Tensor networks as conformal transformations

TL;DR

The paper shows that tensor networks can enact local conformal transformations on a critical spin chain by exploiting the local structure of euclideons and MERA blocks, together with end smoothers that regularize truncated regions. By matching low-energy lattice states to CFT operators via Cardy and Koo–Saleur, the authors construct finite, non-uniform euclidean-time evolutions $V_0^{\text{CFT}}$ and local rescalings $V_1^{\text{CFT}}$ whose matrix elements are diagonally organized in conformal towers and reproduce CFT predictions for non-uniform deformations. In numerical tests on the critical Ising chain, the lattice gates $V$ closely mirror the CFT transformations, validating a concrete, geometry-aware interpretation of tensor networks as curved-space path integrals and suggesting applications to non-perturbative QFT in curved spacetimes and holography. The framework is extendable to non-critical (massive) theories by allowing scale- and time-dependent tensors, offering a versatile route to explore conformal and near-conformal physics with localized operations.

Abstract

Tensor networks are often used to accurately represent ground states of quantum spin chains. Two popular choices of such tensor network representations can be seen to implement linear maps that correspond, respectively, to euclidean time evolution and to global scale transformations. In this paper, by exploiting the local structure of the tensor networks, we explain how to also implement local or non-uniform versions of both euclidean time evolution and scale transformations. We demonstrate our proposal with a critical quantum spin chain on a finite circle, where the low energy physics is described by a conformal field theory (CFT), and where non-uniform euclidean time evolution and local scale transformations are conformal transformations acting on the Hilbert space of the CFT. We numerically show, for the critical quantum Ising chain, that the proposed tensor networks indeed transform the low energy states of the periodic spin chain in the same way as the corresponding conformal transformations do in the CFT.

Figures (9)

• Figure 1: Tensor networks $\mathcal{T}$ and $\mathcal{W}$ for an euclidean time evolution $e^{-H}$ by unit time, made of euclideons $e$, and a scale transformation by a scale factor $2$, made of disentanglers $u$ and isometries $w$.
• Figure 2: Examples of proposed tensor networks for (a) non-uniform euclidean time evolution, made of euclideons $e$ and smoothers $e_L$ and $e_R$, and (b) local scale transformation, made of disentanglers $u$ and isometries $w$, as well as smoothers $u_L$ and $u_R$. (c) A more general conformal transformation mixing non-uniform euclidean time evolution and rescaling is possible by combining the above building blocks.
• Figure 3: (a) When regarding the circle as embedded in a spacetime cylinder at euclidean time $\tau=0$, we can consider two types of deformations of the circle by conformal transformations: time evolution (left), which displaces the points in the circle in the time direction, and deformations of the circle at constant time (right), which map points of the circle into other points of the same circle. (b) Uniform conformal deformations include time evolution and a periodic translation, as generated by the hamiltonian $H^{\hbox{\tiny CFT}}$ and the momentum operator $P^{\hbox{\tiny CFT}}$ in Eq. (\ref{['eq:HP']}). (c) Non-uniform deformations are generated by $Q_0$ and $Q_1$ in Eqs. (\ref{['eq:Q0']})-(\ref{['eq:Q1']}), which can be expanded in Fourier modes $H_n$ and $P_n$, shown here for $n=1,2$.
• Figure 4: (a) Tensor network for a non-uniform euclidean time evolution $V$ on $N=24$ spins together with the euclidean time evolution profile which it is expected to approximate SupplMat. (b) Result of applying $V$ to the ground state $\hbox{$| \mathbb{I} \rangle$}$ (i.e. the identity primary state), the stress-tensor states $|T\rangle$, $|\overline{T}\rangle$, and the primary states $|\sigma\rangle$ and $|\varepsilon\rangle$. Shown are the largest matrix elements $V_{\alpha\beta}$, see Eq. (\ref{['eq:amplitudes']}), with other states in the same conformal tower as the initial state. Also shown are the corresponding matrix elements, computed perturbatively, of a conformal transformation $V_0^{\hbox{\tiny CFT}}$ in the Ising CFT SupplMat. Spurious tower-mixing matrix elements of $V$ (not shown) were found to have a maximum magnitude of $10^{-2}$ and to decrease with system size PathIntegral.
• Figure 5: (a) Tensor network for a non-uniform scale transformation $V$ on a chain of $N=8$ effective sites (resulting from a scale-invariant MERA siMERA with each effective site represented by a vector space of dimension $\chi=8$) together with the non-uniform translation profile it is expected to approximate SupplMat. (b) Result of applying $V$ to the ground state $\hbox{$| \mathbb{I} \rangle$}$, the stress-tensor states $|T\rangle$, $|\overline{T}\rangle$, and the primary operator states $|\sigma\rangle$ and $|\varepsilon\rangle$. Shown are the largest matrix elements $V_{\alpha\beta}$, see Eq. \ref{['eq:amplitudes']}, with other states in the same conformal tower as the initial state, as well as the corresponding matrix elements, computed perturbatively, of a conformal transformation $V_1^{\hbox{\tiny CFT}}$ in the Ising CFT SupplMat. Tower-mixing matrix elements of $V$ involving these starting states were found to have a maximum magnitude of $\sim 5\times 10^{-3}$ ($6.5 \times 10^{-4}$ for elements with $|T\rangle$ and $|\overline{T}\rangle$).