Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $\mathrm{AdS}_3 / \mathrm{CFT}_2$ correspondence

Abstract

One of the many remarkable properties of conformal field theory in two dimensions is its connection to algebraic geometry. Since every compact Riemann surface is a projective algebraic curve, many constructions of interest in physics (which a priori depend on the analytic structure of the spacetime) can be formulated in purely algebraic language. This opens the door to interesting generalizations, obtained by taking another choice of field: for instance, the p-adics. We generalize the AdS/CFT correspondence according to this principle; the result is a formulation of holography in which the bulk geometry is discrete-the Bruhat-Tits tree for PGL(2, Q p )-but the group of bulk isometries nonetheless agrees with that of boundary conformal transformations and is not broken by discretization. We suggest that this forms the natural geometric setting for tensor networks that have been proposed as models of bulk reconstruction via quantum error correcting codes; in certain cases, geodesics in the Bruhat-Tits tree reproduce those constructed using quantum error correction. Other aspects of holography also hold: Standard holographic results for massive free scalar fields in a fixed background carry over to the tree, whose vertical direction can be interpreted as a renormalization-group scale for modes in the boundary CFT. Higher-genus bulk geometries (the BTZ black hole and its generalizations) can be understood straightforwardly in our setting, and the Ryu-Takayanagi formula for the entanglement entropy appears naturally.Much attention has been paid of late to ideas that allow certain features of conformal field theory, such as long-range correlations, to be reproduced in lattice systems or other finitary models. As an example, the multiscale entanglement renormalization ansatz (or MERA), formulated by Vidal in [58], provides an algorithm to compute many-qubit quantum states whose entanglement properties are similar to those of the vacuum state in a conformal field theory. In Vidal's method, the states of progressively more distant qubits are entangled using successive layers of a self-similar network of finite tensors.These proposals can typically be thought of as constructing analogues of the CFT vacuum state using a quantum circuit with an additional "spatial direction," consisting of the successive computational layers of the circuit, and corresponding roughly to the distance scale up to which long-range entanglement has been introduced. As such, they are strongly suggestive of the AdS/CFT correspondence [30,36,61], in which a ddimensional conformal field theory is related to a gravitational theory in d+1-dimensional negatively curved spacetime, and the extra direction can be interpreted as a renormalization scale (or equivalently a length scale) from the perspective of the boundary theory. Furthermore, the construction of the layers (in which the number of tensors scales exponentially with the number of layers) bears comparison with the geometry of hyperbolic space. It was thus natural to search for a connection with ...

Tensor network and (p-adic) AdS/CFT

“…This is in contrast to [26,27], which assumed a p-adic AdS/CFT correspondence and then derived various consequences. In our tensor network construction, we do not assume p-adic AdS/CFT, but aspects of the AdS/CFT correspondence emerge from the tensor network.…”

“…Therefore a tensor network based on the Bruhat-Tits tree would have much more symmetry than its counterpart living on a regular tessellation. This is inspired by recent proposals for the p-adic AdS/CFT correspondence [26][27][28], which generalize the AdS/CFT dictionary to the situation where the boundary theory lives…”

“…A possible connection between p-adic AdS/CFT and tensor networks was first studied in [26], whose construction is based on perfect tensors and on embedding the Bruhat-Tits tree in the geometric tiling (in particular, the HAPPY tiling) of the bulk. The construction in the current paper is different in that (1) we are not using perfect tensors, building on earlier work that imperfect tensors are necessary to furnish non-trivial correlation functions [21]; and (2) we view the Bruhat-Tits tree as an abstract tree, and hence do not need to embed the Bruhat-Tits tree in a real bulk space.…”

“…We emphasize its difference from tensor networks based on regular tessellations of AdS. It can be viewed as an explicit realization of p-adic AdS/CFT recently proposed in [26,27]. …”

“…This consideration is inspired by the recently discussed p-adic AdS/CFT correspondence [26,27]. The proposed duality is a discrete analogue of AdS/CFT.…”