Quantum Yang-Mills theory: An overview of a program

We present an overview of a program to understand the low-energy physics of quantum Yang-Mills theory from a quantum-information perspective. Our setting is that of the Hamiltonian formulation of pure Yang-Mills theory in the temporal gauge on the lattice. Firstly, inspired by recent constructions for $\mathbb{Z}/2\mathbb{Z}$ lattice gauge theory, in particular, Kitaev’s toric code, we describe the gauge-invariant sector of Hilbert space by introducing a primitive quantum gate: the quantum parallel transporter. We then develop a non-Abelian generalization of Laplace interpolation to present an ansatz for the ground state of pure Yang-Mills theory which interpolates between the weak- and strong-coupling renormalization group fixed points. The resulting state acquires the structure of a tensor network, namely, a multiscale entanglement renormalization ansatz, and allows for the efficient computation of local observables and Wilson loops. Various refinements of the tensor network are discussed leading to several generalizations. Finally, the continuum limit of our ansatz as the lattice regulator is removed is then described. This paper is intended as an abstract for an ongoing program: there are still many open problems.

Figure 1

An illustration of the procedure to identify the gauge-invariant sector ${\mathcal{H}}_{\mathcal{G}}$ for a $3\times 3$ grid. First a maximal tree is found (indicated with the orange edges). Then the remaining edges are parallel transported to the root vertex. Finally, the maximal tree is contracted to the origin.

Figure 2

Two possible embeddings (a) and (b) of the graph associated with the Hamiltonian lattice principal chiral field model. For the gauge groups $G=U(1)$ and $G=SU(2)$ this model is equivalent to the ($1+1$)-dimensional quantum rotor model for the rotation groups $O(2)$ and $O(4)$, respectively.

Figure 3

The fidelity per site $f$ of the MPS ground state of the $G\cong U(1)$ and $G\cong SU(2)$ principal chiral field models (79) at inverse coupling $g^{-2}$ with the fine-grained state $V(\lambda )|\mathrm{\Omega }(g_0^{-2})⟩$ for $g_0^{-2}=0.3$. The initial state of the added loops is $|\psi (\lambda )⟩$. As a reference, we also plot the fidelity with $|\mathrm{\Omega }(g_0^{-2})⟩$. MPS ground states have bond dimension and Fourier mode cutoff $D=22$ and $|n|\le 5$ for $G\cong U(1)$, and $D=14$ and $l\le \frac{3}{2}$ for $G\cong SU(2)$. The interpolated states have $D=64$ and $D=30$, respectively. We see that $f(g_0^{-2},g^{-2},\lambda )$ has a clear maximum for $g^{-2}>g_0^{-2}$, showing that the fine-grained state is closer to the ground state at a larger inverse coupling than to the original ground state at $g_0^{-2}$. Furthermore, the fine-grained states are often significantly better approximations than $|\mathrm{\Omega }(g_0^{-2})⟩$ for smaller values of $\lambda$. Note that some of the error in the fine-grained states is due to finite-entanglement effects and the Fourier-mode cutoff, particularly at larger $g^{-2}$ and for $G\cong SU(2)$.