Multi-Field Relativistic Continuous Matrix Product States

Abstract

Relativistic continuous matrix product states (RCMPS) are a powerful variational ansatz for quantum field theories of a single field. However, they inherit a property of their non-relativistic counterpart that makes them divergent for models with multiple fields, unless a regularity condition is satisfied. This has so far restricted the use of RCMPS to toy models with a single self-interacting field. We address this long standing problem by introducing a Riemannian optimization framework, that allows to minimize the energy density over the regular submanifold of multi-field RCMPS, and thus to retain purely variational results. We demonstrate its power on a model of two interacting scalar fields in $1+1$ dimensions. The method captures distinct symmetry-breaking phases, and the signature of a Berezinskii-Kosterlitz-Thouless (BKT) transition along an $O(2)$-symmetric parameter line. This makes RCMPS usable for a far larger class of problems than before.

Figures (4)

• Figure 1: Representation of the RCMPS manifold $\mathcal{M}$, parameterized by $R_1$, $R_2$, and $Q$ matrices, as a subspace in the full Hilbert space $\mathscr{H}$ of the field theory. The restricted manifold of commuting $R_1$ and $R_2$ matrices is denoted by $\mathcal{M}_{\text{reg}}$.
• Figure 2: Ground state energy density $e_0$ for $g=\lambda$. RCMPS results (markers) show excellent agreement with perturbation theory at order $g^2$ and $g^3$ at weak coupling. Already for moderate coupling $g\geq 0.3$, the RCMPS results are far more precise, even at the lowest $D$ we consider.
• Figure 3: Order parameters $\langle\hat{\phi}_1\rangle$ and $\langle\hat{\phi}_2\rangle$ across the phase transition for (a) $g > \lambda$ and (b) $\lambda > g$, for $D=18$. The method correctly identifies the distinct symmetry-breaking patterns.
• Figure 4: RCMPS entanglement entropy $S$ for $g= \lambda$ as a function of $g$ for different bond dimensions $D$.