Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching

TL;DR

The paper reframes nonperturbative QFT analysis through symmetry-twisted boundary conditions and a Hilbert-space lens, introducing quantum distillation via a graded partition function ZΩ(L) that suppresses excited-state contributions while preserving ground-state information. It demonstrates the mechanism first in quantum mechanics (N-dimensional harmonic oscillator and CP^{N-1} QM) and then in 2d Grassmannian Gr(N,M) sigma models, where mixed ’t Hooft anomalies constrain vacuum structure and persist under adiabatic S^1 compactification with a unique twist. A close connection is drawn between anomaly matching, Hilbert-space distillation, and semi-classical path-integral analysis, including fractional instantons and kink dynamics under small-L and resurgent frameworks. The work further links these ideas to large-N volume independence and flavor-momentum transmutation, suggesting analytic, ground-state–driven control over nonperturbative dynamics across a family of asymptotically free theories.

Abstract

A symmetry-twisted boundary condition of the path integral provides a suitable framework for the semi-classical analysis of nonperturbative quantum field theories (QFTs), and we reinterpret it from the viewpoint of the Hilbert space. An appropriate twist with the unbroken symmetry can potentially produce huge cancellations among excited states in the state-sum, without affecting the ground states; we call this effect "quantum distillation". Quantum distillation can provide the underlying mechanism for adiabatic continuity, by preventing a phase transition under $S^1$ compactification. We revisit this point via the 't Hooft anomaly matching condition when it constrains the vacuum structure of the theory on $\mathbb{R}^d$ and upon compactification. We show that there is a precise relation between the persistence of the anomaly upon compactification, the Hilbert space quantum distillation, and the semi-classical analysis of the corresponding symmetry-twisted path integrals. We motivate quantum distillation in quantum mechanical examples, and then study its non-trivial action in QFT, with the example of the 2D Grassmannian sigma model $\mathrm{Gr}(N,M)$. We also discuss the connection of quantum distillation with large-$N$ volume independence and flavor-momentum transmutation.

Figures (2)

• Figure 1: Schematic illustration of the advantages of the symmetry-twisted partition function $\mathcal{Z}_{\Omega}(L)$ in comparison with the thermal partition function $\mathcal{Z}(\beta)$. In the left column, we count the degrees of freedom of mesons. Mesons give a large, $O(N^2)$ contribution to $\mathcal{Z}(\beta)$ since they are in the adjoint representation of the $SU(N)/\mathbb{Z}_N$ symmetry, while $\mathcal{Z}_{\Omega}(L)$ is affected only by $O(1)$. In the middle column, it is explained by the Kaluza-Klein (KK) or Matsubara decomposition of the fields, and the KK modes in $\mathcal{Z}_{\Omega}(L)$ are much denser than those of $\mathcal{Z}(\beta)$, due to the flavor-momentum transmutation. In the right column, we explain its consequence for the 't Hooft anomaly of the theory, and the 't Hooft anomaly in $2$ dimensions persists in $\mathcal{Z}_{\Omega}(L)$ for any $L$, while it survives in $\mathcal{Z}(\beta)$ only for $\beta\to\infty$.
• Figure 2: Schematic image for the graded partition function (\ref{['eq:zho']}) for an $N$-dimensional isotropic simple harmonic oscillator, with $N=4$, showing the representations with $\lambda=0, 1, 2$, corresponding to energy $E_{\lambda}=\frac{N}{2}+\lambda$. Since the phases $q^n=\exp(2\pi i n/4)$ are attached to the states, most of them cancel with one another, and we obtain the information of the ground states. We will see that a similar structure is also present in certain QFTs.