brief communications NATURE | VOL 426 | 13 NOVEMBER 2003 | www.nature.com/nature 139 COMMUNICATIONS ARISING Astrophysics A constraint on canonical quantum gravity? G amma rays from the g-ray burst (GRB) 021206 have been reported to be strongly linearly polarized 1 , with the estimated degree of polarization (80520%) being close to the absolute maximum of 100% — affording us the opportunity to constrain models of quantum gravity, which has had 10 10 years to act on the photons as they travelled towards us. Here I show that if the effects of quantum gravity are linearly proportional to the ratio of the photon energy to the characteristic scale energy of quantum gravity, then the polarization of photons with energies of about 0.1 MeV should be completely random, contrary to what is observed. I conclude that, should the polar- ization measurement be confirmed, quan- tum gravity effects act with a power that is greater than linearity, or that loop quantum gravity is not viable. Compared with previ- ous methods and results (see ref. 2, for exam- ple), testing of the linear polarization of cosmic g-ray bursts may substantially extend the observational window on the theory of quantum gravity. GRBs are characterized by a highly vari- able flux of high-energy photons that propa- gate over cosmological distances. It has been suggested 3 that they are the best candles in cosmological space, allowing us either to study or to constrain the effects of quantum gravity. These effects are known 3,4 to be pro- portional to the ratio (E/E QG ) n of the photon energy, E, to the Planck energy, E QG 10 19 GeV, and to the distance, D, of the photon’s propagation. The linear case (n41) is the best studied 3,4 , but the quadratic case (n42) has also recently been considered (see preprint at http://xxx.lanl.gov/PS_cache/ gr-qc/pdf/0305/0305057.pdf). For n41, the effect of the energy-dependent refraction of photons in quantum space-time should lead to a measurable difference in arrival time (of the order of milliseconds) for photons with different energies 2 . The linear polarization of g-rays from GRB 021206 allows us to test another possi- ble effect of quantum space-time, which is predicted for canonical quantum gravity in loop representation. In this case, space-time exhibits the property of birefringence 4 :two photons with opposite states of helicity, &1 and 11,have different group velocities v 5 4c(15x(E/E QG ) n ) (1) The factor x is about 1 for loop representa- tion of quantum gravity 3 . A linearly polarized electromagnetic wave may be represented as the superposition of two monochromatic waves with opposite circular polarizations. When a linearly polarized wave propagates inside the substance with birefringence, the plane of polarization rotates along the path because of the difference in group velocity between the two circular components. For the linear case n41, the phase angle, w 1 , of a plane of linear polarization changes along a distance D (in light years) as Dw 1 (E) x(D/hc)E 2 /E QG 10 4 x(E/0.1 MeV) 2 D (2) This angle depends on the photon energy as E 2 . Linear polarization measured within a broad energy range should vanish, provided that the difference in accumulated angles is large for photons with different energies. Two photons with energies of around 0.1 MeV and with a difference of energy of about 0.01% will therefore accumulate a difference of dw x in polarization phase angle after a year of propagation in space with birefringence (see equation (2)). For cosmological GRBs,which have a travel distance of D 10 10 light years, the planes of linear polarization of photons with different energies should be totally randomized. The bulk linear polarization of photons with ener- gies greater than 0.01 eV over a broad energy range must become zero even if they were all originally 100% polarized in a single plane. In the quadratic case of quantum space- time birefringence with n42, the rotation of a plane of linear polarization is rather small for photons with energy of around 0.1 MeV Dw 2 (E) x(D/hc)E 3 /E QG 2 10 –19 x(E/0.1 MeV) 3 D (3) However, the distance D 10 10 light years is so large that even the quadratic case of birefrigence could be tested by polarization measurements of photons with energies greater than 100 MeV. The detection 5 of a high-energy component of GRB941017 (energies up to 200 MeV), which dominates the total fluence of the event, suggests that quadratic space-time birefringence could be tested experimentally in the future by polarimetry of such GRBs. I therefore conclude that either the birefringence of quantum space-time with n41 should be below the level of x?10 114 , or it should be quadratic (n42), assuming that the strong linear polarization of GRBs is confirmed by a second measurement. Igor G. Mitrofanov Space Research Institute, Profsojuznaya str. 84/32, 117997 Moscow, Russia e-mail: imitrofa@space.ru 1. Colburn, W. & Boggs, S. E. Nature 423, 415–417 (2003). 2. Jacobson, T., Liberati, D. & Mattingly, D. Nature 424, 1019–1021 (2003). 3. Amelino-Gamelia, G., Ellis, J., Mavromatos, N. E., Nanopoulos, D. V. & Sarkar, S. Nature 393, 763–765 (1998). 4. Gambini, R. & Pullin, J. Phys.Rev.D59, 124021 (1999). 5. Gonzalez, M. M. et al. Nature 424, 749–751 (2003). observed optical bumps. These should arise from emission by the reverse shocks that form in the refreshed shocks. Jonathan Granot*, Ehud Nakar?, Tsvi Piran? *Institute for Advanced Study, Princeton, New Jersey 08540, USA ?Racah Institute for Physics, The Hebrew University, Jerusalem 91904, Israel e-mail: udini@phys.huji.ac.il 1. Stanek, K. Z. et al. Astrophys. J. 591, L17–L20 (2003). 2. Hjorth, J. et al. Nature 423, 847–850 (2003). 3. Price, P. A. et al. Nature 423, 844–847 (2003). 4. Rees, M. J. & Mészáros, P. Astrophys. J. 496, L1–L4 (1998). 5. Kumar, P. & Piran, T. Astrophys. J. 532, 286–293 (2000). 6. Sari, R. & Mészáros, P. Astrophys. J. 535, L33–L37 (2000). 7. Granot, J., Miller, M., Piran, T., Suen, W. M. & Hughes, P. A. in Gamma-Ray Bursts in the Afterglow Era (eds Costa, E., Frontera, F. & Hjorth, J.) 312–314 (Springer, Berlin, 2001). Competing financial interests: declared none. COMMUNICATIONS ARISING Condensed-matter physics Spurious magnetism in high-T c superconductor O ne challenge in condensed-matter physics is to unravel the interplay between magnetism and superconduc- tivity in copper oxides with a high critical temperature (T c ). Kang et al. 1 claim to have revealed a quantum phase transition from the superconducting to an antiferromagnetic state in the electron-doped material Nd 21x Ce x CuO 4 (NCCO) based on the obser- vation of magnetic-field-induced neutron- scattering intensity at (1/2,1/2,0), (1/2,0,0) and related reflections. Here we argue that the observed magnetic intensity is due to a secondary phase of (Nd,Ce) 2 O 3 . We therefore contend that the effect is spurious and not intrinsic to superconducting NCCO. To achieve superconductivity in NCCO, a rather severe oxygen-reduction procedure has to be applied 2 . We have discovered that the reduction process decomposes a small (0.01–0.10%) volume fraction of NCCO.The resultant (Nd,Ce) 2 O 3 secondary phase has the complex cubic bixbyite structure, common among rare-earth (RE) sesquioxides 3 , with a lattice constant,a c ,that is about 2￡2 times the planar lattice constant of tetragonal NCCO. (Nd,Ce) 2 O 3 is epitaxial with the host lattice, with long-range order parallel to the CuO 2 planes of NCCO,but extending only about 5a c perpendicular to the planes. Because of the relationship between the two lattice con- stants, certain structural reflections from the impurity phase appear at seemingly com- mensurate NCCO positions — that is, the cubic (2,0,0) c reflection can also be indexed as (1/2,1/2,0). However, there is roughly a 10% mismatch between a c and the c-lattice con- stant of NCCO,and therefore (0,0,2) c can also be indexed as (0,0,2.2). ? 2003 Nature Publishing Group There are 32 rare-earth ions in the RE 2 O 3 unit cell,belonging to two crystallographically distinct sites with inequivalent saturated moments 3 . At the (2,0,0) c reflection, the con- tributions from the two rare-earth sites inter- fere destructively, which should lead to a peak in the observed scattering intensity in the paramagnetic phase if the moments saturate at different fields. Although the magnetic structure and spin hamiltonian of epitaxial, quasi-two-dimensional (Nd,Ce) 2 O 3 are unknown, it is possible to devise simple experiments to test whether the field-induced scattering is due to NCCO or (Nd,Ce) 2 O 3 . Kang et al. find that at a temperature of 5 K, the (1/2,1/2,0) (that is, (2,0,0) c ) intensity reaches a peak at a field of about 6.5 T, and argue that this peak is associated with the upper critical field B c2 of NCCO. Figure 1a summarizes the field dependence of an x40.18 superconducting sample of ours in the temperature range 1.9–10 K. Our data agree with those of Kang et al. The figure shows that the intensity scales with B/T and exhibits a peak consistent with two-moment paramagnetism. Furthermore, as the upper critical field of a superconductor increases with decreasing temperature,this implies that the reported correspondence of the peak posi- tion with B c2 at 5 K is coincidental. We do not observe spontaneous neodymium ordering of either (Nd,Ce) 2 O 3 or NCCO down to 1.4 K. Figure 1b, c shows that the field effects reported by Kang et al. are also observable in a non-superconducting, oxygen-reduced, x40.10 sample, both at the previously reported positions and at positions that are unrelated to the NCCO lattice but equivalent in the cubic lattice of (Nd,Ce) 2 O 3 . Not only are the incommensurate positions (0,0,2.2) and (1/4,1/4,1.1) unrelated to the proposed NCCO magnetic order, but the physical situation of the magnetic field applied parallel (in the cases of the (0,0,2.2) and (1/4,1/4,1.1)) or perpendicular (in all other cases) to the CuO 2 planes is fundamentally different in that the upper critical fields for the two geometries differ significantly. Note that (1/2,0,0) and (1/4,1/4,1.1) correspond to (1,1,0) c and (1,0,1) c ,respectively.Care was taken to ensure that in all cases the magnetic field was applied along a cubic axis of (Nd,Ce) 2 O 3 and perpendicular to the scat- tering wavevector. These simple experimental tests demon- strate that the observed field effects in oxy- gen-reduced NCCO result from an epitaxial secondary phase of (Nd,Ce) 2 O 3 . P. K. Mang*, S. Larochelle?, M. Greven*? *Department of Applied Physics, ?Department of Physics, and ?Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, California 94305, USA e-mail: greven@stanford.edu 1. Kang, H. J. et al. Nature 423, 522–525 (2003). 2. Tokura, Y., Takagi, H. & Uchida, S. Nature 337, 345–347 (1989). 3. Moon, R. M., Koehler, W. C., Child, H. R. & Raubenheimer, L. J. Phys. Rev. 176, 722–731 (1968). Kang et al. reply — Mang et al. observe a cubic (Nd,Ce) 2 O 3 impurity phase grown epitaxially in annealed samples of electron- doped Nd 21x Ce x CuO 4 (NCCO). They claim that this impurity phase has long-range order parallel to the CuO 2 planes of NCCO but extending only about 4a c perpendicular to the planes, thus forming a quasi-two- dimensional (Nd,Ce) 2 O 3 lattice matched with the a–b plane of NCCO. Although we have confirmed the presence of such an impurity phase, (Nd,Ce) 2 O 3 in our samples forms a three-dimensional long-range structural order 1 and is unrelated to the quasi-two-dimensional superlattice reflections 1,2 . In the paramagnetic state of (Nd,Ce) 2 O 3 , a field will induce a net moment on magnetic Nd. By arbitrarily scaling the impurity scattering at (0,0,2.2) for fields less brief communications 140 NATURE | VOL 426 | 13 NOVEMBER 2003 | www.nature.com/nature Intensity 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 1.9 K x=0.18 4.2 K 0.18 10 K 0.18 5 K 0.15 0 1 2 3 0 0.2 0.4 0.6 0.8 1 B/T (1/2,0,0) x=0.10 (1/4,1/4,1.1) 0.10 (1/2,0,0) 0.15 0 0.2 0.4 0.6 0.8 1 (1/2,1/2,0) x=0.18 (1/2,1/2,0) 0.10 (0,0,2.2) (1/2,1/2,0) 0.15 I/I maximum I/I maximum a b 0.10 Figure 1 Field and temperature dependence of magnetic scatter- ing. a, Arbitrarily scaled scattering intensity at (1/2,1/2,0) for a superconducting sample of NCCO (nominal cerium concentration x40.18; T c 420 K) as a function of B/T with the field along [0,0,1]. The results are compared with the data of Kang et al. 1 (x40.15; T45 K). b, c, Comparison of the results of Kang et al. with data taken at T44 K for a superconducting sample (x40.18) and a non-superconducting sample (x40.10). Super- conductivity in NCCO can be achieved only for x?0.13. The mag- netic field is applied along [1,1 1 ,0] for (0,0,2.2) and (1/4,1/4,1.1) and along [0,0,1] in all other cases. Data were normalized by maximum intensity. Full details are available from the authors. than 7 T to our c-axis field-induced scattering of NCCO at (1/2,1/2,0),Mang et al.argue that our observed magnetic scattering 2 is due entirely to (Nd,Ce) 2 O 3 .We disagree,however. There are three ways to resolve this impu- rity problem.First,if the magnetic scattering at (1/2,1/2,0) (ref. 2) is due entirely to (Nd,Ce) 2 O 3 , one would expect the field- induced intensity to be identical when B is parallel to the c-axis and when it is parallel to the [1,11,0] axis, as required by the cubic symmetry of (Nd,Ce) 2 O 3 . Experimentally, we find that the field-induced effect at (1/2,1/2,0) is much larger when B is parallel to the c-axis 1 , which is inconsistent with the cubic symmetry of (Nd,Ce) 2 O 3 but consis- tent with the upper critical field of NCCO being much smaller in this geometry 1,2 . Second, as the lattice parameter of (Nd,Ce) 2 O 3 does not match the c-axis lattice parameter of NCCO (ref.1),measurements at non-zero integer L cannot be contaminated by (Nd,Ce) 2 O 3 . Our experiments indicate that the (1/2,1/2,3) peak shows an induced antiferromagnetic component when the field is along the c-axis and hence superconduc- tivity is strongly suppressed 1 ,but not when in the a–b plane and superconductivity is only weakly affected 2 . This is direct proof of the connection between field-induced antiferro- magnetic order and suppression of super- conductivity in NCCO. We also note that the qualitatively different behaviour observed when B is perpendicular to c, in comparison with when it is parallel to c, directly violates the cubic symmetry of (Nd,Ce) 2 O 3 . Finally, an independent report 3 confirms our principal findings 1,2 in studies of annealed superconducting Pr 0.89 LaCe 0.11 CuO 4 (PLCCO), a similar electron-doped material in which the cubic impurity phase (Pr,La,Ce) 2 O 3 has a non-magnetic ground state and no field dependence below 7 T (our unpublished observations).For fields up to 5 T,Fujita et al. 3 find enhanced antiferromagnetic order at (1/2,3/2,0) with increasing field in PLCCO. Above 5T,this order decreases with increasing field, which is consistent with the field dependence of (1/2,1/2,0) of NCCO (ref. 2). The agreement between two different elec- tron-doped systems in two independent experiments 1–3 confirms the quantum phase transition from the superconducting to an antiferromagnetic state in electron-doped, high-T c superconductors 2 . H. J. Kang, Pengcheng Dai*, J. W. Lynn, M. Matsuura, J. R. Thompson, Shou-Cheng Zhang, D. N. Argyriou, Y. Onose, Y. Tokura *Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, and Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6393, USA e-mail: daip@ornl.gov 1. Matsuura, M. et al. Phys.Rev.B68, 144503 (2003). 2. Kang, H. J. et al. Nature 423, 522–525 (2003). 3. Fujita, M., Matsuda, M., Katano, S. & Yamada, K. Physica B (in the press). ? 2003 Nature Publishing Group