Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems

We present an operational procedure to transform global symmetries into local symmetries at the level of individual quantum states, as opposed to typical gauging prescriptions for Hamiltonians or Lagrangians. We then construct a compatible gauging map for operators, which preserves locality and reproduces the minimal coupling scheme for simple operators. By combining this construction with the formalism of projected entangled-pair states (PEPS), we can show that an injective PEPS for the matter fields is gauged into a $\mathsf{G}$-injective PEPS for the combined gauge-matter system, which potentially has topological order. We derive the corresponding parent Hamiltonian, which is a frustration-free gauge-theory Hamiltonian closely related to the Kogut-Susskind Hamiltonian at zero coupling constant. We can then introduce gauge dynamics at finite values of the coupling constant by applying a local filtering operation. This scheme results in a low-parameter family of gauge-invariant states of which we can accurately probe the phase diagram, as we illustrate by studying a ${\mathbb{Z}}_2$ gauge theory with Higgs matter.

Popular Summary

Gauging the global symmetry of a physical system involves introducing new degrees of freedom, referred to as gauge fields, to make the system invariant under localized actions of the symmetry transformation. As a classic example of this process, the fundamental forces in nature can be understood as resulting from gauging the global charge symmetries of the elementary particles that build up the Universe; the resulting gauge fields then act as force carriers between those particles. While there is no unique or unambiguous prescription for gauging a system, the process is typically accomplished by applying the so-called minimal coupling prescription to a Hamiltonian or Lagrangian description of the system. We present a general recipe for gauging a global symmetry at the level of individual quantum states.

We are able to formulate an associated gauging map at the level of Hamiltonians that exactly reproduces the minimal coupling prescription for simple operators by transforming global symmetries into local symmetries. Using a lattice with quantum degrees of freedom at its vertices and a formalism of projected entangled-pair states, we discuss and illustrate how this strategy allows us to build few-parameter families of explicitly gauge-invariant quantum states that can be used to study gauge theories from a new perspective. Our recipe does not reference the underlying Hamiltonian. We then use the formalism of tensor network states, which are currently revolutionizing the way we understand the entanglement structure of quantum many-body states, to reestablish the close relationship between discrete gauge theories and topological order. We provide an example by studying to high accuracy the phase diagram of a simple model gauge theory with ${\mathbb{Z}}_2$ gauge fields and Higgs matter.

We expect that our results will drive studies of gauge theories in three dimensions and gauge theories of fermionic matter.

Figure 1

(a) Definition of the graph $\mathrm{\Lambda }$ with vertices $v$ and oriented edges $e$. (b) Construction of the PEPS $|\psi (A)⟩$ from tensors $A_v$ associated with the vertices $v$ and with virtual bonds along the edges $e$ and physical indices depicted as arrows pointing out of the center of every tensor. (c) Symmetry of a PEPS tensor to ensure global symmetry of the state $|\psi (A)⟩$ under the group action $U_g$. (d) Definition of the tensor $X_v$ used in the construction of the projector $P$ onto the gauge-invariant subspace. (e) Result of acting with the tensor $X_e$ on the physical input state $|1⟩$, which is the only case we need throughout this manuscript. (f) PEPS $|\mathrm{\Psi }(B)⟩$ with vertex tensors $B_v$ and edge tensors $B_e$ obtained from acting with $P$ on $|\psi (A)⟩{\otimes }_e|1{⟩}_e$. (g) Symmetry property of the tensor $X_v$.

Figure 2

Trace of the metric $g(\mathit{\beta })$ with $\mathit{\beta }=({\beta }_z,{\beta }_x)$, as obtained from the fidelity $|⟨{\mathrm{\Psi }}_{\mathit{\beta }}|{\mathrm{\Psi }}_{\mathit{\beta }+\mathit{\delta \beta }}⟩|=\exp (-N{\mathit{\delta \beta }}^{T}g(\mathit{\beta })\mathit{\delta \beta })$, where $N$ is the (infinite) number of sites, as defined in Ref. [80]. An analytic expression for $g$ along the coordinate axes (${\beta }_z=0$ or ${\beta }_x=0$) was obtained in Ref. [81]. The red lines in the right panel indicate the slices studied in Figs. 3 and 4.

Figure 3

Fidelity (per site) $|⟨{\mathrm{\Psi }}_{{\mathit{\beta }}^{(1)}}|{\mathrm{\Psi }}_{{\mathit{\beta }}^{(2)}}⟩{|}^{1/N}$ between any two ground states with parameters $\mathit{\beta }$ along the three different slices indicated in Fig. 2.

Figure 4

Expectation values of the relevant Hamiltonian terms, i.e., ${\tau }^x$ and ${\tau }^z$ on the edges, ${\tau }^x{\tau }^x{\tau }^x{\tau }^x$ around a vertex, and ${\tau }^z{\tau }^z{\tau }^z{\tau }^z$ around a plaquette, along the three different slices indicated in Fig. 2.