A tensor network quotient takes the vacuum to the thermal state

TL;DR

The paper demonstrates that a vacuum MERA can realize discrete local conformal transformations by strategically adding or removing tensors to implement logarithmic maps, and that quotients by discrete scale symmetries produce tensor networks for thermal states on a circle. This is tested in the critical Ising model, where the quotient networks reproduce the thermal spectrum with the predicted reduced inverse temperature, supporting the view that optimized MERA embodies emergent local scale invariance. The work clarifies how MERA encodes conformal structure through a tripartite causal decomposition (spacelike, lightlike, timelike) and connects these ideas to holographic duality, offering a concrete bridge between tensor networks and AdS/CFT notions such as BTZ black holes. Together, these results advance the use of tensor networks to model conformal dynamics and thermally curved geometries in a discretized setting.

Abstract

In 1+1-dimensional conformal field theory, the thermal state on a circle is related to a certain quotient of the vacuum on a line. We explain how to take this quotient in the MERA tensor network representation of the vacuum and confirm the validity of the construction in the critical Ising model. This result suggests that the tensors comprising MERA can be interpreted as performing local scale transformations, so that adding or removing them emulates conformal maps. In this sense, the optimized MERA recovers local conformal invariance, which is explicitly broken by the choice of lattice. Our discussion also informs the dialogue between tensor networks and holographic duality.

Figures (14)

• Figure 1: (a) The conformal map $z \rightarrow w = (\beta/\pi)\log z$ maps the upper half plane to an infinite strip of width $\beta$. An exponentially spaced discretization of the positive real axis on the $z$ plane is mapped to a regular discretization of the real axis on the $w$ plane. (b) A quotient by a scaling transformation by a factor $\exp (2\pi^2L/\beta)$ in the upper half plane produces a topological cylinder. That scaling amounts to a translation by $2\pi L$ in the infinite strip. A quotient by a translation by $2\pi L$ in the infinite strip produces a flat cylinder with tallness $\beta$ and circumference of the base equal to $2\pi L$.
• Figure 2: (a) MERA for the ground state on a uniformly discretized infinite line. (b) We can use disentanglers and isometries to replace the regular discretization with an exponential discretization. This produces infinitely many sites near the origin, effectively disconnecting the positive and negative parts of the real axis. This local coarse-graining is a lattice version of the logarithmic scaling map (\ref{['log2']}). (c) A logarithmic scaling map with a different prefactor.
• Figure 3: (a) A global scaling by $\lambda = 1/2$ identifies tensors in MERA. This becomes an exact symmetry of the network after applying a logarithmic scaling map, see figs. \ref{['fig:local']}(b)-(c). (b) The quotient of MERA by a scale factor $\lambda = (1/2)^k$ for $k=2$.
• Figure 4: (Left) Spectrum of singular values $\lambda_{\alpha}$ of the quotient network (understood as a matrix between left and right open indices) for $k=1,2, \cdots, 16$ in the critical Ising model. The spectrum was obtained from the timelike-identified part of the quotient. For each singular value $\lambda_{\alpha}$, we plot the expression $-(k/2\pi\Delta_{\alpha})\log(\lambda_{\alpha}/\lambda_0)$, where $\Delta_{\alpha}$ are the exact scaling dimensions of the Ising CFT. In the absence of approximations, eqs. (\ref{['eq:spectrum']}) and (\ref{['redbeta']}) would set this quantity to $\pi/\log 2$. (Right) The spectrum after including the nearly isometric, lightlike-identified regions in the quotient is almost identical except for small values of $k$.
• Figure 5: Adding or removing disentanglers and isometries locally implements a discrete local scale transformation. The open indices after the transformation live on a cut through MERA, which is piecewise horizontal (spacelike) or at 45$^\circ$ (lightlike) but never timelike.