Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations

Abstract

As an extended companion paper to [1], we elaborate in detail how the tensor network construction of a p-adic CFT encodes geometric information of a dual geometry even as we deform the CFT away from the fixed point by finding a way to assign distances to the tensor network. In fact we demonstrate that a unique (up to normalizations) emergent graph Einstein equation is satisfied by the geometric data encoded in the tensor network, and the graph Einstein tensor automatically recovers the known proposal in the mathematics literature, at least perturbatively order by order in the deformation away from the pure Bruhat-Tits Tree geometry dual to pure CFTs. Once the dust settles, it becomes apparent that the assigned distance indeed corresponds to some Fisher metric between quantum states encoding expectation values of bulk fields in one higher dimension. This is perhaps a first quantitative demonstration that a concrete Einstein equation can be extracted directly from the tensor network, albeit in the simplified setting of the p-adic AdS/CFT. * Chen and Liu are co-first authors of the manuscript.

Effective field theory on a finite boundary of the Bruhat-Tits tree

“…Relations between T p and tensor network are studied in [16][17][18]. The p-adic version of the Anti-de Sitter/Conformal Field Theory duality [19][20][21] is proposed in [10,22]( p-adic AdS/CFT), which are followed by lots of works, such as [23][24][25][26][27][28][29][30][31][32][33].…”

Thread/State correspondence: from bit threads to qubit threads

“…More recently, many enlightening ideas from other fields, such as condensed matter physics, quantum information theory, network flow optimization theory, etc., have entered and benefited the study of the holographic gravity. One of the most striking examples is that inspired by the tensor network method originally used in condensed matter physics as a numerical simulation tool to investigate the wave functions of quantum many-body systems, various holographic tensor network (TN) models have been constructed as toy models of the holographic duality, such as MERA (multiscale entanglement renormalization ansatz) tensor network [15][16][17][18], perfect tensor network [19], random tensor network [20], p-adic tensor network [21][22][23], OSED (one-shot entanglement distillation) tensor network [24][25][26] and so on. For more research on tensor networks in the holographic context, see e.g.…”