The holographic model for S-wave high T c superconductors developed by Hartnoll, Herzog, and Horowitz is generalized to describe D-wave superconductors. The 3 þ 1 dimensional gravitational theory consists of a symmetric, traceless second-rank tensor field and a Uð1Þ gauge field in the background of the anti-de Sitter black hole. Below T c the tensor field, which carries the Uð1Þ charge, undergoes the Higgs mechanism and breaks the Uð1Þ symmetry of the boundary theory spontaneously. The phase transition characterized by the D-wave condensate is second order with the mean field critical exponent ¼ 1=2.As expected, the AC conductivity is isotropic below T c , and the system becomes superconducting in the DC limit but has no hard gap.

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1-J2 model on an L × L square lattice with L=8,16 and 32. By treating tensor elements as variational parameters, we obtain the ground states for different J2/J1 values, and investigate staggered magnetizations, nearest-neighbor spinspin correlations and plaquette order parameters. In addition to the well-known Néel order and collinear order at low and high J2/J1, we observe a plaquette-like order at J2/J1 ≈ 0.5. A continuous transition between the Néel order and the plaquette-like order near J c 1 2 ≈ 0.40J1 is observed. The collinear order emerges at J c 2 2 ≈ 0.62J1 through a first-order phase transition.

We present a method for contracting a square-lattice tensor network in two dimensions, based on auxiliary tensors accomplishing successive truncations (renormalization) of 8-index tensors for 2 × 2 plaquettes into 4-index tensors. Since all approximations are done on the wave function (which also can be interpreted in terms of different kind of tensor network), the scheme is variational, and, thus, the tensors can be optimized by minimizing the energy. Test results for the quantum phase transition of the transverse-field Ising model confirm that even the smallest possible tensors (two values for each tensor index at each renormalization level) produce much better results than the simple product (mean-field) state.