Stripe Antiferromagnetic Order Research Articles

Interplay of Magnetic Interaction and Electronic Structure in New Structure RE-12442 Type Hybrid Fe-Based Superconductors

We present detailed first-principles density functional theory-based studies on RbRE2Fe4As4O2 (RE = Sm, Tb, Dy, Ho) hybrid 12442-type iron-based superconducting compounds with particular emphasis on competing magnetic interactions and their effect on possible magneto-structural coupling and electronic structure. The stripe antiferromagnetic (sAFM) pattern across the xy plane emerges as the most favorable spin configuration for all the four compounds, with close competition among the different magnetic orders along the z-axis. The structural parameters, including arsenic heights, Fe-As-Fe angle, and other relevant factors that influence superconducting Tc and properties, closely match the experimental values in stripe antiferromagnetic arrangement of Fe spins. Geometry optimization with inclusion of explicit magnetic ordering predicts a spin–lattice coupling for all the four compounds, where a weak magneto–structural transition, a tetragonal-to-orthorhombic structural transition, takes place in the relaxed stripe antiferromagnetic spin configuration. Absence of any experimental evidence of such structural transition is possibly an indication of nematic transition in RE-12442 compounds. As a result of structural distortion, the lattice contracts (expands) along the direction with parallel (anti-parallel) alignment of Fe spins. Introduction of stripe antiferromagnetic order in Fe sub-lattice reconstructs the low-energy band structure, which results in significantly reduced number of bands crossing the Fermi level. Moreover, the dispersion of bands and their orbital characteristics also are severely modified in the stripe antiferromagnetic phase similar to BaFe2As2. Calculations of exchange parameters were performed for all the four compounds. Exchange coupling along the anti-parallel alignment of Fe spins J1a is larger than that for the parallel aligned spins J1b. A crossover between the super-exchange-driven in-plane next-nearest-neighbor exchange coupling J2 and in-plane exchange coupling J1a due to lanthanide substitution was found. A large super-exchange-driven next-nearest-neighbor exchange interaction is justified using the construction of 32 maximally localized Wannier functions, where the nearest-neighbor Fe-As hopping amplitudes were found to be larger than the nearest- and the next-nearest-neighbor Fe-Fe hopping amplitudes. We compare the hopping parameters in the stripe antiferromagnetic pattern with non-magnetic configuration, and increased hopping amplitude was found along the anti-parallel spin alignment with more majority-spin electrons in Fe dxz and dxy but not in Fe dyz. On the other hand, the hopping amplitudes are increased in stripe antiferromagnetic phase along the parallel spin alignment with more majority-spin electrons in only Fe dyz. This difference in hopping amplitudes in the stripe antiferromagnetic order enables more isotropic hopping.

Effect of iron vacancies on magnetic order and spin dynamics of the spin ladder BaFe2−δS1.5Se1.5

Quasi-one-dimensional iron chalcogenides possess various magnetic states depending on the lattice distortion, electronic correlations, and presence of defects. We present neutron diffraction and inelastic neutron scattering experiments on the spin ladder compound BaFe$_{2-\delta}$S$_{1.5}$Se$_{1.5}$ with $\sim$6% iron vacancies. The data reveal that long-range magnetic order is absent, while the characteristic magnetic excitations that correspond to both the stripe- and block-type antiferromagnetic correlations are observed. First-principles calculations support the existence of both stripe and block-type antiferromagnetic short-range order in the experimental sample. The disappearance of long-range magnetic order may be due to the competition between these two magnetic orders, which is greatly enhanced for a certain concentration of iron vacancies, which we calculate to be about 6%, consistent with the measured iron vacancy concentration. Our results highlight how iron vacancies in the iron-based spin ladder system strongly influence the magnetic ground state.

Quantum phases and transitions of spin-1/2 quantum compass chain

Quantum phases (QPs) and quantum phase transitions (QPTs) are very important parts of the strongly correlated quantum many-body systems in condensed matter. To study the QPs and QPTs, the systems should include rich quantum phase diagram. In this sense, the corresponding quantum spin models should have strong quantum fluctuation, strong geometric frustration, complicated spin-spin exchange or orbital degrees of freedom, which induces a variety of spontaneous symmetry breaking (SSB) or hidden spontaneous symmetry breaking. The QPs induced by the SSB can be characterized by local order parameters, a concept that originates from Landau-Ginzburg-Wilson paradigm (LGW). However, there is also a novel class of topological QPs beyond LGW, which has aroused one’s great interest since the Haldane phase was found. Such QPs can be characterized only by topological long-range nonlocal string correlation order parameters instead of local order parameters. In this paper, we investigate a spin-1/2 quantum compass chain model (QCC) with orbital degrees of freedom in <i>x</i>, <i>y</i> and <i>z</i> components. The prototype of QCC is the quantum compass model including novel topological QPs beyond LGW, and consequently one can also anticipate the existence of novel topological QPs in QCC. However, very little attention has been paid to the QPs and QPTs for QCC, which deserves to be further investigated. By using the infinite time evolving block decimation in the presentation of matrix product states, we study the QPs and QPTs of QCC. To characterize QPs and QPTs of QCC, the ground state energy, local order parameter, topological long-range nonlocal string correlation order parameters, critical exponent, correlation length and central charge are calculated. The results show the phase diagram of QCC including local antiferromagnetic phase, local stripe antiferromagnetic phase, oscillatory odd Haldane phase and monotonic odd Haldane phase. The QPTs from oscillatory odd Haldane phase to local stripe antiferromagnetic phase and from local antiferromagnetic phase to monotonic odd Haldane phase are continuous; on the contrary, QPTs from local stripe antiferromagnetic phase to local antiferromagnetic phase and from oscillatory odd Haldane phase to monotonic odd Haldane phase are discontinuous. The crossing point where the line of continuous QPTs meets with the line of discontinuous QPTs is the multiple critical point. The critical exponents <i>β</i> of local antiferromagnetic order parameter, local stripe antiferromagnetic order parameter, topological long-range nonlocal oscillatory odd string correlation order parameter, and topological long-range nonlocal monotonic odd string correlation order parameter are all equal to 1/8. Moreover, <inline-formula><tex-math id="M3">\begin{document}$\beta =1/8$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M3.png"/></alternatives></inline-formula> and the central charges <inline-formula><tex-math id="M4">\begin{document}$c = 1/2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20211433_M4.png"/></alternatives></inline-formula> at the critical points show that the QPTs from local phases to nonlocal phases belong to the Ising-type universality class.

A first-principle perspective on electronic nematicity in FeSe

Electronic nematicity is an important order in most iron-based superconductors, and FeSe represents a special example, in which nematicity disentangles from spin ordering. A first-principle description of this order remains elusive. Here, we show that by carefully searching the paramagnetic energy landscape within the density functional theory, a nematic solution stands out at either the +U or hybrid functional level with the lowest energy. The band structure and Fermi surface can be well compared with the recent experimental results. Symmetry analysis assigns the dominant order parameter to the Eu irreducible representations of the D4h point group. Distinct from the B1g Ising nematicity as widely discussed in the context of vestigial stripe antiferromagnetic order, the two-component Eu vector order features mixing of the Fe d-orbitals and inversion symmetry breaking, which lead to striking experimental consequences, e.g., missing of an electron pocket.

Structural, magnetic, and electronic evolution of the spin-ladder system BaFe2S3-x Se x with isoelectronic substitution.

We report experimental studies of a series of BaFe2S3-x Se x (0 ⩽ x ⩽ 3) single crystals and powder specimens using x-ray diffraction, neutron-diffraction, muon-spin-relaxation, and electrical transport measurements. A structural transformation from Cmcm (BaFe2S3) to Pnma (BaFe2Se3) was identified around x = 0.7 - 1. Neutron-diffraction measurements on the samples with x = 0.2, 0.4, and 0.7 reveal that the Néel temperature of the stripe antiferromagnetic order is gradually suppressed from ~120 to 85 K, while the magnitude of the ordered Fe2+ moments shows very little variation. Similarly, the block antiferromagnetic order in BaFe2Se3 remains robust for 1.5 ⩽ x ⩽ 3 with negligible variation in the ordered moment and a slight decrease of the Néel temperature from 250 K (x = 3) to 225 K (x = 1.5). The sample with x = 1 near the Cmcm and Pnma border shows coexisting, two-dimensional, short-range stripe- and block-type antiferromagnetic correlations. The system remains insulating for all x, but the thermal activation gap shows an abrupt increase when traversing the boundary from the Cmcm stripe phase to the Pnma block phase. The results demonstrate that the crystal structure, magnetic order, and electronic properties are strongly coupled in the BaFe2S3-x Se x system.

Nonsuperconducting electronic ground state in pressurized BaFe2S3 and BaFe2S2.5Se0.5

We report a comprehensive study of the spin ladder compound BaFe$_2$S$_{2.5}$Se$_{0.5}$ using neutron diffraction, inelastic neutron scattering, high pressure synchrotron diffraction, and high pressure transport techniques. We find that BaFe$_2$S$_{2.5}$Se$_{0.5}$ possesses the same $Cmcm$ structure and stripe antiferromagnetic order as does BaFe$_2$S$_3$, but with a reduced N{\'{e}}el temperature of $T_N=98$ K compared to 120 K for the undoped system, and a slightly increased ordered moment of 1.40$\mu_B$ per iron. The low-energy spin excitations in BaFe$_2$S$_{2.5}$Se$_{0.5}$ are likewise similar to those observed in BaFe$_2$S$_{3}$. However, unlike the reports of superconductivity in BaFe$_2$S$_3$ below $T_c \sim 14$~K under pressures of 10~GPa or more, we observe no superconductivity in BaFe$_2$S$_{2.5}$Se$_{0.5}$ at any pressure up to 19.7~GPa. In contrast, the resistivity exhibits an upturn at low temperature under pressure. Furthermore, we show that additional high-quality samples of BaFe$_2$S$_3$ synthesized for this study likewise fail to become superconducting under pressure, instead displaying a similar upturn in resistivity at low temperature. These results demonstrate that microscopic, sample-specific details play an important role in determining the ultimate electronic ground state in this spin ladder system. We suggest that the upturn in resistivity at low temperature in both BaFe$_2$S$_3$ and BaFe$_2$S$_{2.5}$Se$_{0.5}$ may result from Anderson localization induced by S vacancies and random Se substitutions, enhanced by the quasi-one-dimensional ladder structure.

Strong electronic correlations and Fermi surface reconstruction in the quasi-one-dimensional iron superconductor BaFe2S3

BaFe2S3 is a special iron superconductor with two-leg ladder structure which can help to unravel the role played by the electronic correlations in high-Tc superconductivity. At zero pressure it is insulating with stripe antiferromagnetic (AF) order and superconductivity emerges under pressure. We use a slave-spin technique to analyze the strength of the local correlations in BaFe2S3. We find that at the pressure at which the superconductivity appears the electronic correlations in BaFe2S3 are similar to the ones measured in other iron superconductors. However, at zero pressure the strength of the correlations is strongly enhanced, being particularly severe for the two orbitals with the largest weight at the Fermi level what invalidates nesting as the mechanism for AF. The system is not a Mott insulator at zero temperature, but these two orbitals with mass enhancements m* ~ 12-15 will become incoherent at higher temperatures. Different from what happens in other iron superconductors, at both pressures, the Fermi surface is reconstructed by the electronic correlations.

Stripe order and magnetic anisotropy in the S=1 antiferromagnet BaMoP2O8

Magnetic behavior of BaMoP$_2$O$_8$ with the spatially anisotropic triangular arrangement of the $S=1$ Mo$^{4+}$ ions is explored using thermodynamic measurements, neutron diffraction, and density-functional band-structure calculations. A broad maximum in the magnetic susceptibility around 46\,K is followed by the stripe antiferromagnetic order with the propagation vector ${\mathbf k}=(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ formed below $T_N\simeq 21$ K. This stripe phase is triggered by a pronounced one-dimensionality of the spin lattice, where one of the in-plane couplings, $J_2\simeq 4.6$ meV, is much stronger than its $J_1\simeq 0.4$ meV counterpart, and stabilized by the weak easy-axis anisotropy. The ordered moment of 1.42(9) $\mu_B$ at 1.5 K is significantly lower than the spin-only moment of 2 $\mu_B$ due to a combined effect of quantum fluctuations and spin-orbit coupling.

Antiferromagnetic order in CaK(Fe1−xNix)4As4 and its interplay with superconductivity

The magnetic order in CaK(Fe[1-x]Ni[x])4As4 (1144) single crystals (x = 0.051 and 0.033) has been studied by neutron diffraction. We observe magnetic Bragg peaks associated to the same propagation vectors as found for the collinear stripe antiferromagnetic (AFM) order in the related BaFe2As2 (122) compound. The AFM state in 1144 preserves tetragonal symmetry and only a commensurate, non-collinear structure with a hedgehog spin-vortex crystal (SVC) arrangement in the Fe plane and simple AFM stacking along the c direction is consistent with our observations. The SVC order is promoted by the reduced symmetry in the FeAs layer in the 1144 structure. The long-range SVC order coexists with superconductivity, however, similar to the doped 122 compounds, the ordered magnetic moment is gradually suppressed with the developing superconducting order parameter. This supports the notion that both collinear and non-collinear magnetism and superconductivity are competing for the same electrons coupled by Fermi surface nesting in iron arsenide superconductors.

Domain imaging across the magneto-structural phase transitions in Fe1+yTe

The investigation of the magnetic phase transitions in the parent compounds of Fe-based superconductors is regarded essential for an understanding of the pairing mechanism in the related superconducting compounds. Even though the chemical and electronic properties of these materials are often strongly inhomogeneous on a nanometer length scale, studies of the magnetic phase transitions using spatially resolved experimental techniques are still scarce. Here, we present a real space spin-resolved scanning tunneling microscopy investigation of the surface of Fe1+yTe single crystals with different excess Fe content, y, which are continuously driven through the magnetic phase transition. For Fe1.08Te, the transition into the low-temperature monoclinic phase is accompanied by the appearance of a chevron-patterned structural ordering due to the four 90° rotational domains of the monoclinic lattice. Each of the structural domains contains locally commensurate nanoscale diagonal double stripe antiferromagnetic spin order domains with π-phase slips accross domain boundaries. In the low-temperature phase of Fe1.12Te, on the other hand, the chevron pattern gets rather narrow and less well-defined, and an additional 90° rotated component of the spin-order with local plaquette order emerges. The simultaneous imaging of spin and structural order we show here gives valuable insights into the nature of the magneto-structural domains of Fe1+yTe near the tricritical point, which presumably add to the understanding of the mechanism of superconductivity in the related Fe1+yTexSe1−x material.

Spin waves and magnetic exchange interactions in the spin-ladder compoundRbFe2Se3

We report an inelastic neutron scattering study of the spin waves of the one-dimensional antiferromagnetic spin ladder compound RbFe$_2$Se$_3$. The results reveal that the products, $SJ$'s, of the spin $S$ and the magnetic exchange interactions $J$'s along the antiferromagnetic (leg) direction and the ferromagnetic (rung) direction are comparable with those for the stripe ordered phase of the parent compounds of the iron-based superconductors. The universality of the $SJ$'s implies nearly universal spin wave dynamics and the irrelevance of the fermiology for the existence of the stripe antiferromagnetic order among various Fe-based materials.

Mott localization in a pure stripe antiferromagnetRb1−δFe1.5−σS2

A combination of neutron diffraction and angle-resolved photoemission spectroscopy measurements on a pure antiferromagnetic stripe Rb$_{1-\delta}$Fe$_{1.5-\sigma}$S$_2$ is reported. A neutron diffraction experiment on a powder sample shows that a 98$\%$ volume fraction of the sample is in the antiferromagnetic stripe phase with rhombic iron vacancy order and a refined composition of Rb$_{0.66}$Fe$_{1.36}$S$_{2}$, and that only 2$\%$ of the sample is in the block antiferromagnetic phase with $\sqrt{5}\times \sqrt{5}$ iron vacancy order. Furthermore, a neutron diffraction experiment on a single crystal shows that there is only a single phase with the stripe antiferromagnetic order with the refined composition of Rb$_{0.78}$Fe$_{1.35}$S$_2$, while the phase with block antiferromagnetic order is absent. Angle-resolved photoemission spectroscopy measurements on the same crystal with the pure stripe phase reveal that the electronic structure is gapped at the Fermi level with a gap larger than 0.325 eV. The data collectively demonstrates that the extra 10$\%$ iron vacancies in addition to the rhombic iron vacancy order effectively impede the formation of the block antiferromagnetic phase; the data also suggest that the stripe antiferromagnetic phase with rhombic iron vacancy order is a Mott insulator.

Spin waves and spatially anisotropic exchange interactions in theS=2stripe antiferromagnetRb0.8Fe1.5S2

An inelastic neutron scattering study of the spin waves corresponding to the stripe antiferromagnetic order in insulating Rb$_{0.8}$Fe$_{1.5}$S$_2$ throughout the Brillouin zone is reported. The spin wave spectra are well described by a Heisenberg Hamiltonian with anisotropic in-plane exchange interactions. Integrating the ordered moment and the spin fluctuations results in a total moment squared of $27.6\pm4.2\mu_B^2$/Fe, consistent with $\mathrm{S \approx 2}$. Unlike $X$Fe$_2$As$_2$ ($X=$ Ca, Sr, and Ba), where the itinerant electrons have a significant contribution, our data suggest that this stripe antiferromagnetically ordered phase in Rb$_{0.8}$Fe$_{1.5}$S$_2$ is a Mott-like insulator with fully localized $3d$ electrons and a high-spin ground state configuration. Nevertheless, the anisotropic exchange couplings appear to be universal in the stripe phase of Fe pnictides and chalcogenides.

Neutron-scattering measurements of spin excitations in LaFeAsO and Ba(Fe(0.953)Co(0.047))(2)As(2): evidence for a sharp enhancement of spin fluctuations by nematic order.

Inelastic neutron scattering is employed to investigate the impact of electronic nematic order on the magnetic spectra of LaFeAsO and Ba(Fe(0.953)Co(0.047))(2)As(2). These materials are ideal to study the paramagnetic-nematic state, since the nematic order, signaled by the tetragonal-to-orthorhombic transition at T(S), sets in well above the stripe antiferromagnetic ordering at T(N). We find that the temperature-dependent dynamic susceptibility displays an anomaly at T(S) followed by a sharp enhancement in the spin-spin correlation length, revealing a strong feedback effect of nematic order on the low-energy magnetic spectrum. Our findings can be consistently described by a model that attributes the structural or nematic transition to magnetic fluctuations, and unveils the key role played by nematic order in promoting the long-range stripe antiferromagnetic order in iron pnictides.

Theoretical investigation of the electronic and magnetic properties of the orthorhombic phase ofBa(Fe1−xCox)2As2

We present a comprehensive study on the low-temperature orthorhombic phase of Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$ based on the Korringa-Kohn-Rostoker-Green function approach. Using this bandstructure method in combination with the coherent potential approximation alloy theory we are able to investigate the evolution of the magnetic and electronic properties of this prototype iron pnictide for arbitrary concentrations $x$, while dealing with the chemical disorder without uncontrolled simplifications by using solely a rigid band shift or the virtual crystal approximation. We discuss the development of the site resolved magnetic moments for the experimentally observed stripe antiferromagnetic order together with the strong electronic anisotropy of the Fermi surface and compare it with angle-resolved photoemission spectroscopy measurements of detwinned crystals. We furthermore calculate magnetic exchange coupling parameters $J_{ij}$ and use them for Monte-Carlo simulations on the basis of the classical Heisenberg model to get an insight on the temperature dependence of the magnetic ordering on the cobalt concentration.

Crossover from spin waves to diffusive spin excitations in underdoped Ba(Fe1−xCox)2As2

Using inelastic neutron scattering, we show that the onset of superconductivity in underdoped Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ coincides with a crossover from well-defined spin waves to overdamped and diffusive spin excitations. This crossover occurs despite the presence of long-range stripe antiferromagnetic order for samples in a compositional range from x=0.04-0.055, and is a consequence of the shrinking spin-density wave gap and a corresponding increase in the particle-hole (Landau) damping. The latter effect is captured by a simple itinerant model relating Co doping to changes in the hot spots of the Fermi surface. We argue that the overdamped spin fluctuations provide a pairing mechanism for superconductivity in these materials.

Neutron scattering measurements of spatially anisotropic magnetic exchange interactions in semiconducting K0.85 Fe1.54Se2 (TN = 280 K.

We use neutron scattering to study the spin excitations associated with the stripe antiferromagnetic order in semiconducting K(0.85)Fe(1.54)Se(2) (T(N) = 280 K). We show that the spin-wave spectra can be accurately described by an effective Heisenberg Hamiltonian with highly anisotropic inplane couplings at T = 5 K. At high temperature (T = 300 K) above T(N), short-range magnetic correlation with anisotropic correlation lengths are observed. Our results suggest that, despite the dramatic difference in the Fermi surface topology, the inplane anisotropic magnetic couplings are a fundamental property of the iron-based compounds; this implies that their antiferromagnetism may originate from local strong correlation effects rather than weak coupling Fermi surface nesting.

Short-range order of Br and three-dimensional magnetism in (CuBr)LaNb2O7

We present a comprehensive study of the crystal structure, magnetic structure, and microscopic magnetic model of (CuBr)LaNb2O7, the Br analog of the spin-gap quantum magnet (CuCl)LaNb2O7. Despite similar crystal structures and spin lattices, the magnetic behavior and even peculiarities of the atomic arrangement in the Cl and Br compounds are very different. The high-resolution x-ray and neutron data reveal a split position of Br atoms in (CuBr)LaNb2O7. This splitting originates from two possible configurations developed by [CuBr] zigzag ribbons. While the Br atoms are locally ordered in the 'ab' plane, their arrangement along the 'c' direction remains partially disordered. The predominant and energetically more favorable configuration features an additional doubling of the 'c' lattice parameter that was not observed in (CuCl)LaNb2O7. (CuBr)LaNb2O7 undergoes long-range antiferromagnetic ordering at TN=32 K, which is nearly 70% of the leading exchange coupling J4~48 K. The Br compound does not show any experimental signatures of low-dimensional magnetism, because the underlying spin lattice is three-dimensional. The coupling along the 'c' direction is comparable to the couplings in the 'ab' plane, even though the shortest Cu--Cu distance along 'c' (11.69 A) is three times larger than nearest-neighbor distances in the 'ab' plane (3.55 A). The stripe antiferromagnetic long-range order featuring columns of parallel spins in the 'ab' plane and antiparallel spins along 'c' is verified experimentally and confirmed by the microscopic analysis.

Orbital-selective magnetism in the spin-ladder iron selenides Ba1−xKxFe2Se3

Here we show that the 2.80(8) {\mu}B/Fe block antiferromagnetic order of BaFe2Se3 transforms into stripe antiferromagnetic order in KFe2Se3 with a decrease in moment to 2.1(1) {\mu}B/Fe. This reduction is larger than expected from the change in electron count from Ba$^{2+}$ to K$^{+}$, and occurs with the loss of the displacements of Fe atoms from ideal positions in the ladders, as found by neutron pair distribution function analysis. Intermediate compositions remain insulating, and magnetic susceptibility measurements show a suppression of magnetic order and probable formation of a spin-glass. Together, these results imply an orbital-dependent selection of magnetic versus bonded behavior, driven by relative bandwidths and fillings.

Orbitally and magnetically induced anisotropy in iron-based superconductors

Recent experimental developments in the iron pnictides have unambiguously demonstrated the existence of in-plane electronic anisotropy in the absence of the long-range magnetic order. Such anisotropy can arise from orbital ordering, which is described by an energy splitting between the two otherwise degenerate ${d}_{xz}$ and ${d}_{yz}$ orbitals. By including this phenomenological orbital order into a five-orbital Hubbard model, we obtain the mean-field solutions where the magnetic order is determined self-consistently. Despite sensitivity of the resulting states to the input parameters, we find that a weak orbital order that places the ${d}_{yz}$ orbital slightly higher in energy than the ${d}_{xz}$ orbital, combined with intermediate on-site interactions, produces band dispersions that are compatible with the photoemission results. In this regime, the stripe antiferromagnetic order is further stabilized and the resistivity displays the observed anisotropy. We also calculate the optical conductivity and show that it agrees with the temperature evolution of the anisotropy seen experimentally.