Renormalization group circuits for gapless states

Brian Swingle, John McGreevy, and Shenglong Xu

Phys. Rev. B 93, 205159 – Published 31 May 2016

Abstract

We show that a large class of gapless states are renormalization group fixed points in the sense that they can be grown scale by scale using local unitaries. This class of examples includes some theories with a dynamical exponent different from one, but does not include conformal field theories. The key property of the states we consider is that the ground-state wave function is related to the statistical weight of a local statistical model. We give several examples of our construction in the context of Ising magnetism.

Received 24 February 2016
Revised 26 April 2016

DOI:https://doi.org/10.1103/PhysRevB.93.205159

Physics Subject Headings (PhySH)

Quantum criticality Quantum entanglement

Ising model Renormalization group Tensor network methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Brian Swingle¹, John McGreevy², and Shenglong Xu²

¹Department of Physics, Stanford University, Palo Alto, California 94305, USA
²Department of Physics, University of California at San Diego, La Jolla, California 92093, USA

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Vol. 93, Iss. 20 — 15 May 2016

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Images

Figure 1
Temperature flow of real-space RG for 1D classical Ising model. Blue dash line represents the change of $β$ , which labels the square root state after successive RG transformation.
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Figure 2
Sketch of the RG circuit for the quantum Ising chain. The blue dots in the bottom layer represent the initial square root state. Green triangles are unitary operators $u_{i}^{s_{l} s_{r}}$ [Eq. (2.15)] and yellow squares are projection operators $P_{i}^{s}$ , serving as controllers. Horizontal dash lines contract the indices. After each iteration, the state on the odd sites becomes product state $| X 〉$ and distangled from the state on the even sites, which forms a new square root state with renormalized $β$ . Eventually, the initial state is transformed by local unitary operators to a complete untangled state $| X 〉$ (top layer).
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Figure 3
Transformation of a single site operator under unitary-RG transformation. Green triangles are unitary operators $u_{i}^{s_{l} s_{r}}$ and yellow squares are projection operators $P_{i}^{s}$ , serving as controllers. Dash lines contract the indices. (a) The application of $U^{(1)}$ on a local operator $O$ (red diamond) that has its support on a single site. After the transformation, the renormalized new operator, denoted as $O^{(1)}$ has its support on two sites, in the form of two projection operators linked by a matrix $m$ determined by the initial operator $O$ . (b) We apply unitary transformation $U^{(2)}$ to $O^{(1)}$ . The middle site in (a) is omitted since it is disentangled with the rest of the Hilbert space. There are two equivalent situations. In the first case, $u^{s_{l} s_{r}}$ is applied to the right projection operator in $O^{(1)}$ ; in the second case, $u^{s_{l} s_{r}}$ is applied to the left projection operator. After the transformation, a new operator $O^{(2)}$ is obtained, which shares the same form as $O^{(1)}$ but with renormalized $\tilde{m}$ . Then this procedure can be repeated.
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Figure 4
A sketch of the ingredients of the RG circuit for a 2D quantum Ising square root state. In the TRG representation of the triangular lattice Ising model, the tensors act on vector spaces associated with the links of the lattice; we have found it convenient to draw directly the resulting link lattice, which in this case is the kagome lattice. The rewiring step $U_{1}$ is at left and the disentangling step $U_{2}$ is at right. Orange blocks are controllers, green blocks are unitary transformations that depend on the controllers that connected to them. White balls represent disentangled sites. This color choice is consistent with the figures above for the one-dimensional case.
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