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We develop the statistical mechanics of the Hopfield model in a transverse field to investigate how quantum fluctuations affect the macroscopic behavior of neural networks. When the number of embedded patterns is finite, the Trotter decomposition reduces the problem to that of a random Ising model. It turns out that the effects of quantum fluctuations on macroscopic variables play the same roles as those of thermal fluctuations. For an extensive number of embedded patterns, we apply the replica method to the Trotter-decomposed system. The result is summarized as a ground-state phase diagram drawn in terms of the number of patterns per site, $\alpha$, and the strength of the transverse field, $\Delta$. The phase diagram coincides very accurately with that of the conventional classical Hopfield model if we replace the temperature T in the latter model by $\Delta$. Quantum fluctuations are thus concluded to be quite similar to thermal fluctuations in determination of the macroscopic behavior of the present model.Comment: 34 pages, LaTeX, 9 PS figures, uses jpsj.st

Phase diagram of vortex states of high-Tc superconductors with sparse and weak columnar defects is obtained by large-scale Monte Carlo simulations of the three-dimensional anisotropic, frustrated XY model. The Bragg-Bose glass phase characterized by hexagonal Bragg spots and the diverging tilt modulus is observed numerically for the first time at low density of columnar defects. As the density of defects increases, the melting temperature increases owing to "selected pinning" of flux lines. When the density of defects further increases, the transition to the Bose glass phase occurs. The interstitial liquid region is observed between these two glass phases and the vortex liquid phase. PACS numbers: 74.25.Qt, 74.62.Dh, 74.25.Dw In vortex states of high-T c superconductors, columnar defects introduced by heavy-ion irradiation are strong pinning centers of flux lines, and dramatically enhance the critical current.[1] Nelson and Vinokur [2] analyzed these defects and derived a phase transition between the Bose glass (BG) and vortex liquid (VL) phases. They also showed that this system can be mapped to twodimensional interacting bosons in a random potential and that this phase transition corresponds to a localization transition of interacting bosons. In the case of dense and strong columnar defects, every flux line is trapped by randomly-distributed columnar defects at low temperatures, and the BG phase naturally appears. Many numerical studies [3,4,5] have been made in this case.In the case of sparse and strong columnar defects, part of flux lines are not trapped by defects, and physical properties are more complicated. At low temperatures untrapped flux lines are also frozen owing to interactions with trapped flux lines, and the BG phase is stable. At intermediate temperatures the defects still trap some flux lines but other interstitial ones are freely moving around, and the resistivity becomes finite. This region is called as the interstitial liquid (IL).[6] At high temperatures all flux lines are depinned and the VL phase appears. It is not clarified yet whether there exists a phase transition or merely a crossover between the IL and VL. A numerical simulation was attempted [7] in this case, and the existence of the BG phase was confirmed. Two kinds of BG phases for dense or sparse columnar defects are divided by the Mott insulator phase at the matching field. Recently the transition between these two (strong and weak) BG phases was observed numerically. [8] In pure systems, the triangular flux-line lattice (FLL) is formed below the melting temperature T m . Is such order completely destroyed by introducing infinitesimal columnar defects? In vortex states with point defects, the long-range order of vortex positions is destroyed by ; Electronic address: nonomura.yoshihiko@nims.go.jp infinitesimal defects, but the Bragg glass phase [9,10,11] characterized by the quasi long-range order of vortex positions and Bragg spots of the structure factor is stable up to a considerable density of point defects. In the presen...

Nonomura and Hu Reply:In the preceding Comment [1], Olsson and Teitel questioned the possible vortex slush (VS) phase in the frustrated XY model with point defects reported by the present authors [2]. The VS phase was originally proposed in order to explain an experiment of irradiated YBa 2 Cu 3 O 7 (YBCO) [3]. This phase was also observed in an optimally doped pristine YBCO [4], where the VS phase locates above the Bragg glass (BG) phase in the H-T phase diagram. In Monte Carlo simulations of the frustrated XY model by the present authors [2], a first-order transition was observed between the vortex liquid (VL) and VS phases up to a certain density of point defects. The structure factor in the VS phase shows obscure Bragg peaks, which was interpreted as a short-range order in the ab plane. In comparison with the BG phase for lower density of point defects, the VS phase shows a much larger density of dislocations in the ab plane and the vanishing helicity modulus along the z axis.Olsson and Teitel (OT) simulated the same model for the same parametrization as used in Ref. [2]. On the basis of the structure factors observed in layer by layer, they argued that the VS phase observed by the present authors may be an artifact of a finite system size, and that this region may be included in the BG phase in the thermodynamic limit. However, their argument in the Comment is not justified sufficiently by the provided numerical data for the following reasons.First, they observed strong hysteresis behavior by sweeping the pinning strength of point defects in the VS region, and took this behavior as the evidence for a wide coexistence region of the first-order melting of the Bragg glass. However, this behavior can alternatively be interpreted as a merging of the consequent VL-VS and VS-BG first-order phase transitions due to a small system size, with which the two-step behavior in the hysteresis curve of the peak value of the structure factor in Fig. 1b of Ref.[1] looks consistent. It should also be pointed out that the hysteresis behavior may be enhanced by their -sweeping procedure. Since the annealing process is not included in this procedure in spite of possible drastic changes of the configurations of flux lines caused by varying , Monte Carlo steps necessary for equilibration in this procedure may be much larger than those in the temperature sweeping used in our previous Letter [2].Second, they argued that the energy loss due to a mismatch in different layers is proportional to J z L 2 with the transverse system size L. This scaling argument is based on the assumption that the mismatch characterized by the change of peak positions of Bragg peaks occurs as abruptly as a domain wall of the Ising model. However, the mismatches displayed in Fig. 2 of Ref.[1] relax across a number of layers. When the relaxation takes place across L w layers, the energy loss is proportional to J z L 2 =L w . It is natural to expect that L w depends on the thickness of the system L z . Provided L w is proportional to L z L,