Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry protected topological (SPT) phase, a bulk critical entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. For the S = 1 SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical entanglement spectrum has a residual entropy per lattice site sr = 0.67602, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted c ≈ 1.01 ± 0.01. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse square long-range interaction. Introduction.-Topological properties of lowdimensional quantum many-body systems have been attracting considerable interest in quantum information sciences, condensed matter physics and quantum field theories. It is understood that important information of a topological phase is encoded in the von Neumann entanglement entropy of its ground state [1][2][3].In a seminal paper, Li and Haldane[4] introduced the entanglement spectrum (ES) from the eigenvalues of the reduced density matrix upon tracing out a subsystem, and the low-lying part bears a remarkable similarity to the physical edge spectrum of the topological state [5][6][7][8]. Recently, Hsieh and Fu[9] have suggested that a symmetric extensive bipartition of the ground state wave function for a topological phase leads to a bulk critical ES that resembles the excitation spectrum of the critical point separating the topological phase from the trivial gapped phase. They used an example of noninteracting fermion Chern insulator to illustrate the emergent critical ES to describe the phase transition in the integer quantum Hall effect.
1. Tryptophan (Trp), besides its role as an essential amino acid in protein synthesis, may also have other important effects on laying hens under summer conditions. 2. Babcock Brown layers (n = 768), 40 weeks of age, were allocated to 4 treatment groups, each of which included 6 replicates of 32 hens. Each group received the same basal diet, formulated with maize and soybean meal, for 8 weeks. Hens were fed on the basal diet with 0·0, 0·2, 0·4, and 0·8 g/kg L-Trp to achieve dietary concentrations of 1·7, 1·9 g/kg, 2·1 g/kg or 2·5 g/kg of Trp, respectively. 3. Supplementing L-Trp had no affect on laying performance. Adding 0·2 or 0·4 g/kg L-Trp improved egg shell strength compared with those fed on the control diet. Serum albumin concentration increased at 0·4 g/kg compared with those receiving 0·0 or 0·8 g/kg Trp. The addition of Trp at 0·4 g/kg increased serum IgM concentration quadratically. Serum superoxide dismutase (T-SOD) increased linearly and quadratically at 0·4 g/kg. 4. In conclusion, we suggest that 0·2 to 0·4 g/kg Trp may have beneficial effects on laying hens under conditions of high temperature and humidity.
Unsupervised machine learning via a restricted Boltzmann machine is an useful tool in distinguishing an ordered phase from a disordered phase. Here we study its application on the two-dimensional Ashkin-Teller model, which features a partially ordered product phase. We train the neural network with spin configuration data generated by Monte Carlo simulations and show that distinct features of the product phase can be learned from non-ergodic samples resulting from symmetry breaking. Careful analysis of the weight matrices inspires us to define a nontrivial machine-learning motivated quantity of the product form, which resembles the conventional product order parameter.PACS numbers:
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