Tensor networks as path integral geometry

TL;DR

The paper reframes a class of tensor networks describing critical spin chains as discrete Euclidean path integrals on curved spacetime, assigning a geometry to the networks via two compatibility rules. It derives a precise link between lattice strips and continuum conformal maps through the generator Q = Q0 + iQ1, rooted in the Euclidean stress tensor and Ward identities, and demonstrates how to refine the network to approximate continuous path-integral geometries. By mapping TN matrix elements to Virasoro-based CFT transformations and performing numerical comparisons in the critical Ising model, it shows that TN maps converge toward the corresponding continuum path-integral maps as the UV cutoff is reduced. This work provides a concrete bridge between tensor-network geometries and spacetime path integrals, offering a framework to study holographic-inspired structures and curved-space QFT on lattice models.

Abstract

In the context of a quantum critical spin chain whose low energy physics corresponds to a conformal field theory (CFT), it was recently demonstrated [A. Milsted G. Vidal, arXiv:1805.12524] that certain classes of tensor networks used for numerically describing the ground state of the spin chain can also be used to implement (discrete, approximate versions of) conformal transformations on the lattice. In the continuum, the same conformal transformations can be implemented through a CFT path integral on some curved spacetime. Based on this observation, in this paper we propose to interpret the tensor networks themselves as a path integrals on curved spacetime. This perspective assigns (a discrete, approximate version of) a geometry to the tensor network, namely that of the underlying curved spacetime.

Figures (8)

• Figure 1: (a) Manifold $\mathcal{M}$ that can be foliated into euclidean time slices $\Sigma_{\tau}$. (b) The transition between time slices $\Sigma_{\hbox{\tiny in}}$ and $\Sigma_{\hbox{\tiny out}}$, where $\tau_{\hbox{\tiny out}} - \tau_{\hbox{\tiny in}} = \epsilon \ll 1$, is characterized by lapse and shift functions $\alpha(\tau,x)$ and $\beta(\tau,x)$ that are proportional to the profile functions $a(\tau,x)$ and $b(\tau,x)$ in Eqs. \ref{['eq:Q0']} and \ref{['eq:Q1']}, respectively (see appendix. \ref{['sec:AppI']}). (c) Profile functions $a=1$ and $b=0$ for the euclidean plane E$_2$ in the coordinates $\tau,x$ of Eq. \ref{['eq:gE2']}. (d) Profile functions $a=1$ and $b=x/L$ for the hyperbolic plane H$_2$ in the coordinates $\tau,x$ of Eq. \ref{['eq:gH2']}.
• Figure 2: (a) Layer $\mathcal{T}$ of euclideons $e$ that implements a uniform euclidean time evolution $e^{-H}$ and double layer $\mathcal{W}$ of disentanglers $u$ and isometries $w$ that implements a uniform rescaling of space $2^{-iD}$. (b) Tensor network TN$_1 = \mathcal{T}^{n}$ implementing $e^{-nH}$, which by rules 1-2 corresponds to a path integral on flat euclidean spacetime E$_2$. (c) Tensor network TN$_2 = (\mathcal{W}\mathcal{T})^n$ implementing $2^{ -n(H+iD)}$, which by rules 1-2 corresponds to a path integral on the hyperbolic plane H$_2$. (d) An euclideon $e$ represents a (path integral on a) $a_{\hbox{\tiny UV}} \times a_{\hbox{\tiny UV}}$ square patch of euclidean spacetime. By contracting several euclideons together, we obtain a larger patch of flat spacetime. A layer $\mathcal{W}$ of disentanglers and isometries allows us to glue the top of two euclideons with the bottom of one euclideon.
• Figure 3: (a) Sequence of tensor networks, made of euclideons $e$ and smoothers $e_L$ and $e_R$, that approximate the continuous profile $f(x) = \alpha(1-\cos\left(x\right))$, with $\alpha=2\pi/32$. The x-axis labels sites $1$ to $N$ with lattice spacing $2\pi/N$. The corresponding linear maps $V_N$ are expected to act as improving approximations to $V=e^{-Q}$. (b) The difference between the numerical matrix elements $\hbox{$\langle \phi_\alpha |$} V_N \hbox{$| \phi_{\beta} \rangle$}$, for the first 41 low-energy states of the quantum critical spin chain (critical transverse field Ising model), and the corresponding CFT CFT matrix elements of $V$. We plot the mean absolute difference within each conformal tower, as well as between towers (mixing). We observe convergence in $N$ up to an error $\mathcal{O}(\alpha^5)$ made in computing the matrix elements of $V$ to order $\alpha^4$. For these computations we used periodic Matrix Product States MPSMPS_critpMPSpuMPS and lattice Virasoro generators LatVira. See appendix \ref{['sec:app_refinement']}.
• Figure 4: Points $p\in\Sigma_{\tau_0}$ and $q\in\Sigma_{\tau_0+\epsilon}$ expressed both in the $x^{\mu}=(\tau,x)$ coordinates (in black) and in auxiliary conformal coordinates $y^{\mu} = (y^{0},y^{1})$ (in blue).
• Figure 5: The transition vector $\xi^{\mu}$ between slice $\Sigma_{\tau_0}$ and $\Sigma_{\tau_0+\epsilon}$ expressed in coordinates $x^{\mu}=(\tau,x)$, namely $\xi^{\mu}(x) = (1,0)$, and auxiliary conformal coordinates $y^{\mu} = (y^0,y^1)$, namely $\tilde{\xi}^{\mu}(x) = (a(x), b(x))$. The so-called lapse $\alpha(x) \equiv \epsilon a(x)$ and shift $\beta(x) \equiv \epsilon b(x)$ are, respectively, the projections of $\epsilon \xi^{\mu}$ onto the unit vector $n^{\mu}$ normal to the slice $\Sigma_{\tau_0}$ and the projections of $\epsilon \xi^{\mu}$ onto the time slice $\Sigma_{\tau_0}$ itself.