Thermoelectric effects in superconductors, and their fundamental importance, were first discussed 70 years ago by Ginzburg. These effects originate in charge transfer by thermal quasiparticles, which is compensated by the counterflow of superconducti
Abstract. We present a new conceptual abstraction in symmetry breaking-the GE-tree. The construction and traversal of a GE-tree breaks all symmetries in any constraint satisfaction or similar problem. We give a polynomial-time algorithm for this cons
Symmetries in constraint satisfaction or combinatorial optimization problems can cause considerable diculties for exact solvers.
Symmetries in constraint satisfaction or combinatorial optimization problems can cause considerable dificulties for exact solvers. One way to overcome the problem is to employ sophisticated models with no or at least less symmetries. However, this of
We introduce the study of Conditional symmetry breaking in constraint programming. This arises in a sub-problem of a constraint sat- isfaction problem, where the sub-problem satisfies some condition under which additional symetries hold. Conditional
ARTICLE IN PRESS Physica B 404 (2009) 507–509
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Search for time-reversal symmetry breaking in unconventional superconductors Aharon Kapitulnik a,b,, Jing Xia a,b, Elizabeth Schemm a,b a b
Department of Physics, Stanford University, Stanford, CA 94305, USA Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
abstract We have used polar Kerr effect to test for time-reversal symmetry breaking effects in unconventional superconductors. In particular, high precision measurements of the spin-triplet superconductor Sr2 RuO4 4 using a Sagnac interferometer with a zero-area Sagnac loop showed a non-zero Kerr rotations as big as 65 nrad appearing below T c in large domains. & 2008 Elsevier B.V. All rights reserved.
One can describe unconventional superconductors as materials that display superconductivity but that do not conform to BCS or Migdal–Eliashberg theories. A more pragmatic deﬁnition has been widely used to classify superconductors in which the order parameter is averaged over the entire Fermi surface of the material and if the result is zero that superconductor is deemed unconventional . Obviously non-S-wave superconductors in which the phase of the order parameter changes sign are unconventional. Although many early discovered superconductors turned to be unconventional, it was not until high-T C superconductors were discovered and their order parameter was shown to have a dx2 y2 symmetry that the notion of unconventional superconductivity caught-up the community. S-wave superconductors inherently preserve time-reversal symmetry, however, unconventional superconductors can be found in forms that break time-reversal symmetry. Magneto-optical (MO) effects are described within quantum theory as the interaction of photons with the electrons spins through spin–orbit interaction (see e.g. ). Macroscopically, a linearly polarized light that interacts with magnetized media can exhibit both ellipticity and a rotation of the polarization state. The leading terms in any MO effects are proportional to the offdiagonal part of the ac conductivity: sxy ðoÞ . A ﬁnite MO effect measured in a material unambiguously points to time-reversal symmetry breaking (TRSB) in that system (e.g. in an applied magnetic ﬁeld, sxy ðoÞ is ﬁnite and proportional to the ﬁeld). Its zero frequency limit is the known Hall coefﬁcient of the material. Measurements of polarization rotation upon transmission through (Faraday) or reﬂection from (Kerr) a TRSB material is a very difﬁcult task, especially if low incident power is required, and
E-mail address: [email protected] (A. Kapitulnik). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.11.058
the signal cannot be modulated for lock-in detection . To overcome these difﬁculties, we invented the MO ﬁber Sagnac interferometer [4–6], which measures the relative phase shift between two beams of light that travel an identical path in opposite directions. The latest version of the Sagnac interferometer was introduced by our group in 2006  in which a polarization maintaining ﬁber supports a ‘‘back and force’’ beams traveling forward in the slow axis and returning in the fast axis of the ﬁber (and vice versa). Similar to the original design , the two main beams travel precisely the same distance from source to detector, except for a small phase difference j, which is solely from the TRS-breaking sample. This scheme provides almost absolute immunity against reciprocal effects while measuring the Kerr angle yK ¼ 2j with a shot-noise limited sensitivity for powers above 3 mW (e.g. at 20 mW the sensitivity is 70 nrad/ pﬃﬃﬃﬃﬃﬃﬃ Hz). We also showed remarkable stability of 20 nrad variation over 35 hours of measurement at a constant temperature. Note also that in this scheme, if a mirror is placed at the back of a transparent sample, the Faraday angle can be measured with double the sensitivity (because of the two passings of the beam through the sample yF ¼ 4j). Sr2 RuO4  is widely believed to be an odd parity [9–12] p-wave superconductor (angular momentum L ¼ 1 and spin S ¼ 1). Many such states are possible for Sr2 RuO4, some of which are TRSB, either because of their spin or orbital (or both) parts of the pair wave function. A speciﬁc prediction was made for Sr2 RuO4 for an order parameter with the symmetry ~ d ¼ D0 z^ ðpx ipy Þ [9,10] in which the spin triplet is in the ruthenium-oxide planes while the L ¼ 1 angular momentum points either up or down of the planes. Such a state breaks time-reversal symmetry, resembling an orbital ferromagnet. An ideal sample will not exhibit a net magnetic moment due to Meissner effect screening of the TRSB moment. Conﬂicting results existed prior to our studies, some supported TRSB effects  and some did not , both interpreted their
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A. Kapitulnik et al. / Physica B 404 (2009) 507–509
Fig. 1. Representative results of training the chirality with an applied ﬁeld. Starting with a þ93 Oe ﬁeld cool, then zero-ﬁeld warm-up (circles). The two solid squares represent the last two points just before the ﬁeld was turned off. Dashed curve is guide to the eye, showing a linear dependence near T C .
results with a ‘‘p þ ip’’ state in mind. Thus, establishing the existence of this effect, in the bulk, without relying on imperfections and defects was of utmost importance. Indeed, in the ﬁrst demonstration of the capabilities of the loopless Sagnac interferometer we tackled the Sr2 RuO4 problem. In this study polar Kerr effect was measured with high precision, observing a non-zero Kerr rotations as big as 65 nrad appearing below T C in large domains . Our results therefore implied unambiguously a broken time-reversal symmetry state in the superconducting state of Sr2 RuO4 . Fig. 1 is a representative of our results. Here we show a measurement performed during a zeroﬁeld warm-up after cooling the sample in a ﬁeld of þ93 Oe. We note that the size of the signal is identical to measurements done after zero-ﬁeld cool-down, thus indicating (together with the two data points taken in a ﬁeld: see Fig. 1) that the effect we measure is genuine and not due to vortices. With renewed interest in the nature of the pseudo-gap in highT C superconductors [16–18], we used the Sagnac interferometer to measure polar Kerr effect of YBa2 Cu3 O6þx single crystals. In our recent paper , we reported high-resolution optical Kerr effect measurements on YBa2 Cu3 O6þx crystals with various hole concentrations (which is a monotonic function of the oxygen concentration, and it also depends on oxygen ordering in the chains ). We identify a sharp phase transition at a temperature T s ðpÞ, below which there is a non-zero Kerr angle, indicating the existence of a phase with broken time-reversal symmetry. Both the magnitude and hole concentration dependence of T s are in close correspondence with those of the pseudo-gap crossover temperature, T , which has been identiﬁed in other physical quantities. In particular T s is substantially larger than the superconducting T C in underdoped materials, but drops rapidly with increasing hole concentration, so that it is smaller than T C in a near optimally doped crystal and extrapolates to zero at a putative quantum critical point under the superconducting dome. Fig. 2 shows the phase diagram obtained from a set of four crystals and two thin ﬁlms. The fact that thin ﬁlms produced under different conditions and with uncontrolled disorder show the same transition as high quality single crystals strengthen our belief that the effect is genuine and independent of disorder. In conclusion we have shown some initial results on the use of a Sagnac Kerr apparatus for the study of broken time-reversal
Fig. 2. The onset of the Kerr-effect signal, T s for single crystals (circles), and for thin ﬁlms (squares). Also shown are the superconducting dome for this set of crystals and the onset of antiferromagnetic order T N . Inset shows a representative Kerr transition at T s for a sample with x ¼ 0:5.
symmetry effects in unconventional superconductors. We ﬁrst demonstrated measurements in the superconducting state of Sr2 RuO4 , showing that TRSB onsets at T C . We then showed that in YBa2 Cu3 O6þx , Kerr effect onsets at a temperature close to the socalled pseudo-gap temperature. Fabrication of the Sagnac system was supported by Stanford’s Center for Probing the Nanoscale, NSF NSEC Grant 0425897. Work at Stanford was supported by the Department of Energy Grant DEAC02-76SF00515. References             
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