We plot the probability to measure the correct result P after N applications of the gate C, Lh to the initial state IV(0)) .~r (IO)bIO)k + j1));jl)k). We have chosen the parameters such that a single gate Ckb fails with probability p = 0.09. Without error correction (dashed line), Nc>p = 1/p = 11 gates can be performed reliably, where P = exp(-N/Nop). Note that we have used the gate C, Lh instead of Cab (Eq. 5) for the

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We report adiabatic specific heat measurements for a high-purity YBa 2 Cu 3 O x twinned single crystal. The order of the transition on the melting line of the vortex system B m ͑T ͒ is investigated for three oxygen concentrations. For x 7.00, no critical end point is observed. For x 6.94, first-order transitions give way to second-order transitions above a tricritical point at a field B cr 9.5 T, and the slope of B m ͑T ͒ changes. The latent heat, the slope of B m ͑T͒, and B cr increase with x. Data for less pure crystals are included for comparison purposes. [S0031-9007(97)05280-0] PACS numbers: 74.25.Bt, 74.60.Ge, 74.62.Dh, 74.72.BkThe high superconducting critical temperature T c , the short coherence length j, the large anisotropy g ϵ ͑m c ͞m ab ͒ 1͞2 , and the high upper critical field B c2 of high temperature superconductors all contribute to enhance thermal fluctuations. As a consequence, the vortex system melts at temperatures significantly below B c2 , a phenomenon that gives rise to anomalies particularly in the magnetoresistance [1], the magnetization M [2,3], and the specific heat C [4][5][6][7]. Another consequence is the absence of any abrupt phase transition on the B c2 ͑T ͒ line, which becomes a crossover.Depending on samples, the vortex melting transition has been reported to be of first or second order. First order means a discontinuity in the first derivatives of the free energy F, i.e., a jump DM in the magnetization and a jump DS in the entropy (or a d function in the specific heat). Second order means a discontinuity in the second derivatives of F, i.e., a break in the slope of the magnetization and a jump DC in the specific heat. Untwinned crystals of YBa 2 Cu 3 O x ͑YBCO x ͒ near to optimal doping [4,8] and overdoped twinned crystals of YBa 2 Cu 3 O 7.00 [5,6] have shown sharp peaks on the melting line B m ͑T ͒ which are attributed to first-order transitions between a vortex lattice and a vortex liquid. The latent heat below these peaks is 0.45k B T to 0.6k B T per vortex per CuO 2 layer on the average, in agreement with the observed M jumps [2,3]. Alternatively, the same crystals in fields #0.5 T [4,8] or #5 T [5,6], a twinned crystal in fields from 3 to 7 T [9], and a massive but less ordered twinned crystal in fields from 1 to 14 T [5,7] have shown specific heat steps practically on the same B m ͑T ͒ line. The latter have been attributed to second-order transitions from a vortex glass to a vortex liquid [5,7]. Finally, it has been reported, on the basis of transport measurements, that the first-order transition line may terminate in a critical point near ഠ10 T on the high field side [10], but calorimetric measurements up to 26.5 T in a fully oxygenated YBa 2 Cu 3 O 7.00 crystal have not detected any critical end point [11].In order to clarify the existence of the critical points on the melting line B m ͑T͒, we have investigated the specific heat of the high-purity crystal of Ref.[6] in oxygendeficient states. The main experimental result of this study is that the range of fields where s...

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In a recent article, Overend, Howson, and Lawrie [1], subsequently referred to as OHL, gave strong evidence that their specific heat data near T, in a magnetic field were better described by low field critical scaling laws than by the lowest Landau level (LLL) high field approximation, at least up to 8 Tesla. The critical exponents and amplitude ratios were found to be consistent with those observed in the A transition of He. OHL based their conclusion on the excellent scaling obtained using the 3D X-Y model, and the impossibility of scaling their data using the LLL approximation.OHL measured the specific heat C of a mg-size single crystal of YBa2Cu307 s (YBCO) in magnetic fields up to 8T, using an ac technique. Such methods yield only relative values. We have recently measured the specific heat of a much larger crystal (0.29g) of YBCO up to 16T, using a high precision adiabatic, continuous heating method.The weakest part of any scaling attempt is the approximation used for the nonsingular part of the specific heat, since the fluctuation component Cf1 is but a few percent of the total heat capacity Ct,t. OHL use a linear baseline (i.e., C", -Cri = a + bT) situatedfar above the specifi heat peak. To avoid such an arbitrary baseline subtraction, we suggest another way to scale the data. Since phonons, and more generally nonsingular contributions, are expected to be insensitive to the field, we have rlC, ",/r) lnB = r)Cfj/r) lnB. The quantity AC",/rl lnB obeys the same scaling laws as Cf& but is not blurred by a large background. The price to be paid for the derivation is a high accuracy of the thermometry versus field. The latter was verified in our case using the triple point of argon.Using the measured quantity r)Cr~j&3 lnB, we tested both the LLL [3] and 3D X-Y approximations [4], as attempted by OHL using Cri -Co. We define T, (B) as r)(C/T)/r)B(T = max. o 20 -QSII%+ IIII' --2O--60-037T 0.75 T 1.5 T Q 3 T o 12 T -0.05 -0.03 -0.01 0.01 0.03 0.05 {TTc) / (TcH ) [ T-0.763] FIG. 2. Scaling plot of B ' 't 'BCt~/6 lnB in the 3D X-I' model. u = -0.0066 and v = 0.664. The plot of r)Cf~/r) lnB vs the scaling variable x = [T-T, (B)]j/[BT] , / corresponding to the LLL approximation is given in Fig. 1. The curves collapse on a single one for average fields higher than 1 T. Below 1 T, as could be expected, either higher Landau levels contribute, making the LLL approximation invalid, or finite size and/or inhomogeneity effects limit the sharpness of the transition. The corresponding 3D X-Y scaling plot is given by B ' /~'r)Cr~/r) lnB vs [T -T,]j(tT, B'/ "] (Fig. 2). The result is somewhat less good but does not differ drastically from that obtained in the LLL scheme. This is due to the negligible variation of the prefactor B~/ (2'1 (0.986 & [B(T)]~/ (2'1 & 1.007 for 0.25 & B & 16 T, which is replaced by 1 in the LLL case, and to the small difference between I/2v = 0.753 and the LLL exponent of 3. 2Summarizing, we state that, based on a method that does not depend on an arbitrary baseline subtraction, one cannot decisively ru...