, possess stable fixed points with universal damping of nodal quasiparticles; the latter leaves the gapped quasiparticles along (1,0), (0,1) essentially undamped.

Mean field theory of a quantum heisenberg spin glass

A full mean-field solution of a quantum Heisenberg spin-glass model is presented in a large- N limit. A spin-glass transition is found for all values of the spin S. The quantum critical regime associated with the quantum transition at S = 0 and the various regimes in the spin-glass phase at high spin are analyzed. The specific heat is shown to vanish linearly with temperature. In the spin-glass phase, intriguing connections between the equilibrium properties of the quantum problem and the out-of-equilibrium dynamics of classical models are pointed out.

Quantum criticality: competing ground states in low dimensions

Small changes in an external parameter can often lead to dramatic qualitative changes in the lowest energy quantum mechanical ground state of a correlated electron system. In anisotropic crystals, such as the high-temperature superconductors where electron motion occurs primarily on a two-dimensional square lattice, the quantum critical point between two such lowest energy states has nontrivial emergent excitations that control the physics over a significant portion of the phase diagram. Nonzero temperature dynamic properties near quantum critical points are described, using simple theoretical models. Possible quantum phases and transitions in the two-dimensional electron gas on a square lattice are discussed.

Nuclear import of IkappaBalpha is accomplished by a ran-independent transport pathway

The inhibitor of kappa B alpha (IkappaBalpha) protein is able to shuttle between the cytoplasm and the nucleus. We have utilized a combination of in vivo and in vitro approaches to provide mechanistic insight into nucleocytoplasmic shuttling by IkappaBalpha. IkappaBalpha contains multiple functional domains that contribute to shuttling of IkappaBalpha between the cytoplasm and the nucleus. Nuclear import of IkappaBalpha is mediated by the central ankyrin repeat domain. Similar to previously described nuclear import pathways, nuclear import of IkappaBalpha is temperature and ATP dependent and is blocked by a dominant-negative mutant of importin beta. However, in contrast to classical nuclear import pathways, nuclear import of IkappaBalpha is independent of soluble cytosolic factors and is not blocked by the dominant-negative RanQ69L protein. Nuclear export of IkappaBalpha is mediated by an N-terminal nuclear export sequence. Nuclear export of IkappaBalpha requires the CRM1 nuclear export receptor and is blocked by the dominant-negative RanQ69L protein. Our results are consistent with a model in which nuclear import of IkappaBalpha is mediated through direct interactions with components of the nuclear pore complex, while nuclear export of IkappaBalpha is mediated via a CRM1-dependent pathway.

Quantum impurity in a nearly critical two-dimensional antiferromagnet

The spin dynamics of an arbitrary localized impurity in an insulating two-dimensional antiferromagnet, across the host transition from a paramagnet with a spin gap to a Neel state, is described. The impurity spin susceptibility has a Curie-like divergence at the quantum-critical coupling, but with a universal effective spin that is neither an integer nor a half-odd integer. In the Neel state, the transverse impurity susceptibility is a universal number divided by the host spin stiffness (which determines the energy cost to slow twists in the orientation of the Neel order). These and numerous other results for the thermodynamics, Knight shift, and magnon damping have important applications in experiments on layered transition metal oxides.

Loss of IkappaB alpha-mediated control over nuclear import and DNA binding enables oncogenic activation of c-Rel

The IkappaB alpha protein is able both to inhibit nuclear import of Rel/NF-kappaB proteins and to mediate the export of Rel/NF-kappaB proteins from the nucleus. We now demonstrate that the c-Rel-IkappaB alpha complex is stably retained in the cytoplasm in the presence of leptomycin B, a specific inhibitor of Crm1-mediated nuclear export. In contrast, leptomycin B treatment results in the rapid and complete relocalization of the v-Rel-IkappaB alpha complex from the cytoplasm to the nucleus. IkappaB alpha also mediates the rapid nuclear shuttling of v-Rel in an interspecies heterokaryon assay. Thus, continuous nuclear export is required for cytoplasmic retention of the v-Rel-IkappaB alpha complex. Furthermore, although IkappaB alpha is able to mask the c-Rel-derived nuclear localization sequence (NLS), IkappaB alpha is unable to mask the v-Rel-derived NLS in the context of the v-Rel-IkappaB alpha complex. Taken together, our results demonstrate that IkappaB alpha is unable to inhibit nuclear import of v-Rel. We have identified two amino acid differences between c-Rel and v-Rel (Y286S and L302P) which link the failure of IkappaB alpha to inhibit nuclear import and DNA binding of a mutant c-Rel protein to oncogenesis. Our results support a model in which loss of IkappaB alpha-mediated control over c-Rel leads to oncogenic activation of c-Rel.

Nuclear localization of IkappaB alpha is mediated by the second ankyrin repeat: the IkappaB alpha ankyrin repeats define a novel class of cis-acting nuclear import sequences

The ability of the IkappaB alpha protein to sequester dimeric NF-kappaB/Rel proteins in the cytoplasm provides an effective mechanism for regulating the potent transcriptional activation properties of NF-kappaB/Rel family members. IkappaB alpha can also act in the nucleus as a postinduction repressor of NF-kappaB/Rel proteins. The mechanism by which IkappaB alpha enters the nucleus is not known, as IkappaB alpha lacks a discernible classical nuclear localization sequence (NLS). We now report that nuclear localization of IkappaB alpha is mediated by a novel nuclear import sequence within the second ankyrin repeat. Deletion of the second ankyrin repeat or alanine substitution of hydrophobic residues within the second ankyrin repeat disrupts nuclear localization of IkappaB alpha. Furthermore, a region encompassing the second ankyrin repeat of IkappaB alpha is able to function as a discrete nuclear import sequence. The presence of a discrete nuclear import sequence in IkappaB alpha suggests that cytoplasmic sequestration of the NF-kappaB/Rel-IkappaB alpha complex is a consequence of the mutual masking of the NLS within NF-kappaB/Rel proteins and the import sequence within IkappaB alpha. Nuclear import may be a conserved property of ankyrin repeat domains (ARDs), as the ARDs from two other ARD-containing proteins, 53BP2 and GABPbeta, are also able to function as nuclear import sequences. We propose that the IkappaB alpha ankyrin repeats define a novel class of cis-acting nuclear import sequences.

A threshold nuclear level of the v-Rel oncoprotein is required for transformation of avian lymphocytes

The net distribution of eukaryotic transcription factors between the cytoplasm and the nucleus provides an effective mechanism for controlling gene expression. We have utilized cis-acting signals for both nuclear import and nuclear export to experimentally manipulate the distribution of the v-Rel oncoprotein between the nucleus and the cytoplasm. The respective abilities of the v-Rel oncoprotein to localize to the nucleus in chicken embryo fibroblasts, to activate kappaB-dependent transcription in yeast, and to transform avian lymphoid cells were each markedly reduced by the fusion of a cis-acting nuclear export signal onto v-Rel. Our results demonstrate that a threshold nuclear function of v-Rel is required for manifestation of its oncogenic properties. In contrast, while increased expression of the avian IkappaB-alpha protein was able to prevent nuclear localization of v-Rel in chicken embryo fibroblasts, coexpression of IkappaB-alpha with v-Rel in the target cell for v-Rel mediated transformation did not reduce the ability of v-Rel to transform avian lymphoid cells or alter the distribution of v-Rel between the nucleus and the cytoplasm in v-Rel-transformed cells. Our results suggest that the ability of IkappaB-alpha to inhibit nuclear localization of v-Rel is affected by cell-type specific differences between fibroblasts and lymphoid cells.

PEST-dependent cytoplasmic retention of v-Rel by I(kappa)B-alpha: evidence that I(kappa)B-alpha regulates cellular localization of c-Rel and v-Rel by distinct mechanisms

Association of c-Rel with the inhibitor of kappaB-alpha (IkappaB-alpha) protein regulates both cellular localization and DNA binding. The ability of v-Rel, the oncogenic viral counterpart of avian c-Rel, to evade regulation by p40, the avian IkappaB-alpha protein, contributes to v-Rel-mediated oncogenesis. The yeast two-hybrid system was utilized to dissect Rel:IkappaB-alpha interactions in vivo. We find that distinct domains in c-Rel and v-Rel are required for association with p40. Furthermore, while the ankyrin repeat domain of p40 is sufficient for association with c-Rel, both the ankyrin repeat domain and the PEST domain are required for association with v-Rel. Two amino acid differences between c-Rel and v-Rel that are principally responsible for PEST-dependent association of v-Rel with p40 were identified. These same amino acids were principally responsible for PEST-dependent cytoplasmic retention of v-Rel by p40. The presence of mutations in c-Rel that were sufficient to confer PEST-dependent association of the mutant c-Rel protein with p40 did not increase the weak oncogenicity of c-Rel. However, the introduction of these two c-Rel-derived amino acids into v-Rel markedly reduced the oncogenicity of v-Rel. Deletion of the NLS of either c-Rel or v-Rel did not abolish association with p40, but did confer PEST-dependent association of c-Rel with p40. Surprisingly, deletion of the nuclear localization signal in v-Rel did not affect oncogenicity by v-Rel. Analysis of several mutant c-Rel and v-Rel proteins demonstrated that association of Rel proteins with p40 is necessary but not sufficient for cytoplasmic retention. These results are not consistent with the hypothesis that p40 regulates cellular localization of v-Rel and c-Rel by the same mechanism. Rather, these results support the hypothesis that p40 regulates cellular localization of v-Rel and c-Rel by distinct mechanisms.

I kappa B-alpha-mediated inhibition of nuclear transport and DNA-binding by Rel proteins are separable functions: phosphorylation of C-terminal serine residues of I kappa B-alpha is specifically required for inhibition of DNA-binding

I kappa B-alpha inhibits both DNA-binding and nuclear translocation of dimeric Rel complexes that contain either the RelA or c-Rel proteins. These inhibitory functions of I kappa B-alpha proteins are regulated by both constitutive and inducible phosphorylation. We have mapped the constitutive phosphorylation sites of p40, the avian I kappa B-alpha protein, to a C-terminal acidic serine-rich region that contains four serine residues. Deletions or point mutations that significantly alter the overall negatively charged character of this region abolish association of p40 with Rel proteins in vitro. Serine-to-alanine amino acid substitutions in this region modulate the association of p40 with Rel proteins in vitro and abolish p40-mediated inhibition of DNA-binding by c-Rel. Substitution of aspartic acid residues for the phosphorylated serine residues has no effect on p40-mediated inhibition of DNA-binding. In contrast, the C-terminal acidic serine-rich region is not required for p40-mediated inhibition of nuclear translocation of Rel proteins. Our results demonstrate that p40-mediated inhibition of nuclear translocation and inhibition of DNA-binding by Rel proteins are separable functions. Our results suggest that the phosphorylation status of C-terminal serine residues of I kappa B-alpha proteins will be an important aspect of the autoregulatory feedback loop that enforces temporal control of Rel-regulated gene expression.