Field digitization scaling in a $\mathbb{Z}_N \subset U(1)$ symmetric model

Abstract

The simulation of quantum field theories, both classical and quantum, requires regularization of infinitely many degrees of freedom. However, in the context of field digitization (FD) -- a truncation of the local fields to $N$ discrete values -- a comprehensive framework to obtain continuum results is currently missing. Here, we propose to analyze FD by interpreting the parameter $N$ as a coupling in the renormalization group (RG) sense. As a first example, we investigate the two-dimensional classical $N$-state clock model as a $\mathbb{Z}_N$ FD of the $U(1)$-symmetric $XY$-model. Using effective field theory, we employ the RG to derive generalized scaling hypotheses involving the FD parameter $N$, which allows us to relate data obtained for different $N$-regularized models in a procedure that we term $\textit{field digitization scaling}$ (FDS). Using numerical tensor-network calculations at finite bond dimension $χ$, we further uncover an unconventional universal crossover around a low-temperature phase transition induced by finite $N$, demonstrating that FDS can be extended to describe the interplay of $χ$ and $N$. Finally, we analytically prove that our calculations for the 2D classical-statistical $\mathbb{Z}_N$ clock model are directly related to the quantum physics in the ground state of a (2+1)D $\mathbb{Z}_N$ lattice gauge theory which serves as a FD of compact quantum electrodynamics. Our study thus paves the way for applications of FDS to quantum simulations of more complex models in higher spatial dimensions, where it could serve as a tool to analyze the continuum limit of digitized quantum field theories.

Figures (7)

• Figure 1: Field digitization scaling in the $N$-state clock model. (a) We investigate the $N$-state clock model with nearest-neighbor-interacting, discrete angles $\vartheta=2\pi n/N$, ($n=0,\dots, N-1$) on a 2D square lattice. In the limit $N\rightarrow \infty$, these $\mathbb{Z}_N$-symmetric models give rise to the $U(1)$-invariant $XY$-model, which is gapless for $T < T_{\rm H}$. (b) Field digitization to $N$ discrete angles is relevant only at low temperatures, leading to an ordered, gapped phase at $T\leq T_{\rm L}(N)$. (c) In this regime we uncover a self-similar scaling behavior among different $N$-regularized models. (Left) Magnetization $M$ for different $N$ versus temperature $T$. Dashed colored vertical lines represent $T_{\rm L}(N)$. (Right) Based on the behavior of the correlation length $\xi(T, N)$ [see Eq. \ref{['eq:corr_infty_num']} and Fig. \ref{['fig:Fig2']}], we formulate an $N$-dependent scaling Ansatz [Eq. \ref{['eq:Ansatz_N']}] for local observables. Upon rescaling both axes accordingly, we observe a collapse of $M$ for different $N$ and $T$. Here, $a=1.5, b=1, \Delta_M(N)=2/N^2$.
• Figure 2: Relevant field digitization and universal behavior. In the gapped, ordered phase we uncover a universal scaling form of the correlation length. Shown is the extrapolated correlation length $\xi_\infty=\xi(\chi\rightarrow \infty)$ as a function of the reduced temperature $t=(T-T_{\rm L})/T_{\rm L}$ for field dimensions $N=6,7,8,9$. Data points collapse on a single curve [dashed black line, see Eq. \ref{['eq:corr_infty_num']}] upon rescaling with $N$. (Inset) Unrescaled correlation length $\xi_\infty$ versus temperature $T$. Colored vertical dashed lines represent $T_\mathrm{L}(N)$.
• Figure 3: Emergent $U(1)$ symmetry at finite $N$. In the gapless, critical phase the field digitization $N$ is an irrelevant perturbation: (Inset) the trivial collapse in $N$ reveals the emergent $U(1)$ symmetry. Instead, the finite $\chi$ truncates the otherwise diverging correlation length $\xi$, shown versus $T$ for different $N$ and increasing bond dimension $\chi$ (increasing intensity). (Main) Upon rescaling the vertical axis as $\xi/\chi^\kappa$, we observe a collapse for different $\chi$ and $N$ as a function of temperature $T$.
• Figure 4: Crossover regime. Close to the CP $T=T_{\rm L}$ both truncations in $N$ and $\chi$ are relevant --- rescaling according to Eq. \ref{['eq:scalingAnsatz']} leads to a scaling collapse. (Main) Rescaled magnetization $M$ is shown as a function of the parameter $x=t/N \cdot \log^2{(\chi^\kappa/\xi_0(N))}$ for several $N$ (different colors and shapes) and bond dimension $\chi$ (increasing intensity, from $\chi=60$ up to a maximum of $\chi=192$ for $N=6,7$). (Inset) Unrescaled magnetization $M$ versus temperature $T$. Vertical dashed lines represent $T_\mathrm{L}(N)$. The black shaded arrow indicates increasing $\chi$.
• Figure 5: Connecting the classical $N$-state clock model to a (2+1)D $\mathbb{Z}_N$ LGT. The tensor-network encoding of the 2D classical partition function $Z_N$ [(a)] captures the quantum ground state of a $\mathbb{Z}_N$ LGT with Hamiltonian given in Eq. \ref{['eq:LGT_Ham']} on the dual lattice [(b)]. (a) On each site $p$ of a 2D lattice we define the microscopic, discrete angle $\vartheta_p$ [see Eq. \ref{['eq:clock_Ham']}]. The partition function $Z_N$ [Eq. \ref{['eq:partition_clock']}] on this lattice can be encoded in a 2D tensor network, as depicted in the background. The dashed lines represent the dual lattice, each site of which is mapped to a plaquette of the original lattice (and vice versa). (b) On the dual lattice we define plaquette [$U_p=X_{p,1}X_{p,2}X_{p,3}^\dagger X_{p,4}^\dagger$] and Gauss' law operators [$G_s=Z_{s,1}Z_{s,2}Z_{s,3}^\dagger Z_{s,4}^\dagger$], where the labels $1,..,4$ refer to links near plaquettes $p$ and vertices $s$ as indicated. Similarly, we define the operators $D_{p}^{(\prime)}=f(Z_{p,l})$, with $l=1,.., 4$ (see EM \ref{['sec:app_LGT']}). The operator $Z_l$ on the link $l=\langle p, p'\rangle$ between two plaquettes $p, p'$ is defined by $\vartheta_{p}$ and $\vartheta_{p'}$ as $Z_l\equiv \exp\{i(\vartheta_{p}-\vartheta_{p'})\}$.