On the Symmetries of Renormalization Group Fixed Points

Wu Xintian, Chang-De Gong

The replica trick used in the renormalization group (RG) for the disordered spin systems is discussed. The recursion relations of RG are derived in a more general situation in which the coupling constants can have arbitrary symmetry to the replica indices. It is shown that the random fixed point found by Luther, Grinstein and Lubensky is unstable. This instability implies that the replica symmetry should be broken in the RG for the disordered systems.

Jerzy Lukierski B

Sukhwinder Singh, Guifré Vidal

Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multiscale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. Here we propose the use of symmetric tensors as a mechanism to build a symmetry-protected RG flow, and discuss two important applications of this construction. First, we argue that symmetry-protected entanglement renormalization produces the proper structure of RG fixed points, namely, a fixed-point for each symmetry-protected phase. Second, in the context of holography, we show that by using symmetric tensors, a global symmetry at the boundary becomes a local symmetry in the bulk, thus explicitly realizing in the MERA a characteristic feature of the AdS/CFT correspondence.

J. Lukieeski

Jonathan F. San Miguel, Arpit Dua, Dominic J. Williamson

We introduce the subsystem symmetry-preserving real-space entanglement renormalization group and apply it to study bifurcating flows generated by linear and fractal subsystem symmetry-protected topological phases in two spatial dimensions. We classify all bifurcating fixed points that are given by subsystem symmetric cluster states with two qubits per unit cell. In particular, we find that the square lattice cluster state is a quotient-bifurcating fixed point, while the cluster states derived from Yoshida's first-order fractal spin liquid models are self-bifurcating fixed points. We discuss the relevance of bifurcating subsystem symmetry-preserving renormalization group fixed points for the classification and equivalence of subsystem symmetry-protected topological phases.

Giancarlo Benettin, C. Di Castro, Giovanni Jona-Lasinio +2 more