Applying the Variational Principle to ($1+1$)-Dimensional Quantum Field Theories

We extend the recently introduced continuous matrix product state variational class to the setting of ($1+1$)-dimensional relativistic quantum field theories. This allows one to overcome the difficulties highlighted by Feynman concerning the application of the variational procedure to relativistic theories, and provides a new way to regularize quantum field theories. A fermionic version of the continuous matrix product state is introduced which is manifestly free of fermion doubling and sign problems. We illustrate the power of the formalism by studying the momentum occupation for free massive Dirac fermions and the chiral symmetry breaking in the Gross-Neveu model.

Figure 1

Hypothetical momentum distribution of an optimal cMPS for a free fermionic theory: high-frequency degrees of freedom are well approximated up to a cutoff $\Lambda$, after which the momentum occupation decays as $k^{-4}$. Also shown is the effect of a scale transformation.

Figure 2

Momentum occupation of the antiparticle levels $n^{--}(k)$, the particle levels $n^{++}(k)$ and the mixing $|n^{+-}(k)|$ in a cMPS approximation of the Dirac field with mass $m/\Lambda =1/10$. The auxiliary space of the cMPS is ${\mathbb{C}}^2\otimes {\mathbb{C}}^2\otimes {\mathbb{C}}^D$. The vertical line indicates the position of the exact cutoff.

Figure 3

Expectation value of $\sigma =⟨\chi |{\widehat{\psi }}^{†}{\sigma }^z\widehat{\psi }|\chi ⟩$ in the Gross-Neveu model as function of $\lambda (\Lambda )$ for $N=\infty$. The solid line is the exact result $|\lambda \sigma |/\Lambda \approx 2e^{-\pi /\lambda }$ and follows from Eq. (3). The dotted line shows a fit of the form $c_1{e}^{-c_2/\lambda }$ to the numerical results for ${\lambda }^{-1}\le 1$ at $D=16$ and results in $c_2={3.142}_{-0.047}^{+0.047}$ and $c_1={2.057}_{-0.072}^{+0.074}$.