Abstract
Recent studies of holographic tensor network models defined on regular tessellations of hyperbolic space have not yet addressed the underlying discrete geometry of the boundary. We show that the boundary degrees of freedom naturally live on a novel structure, a conformal quasicrystal, that provides a discrete model of conformal geometry. We introduce and construct a class of one-dimensional conformal quasicrystals, and discuss a higher-dimensional example (related to the Penrose tiling). Our construction permits discretizations of conformal field theories that preserve an infinite discrete subgroup of the global conformal group at the cost of lattice periodicity.
Highlights
A central topic in theoretical physics over the past two decades has been holography: the idea that a quantum theory in a bulk space may be precisely dual to another living on the boundary of that space
We show that the boundary degrees of freedom naturally live on a novel structure, a “conformal quasicrystal,” that provides a discrete model of conformal geometry
Physicists have been interested in the possibility that discrete models of holography would permit them to understand this idea in greater detail, by bringing in the tools of condensed matter physics and quantum information theory—much as lattice gauge theory led to conceptual and practical progress in understanding gauge theories in the continuum [5]
Summary
A central topic in theoretical physics over the past two decades has been holography: the idea that a quantum theory in a bulk space may be precisely dual to another living on the boundary of that space. There has been a common expectation, based on an analogy with AdS/CFT [1,2,3], that TNs living on discretizations of a hyperbolic space define a lattice state of a critical system on the boundary and vice versa. We argue that the d.o.f. on the boundary of a regular tessellation of hyperbolic space naturally live on a remarkable structure—a “conformal quasicrystal” (CQC) —built entirely from the data of the bulk tessellation This provides a new clue about the type of boundary theory that should appear in a discrete version of holography. Fractional and negative exponents behave in the usual way, e.g., AxA−y 1⁄4 Ax−y for all x; y ∈ Q
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