Matrix Product States for Gauge Field Theories

The matrix product state formalism is used to simulate Hamiltonian lattice gauge theories. To this end, we define matrix product state manifolds which are manifestly gauge invariant. As an application, we study ($1+1$)-dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model, and are able to determine very accurately the ground-state properties and elementary one-particle excitations in the continuum limit. In particular, a novel particle excitation in the form of a heavy vector boson is uncovered, compatible with the strong coupling expansion in the continuum. We also study full quantum nonequilibrium dynamics by simulating the real-time evolution of the system induced by a quench in the form of a uniform background electric field.

Figure 1

Results for $m/g=0.25$, $x=100$. (a) Distribution of the (base-10) logarithm of the Schmidt coefficients $\lambda$ in every charge sector for $D=(5,20,48,70,62,34,10)$. (b) Difference for the estimated energies of the excited states for various bond dimension with respect to these with $D=(5,20,48,70,62,34,10)$ for the vector sector $\gamma =-1$. Only the first two excitations are stable under variation over $D$.

Figure 2

Results for $m/g=0.25$, $x=100$, all quantities are in units $g=1$. (a) Difference of $E(t)\equiv ⟨{\mathrm{\Psi }}_0(t)|L(1)+L(2)|{\mathrm{\Psi }}_0(t)⟩/2$ for various tolerances $\epsilon$ with respect to the estimated value for $\epsilon ={\epsilon }_0=2\times 10^{-6}$ ($\alpha =0.3$). (b) $E(t)/\alpha$ for different values of $\alpha$. (c) $E^{sq}(t)/{\alpha }^2$, with $E^{sq}(t)\equiv ⟨{\mathrm{\Psi }}_0(t)|L^2(1)+L^2(2)|{\mathrm{\Psi }}_0(t)⟩/2$, for the same set of $\alpha$ values as in (b). (d): For $\alpha =0.3$, the different energy densities of, respectively, the first ($H_g$), second ($H_f$), and third term ($H_i$) of Eq. (2) but with $L(n)\to L(n)+\alpha$ (we subtracted the values at $t=0$ without background field). The straight blue line is the total energy density obtained as the sum of the three terms.