An efficient finite-resource formulation of non-Abelian lattice gauge theories beyond one dimension

Abstract

Non-Abelian gauge theories provide the most accurate description of fundamental interactions, showing remarkable agreement with experimental data in cosmology and particle physics. Highly precise predictions can be made using standard techniques, both in the continuum and in the lattice frameworks. However, classical methods have limitations, particularly when attempting to extrapolate the continuum limit from the study of lattice gauge theories. Complementing classical computations or combining them with quantum computational methods, to improve the predictions towards the continuum limit with current quantum resources, is a formidable open challenge. In this paper, we propose a resource-efficient method to compute the running of the coupling in non-Abelian gauge theories beyond one spatial dimension. We first represent the Hamiltonian on periodic lattices in terms of loop variables and conjugate loop electric fields, exploiting the Gauss law to retain the gauge-independent ones. Then, we identify a local basis for small and large loops variationally to minimize the truncation error while computing the running of the coupling on small tori. Our method enables computations at arbitrary values of the bare coupling and lattice spacing with current quantum computers, simulators and tensor-network calculations, in regimes otherwise inaccessible.

Figures (10)

• Figure 1: (a) Arbitrary periodic lattice represented as the surface of a torus in the three-dimensional space. The fundamental degrees of freedom after the change of variables to the loop (or dual) degrees of freedom are the plaquettes (red square) and the two large loops wrapping the torus (blue circles). (b) Unfolded minimal torus, i.e., the single periodic plaquette, identified by four sites ($\bullet$) and eight links ($-$). In the dual formulation, the physical states of the pure gauge theory are represented by three independent plaquettes (red loops) and two large loops (blue loops). (c) Cartoon picture of the basis for the physical states of the lattice gauge theory on the minimal torus. Every fundamental degree of freedom is associated to a free parameter $g_I$, $I=1,\ldots,5$, that we determine variationally (d) by minimizing the expectation value of the dual Hamiltonian in the local parameters space. (e) Expectation values of the plaquette operator $\langle\square\rangle$ can be computed for all values of the coupling constant $\beta=(2g^2)^{-1}$ with a small number of states. Fast convergence is also appreciated in the energy of the ground state (f).
• Figure 2: (a) Basic canonical transformation to join two consecutive links, $U(\bm{A})$ and $U(\bm{B})$, connecting $\bm{A}\rightarrow\bm{B}$ and $\bm{B}\rightarrow\bm{C}$, respectively. From $U(\bm{A})$ and $U(\bm{B})$ with conjugate electric fields reported in the left-hand side of the graphical equality, two new links $\mathcal{U}(\bm{B})$ and $\mathcal{U}(\bm{AC})$ are introduced, which are still mutually independent. Here and in the following, we use green (red) dots to represent left (right) electric fields. (b) Graphical representation of all the canonical transformations performed in the main text. The left panel shows the initial Kogut--Susskind links. Blue letters represent physical sites, while orange letters are repeated to explain the lattice periodicity. The central panel shows the results of the first set of canonical transformations, i.e., five links and the three independent plaquettes, and for the sake of simplicity we report only the nomenclature for the plaquettes. The right panel is the final outcome after the second set of canonical transformations, i.e., five closed loops (three plaquettes, two large loops) and three open strings ending in the reference point $\bm{D}$. (c) Extension of canonical transformation to a larger torus. The left panel shows the initial links, while in the right panel we report the independent plaquettes (red loops) and the large loops $\mathcal{L}_{x,y}$ (blue loops).
• Figure 3: (a-b) Plot of the distance between local couplings $(\beta_1,\beta_2,\beta_3)=(1/2g_1^2,3/4g_2^2,1/2g_3^2)$, determined through variational optimization, and the bare coupling $\beta=(2g^2)^{-1}$, as a function of the bare coupling $\beta$ for the different values of truncations $L_{max}=1,4$ (left and right subplots, respectively). Instabilities in the data points are associated to the numerical minimization. (c) Energies of the variationally determined ground states as a function of the bare coupling $\beta$ for the different values of truncations $L_{max}=1,4,5$. (d) Relative energy difference $\Delta E_0/E_0=(E_0(\bm{g_0})-E_0(\bm{g_V}))/E_0(\bm{g_V})$ between the initial ansatz $\bm{g}_0$ and the optimal couplings as a function of the bare coupling $\beta$ for the different values of truncations $L_{max}=1,4,5$. The data shows the absence of energy reduction for strong couplings, which explains the numerical instabilities of panels (a-b) for $\beta\lesssim 1$. (e) Energies of both variationally determined ground states and ground states in the electric representation as a function of the bare coupling $\beta$ for the different values of truncations $L_{max}=1,4,5$. We observe that the electric representation is efficient in the strong coupling regime, while it is increasingly worse towards weak coupling. In contrast, our approach efficiently interpolates between strong and weak coupling regimes. (f) Relative energy differences $\delta E/E$ between consecutive values of the truncation $L_{max}$ as a function of the bare coupling $\beta$. (g) Infidelities $1-\mathcal{F}_{\text{GS}}$ between consecutive values of the truncation $L_{max}$ as a function of the bare coupling $\beta$.
• Figure 4: (a) Expectation values of the plaquette operator $\langle\square\rangle$ on the variationally determined ground states as a function of the bare coupling $\beta$, for the different values of truncations $L_{max}=1,4,5$. (b) Relative difference $\frac{\Delta \langle\square\rangle}{\langle\square\rangle}\equiv\frac{\langle\square\rangle(\bm{g_0})-\langle\square\rangle(\bm{g_V})}{\langle\square\rangle(\bm{g_V})}$ in the expectation values of the plaquette operator between the initial ansatz $\bm{g}_0$ and the optimal couplings as a function of the bare coupling $\beta$, for the different values of truncations $L_{max}=1,4,5$. As observed for the energy (see Fig. \ref{['joint_energy_beta_etc']}), the variational approach is more advantageous for smaller $L_{max}$. (c) Expectation values of the plaquette operator $\langle\square\rangle$ on both the variationally determined ground states and ground states in the electric representation as a function of the bare coupling $\beta$, for the different values of truncations $L_{max}=1,4,5$. (d) Relative differences in the expectation value of the plaquette operator $\delta \langle\square\rangle/\langle\square\rangle\equiv(\langle\square\rangle_{L_{max}'}-\langle\square\rangle_{L_{max}})/\langle \square\rangle_{L_{max}'}$ between consecutive values of the truncation $L_{max}$ as a function of the bare coupling $\beta$.
• Figure 5: Graphical representation of canonical transformations to obtain five independent plaquettes $W_{\bm{n}}$ (left plot) and two large loops $\mathcal{L}_{\hat{x},\hat{y}}$ (right plot). In the final lattices, we report only the reference site $\bm{F}$.