Renormalization Group Flows of Hamiltonians Using Tensor Networks

A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015); S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff’s spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.

Figure 1

(a)–(e) Illustration of a single step of the ${\mathrm{TNR}}_{+}$ algorithm. (a) Starting from a bipartite square lattice, (b) we approximate the periodic MPS representing a block of four sites by a rotated version (c) with a different tensor product structure and (d) contract these numerically optimized tensors exactly to (e) arrive at a coarse-grained tilted lattice. (f) Iterating the ${\mathrm{TNR}}_{+}$ procedure in the presence of an open boundary generates a stochastic MERA.

Figure 2

${\mathrm{TNR}}_{+}$ simulations for the critical Ising model and spin ice. (a) Relative error of the free energy per site in function of the ${\mathrm{TNR}}_{+}$ bond dimension ($N=2^{32}$ sites). (b),(c) Scaling dimensions extracted from the linear transfer matrix MPO Eq. (5) in function of the RG step (Ising $D=16$, spin ice $D=12$).

Figure 3

Non-negative tensor elements of normalized fixed point tensors $C_{1,A}^{*}$ obtained from $D=6$ ${\mathrm{TNR}}_{+}$ simulations of the Ising model at (a) $T<T_c$, (b) $T=T_c$, and (c) $T>T_c$.