Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders

TL;DR

This paper introduces Stiefel liquids, a new class of 2+1D quantum critical liquids with unique properties, unifies known critical points, and conjectures the existence of non-Lagrangian states beyond traditional theoretical descriptions.

Contribution

It proposes Stiefel liquids based on nonlinear sigma models, unifies known quantum critical points, and conjectures the existence of non-Lagrangian quantum liquids with potential realizations in frustrated spin systems.

Findings

Stiefel liquids exhibit large emergent symmetries and nontrivial anomalies.

Deconfined quantum critical point and U(1) Dirac spin liquid are special cases of Stiefel liquids.

Some non-Lagrangian Stiefel liquids may be realized in frustrated quantum spin systems.

Abstract

We propose a new type of quantum liquids, dubbed Stiefel liquids, based on $2+1$ dimensional nonlinear sigma models on target space $SO(N)/SO(4)$, supplemented with Wess-Zumino-Witten terms. We argue that the Stiefel liquids form a class of critical quantum liquids with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies. We show that the well known deconfined quantum critical point and $U(1)$ Dirac spin liquid are unified as two special examples of Stiefel liquids, with $N=5$ and $N=6$, respectively. Furthermore, we conjecture that Stiefel liquids with $N>6$ are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian. Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field theory familiar in the study of exotic quantum liquids in condensed matter physics. The intrinsic absence of (conventional or parton-like) mean-field construction also means that, within the traditional approaches, it will be difficult to decide whether a non-Lagrangian state can actually emerge from a specific UV system (such as a lattice spin system). For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems. Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or Kagome lattice, through the intertwinement between non-coplanar magnetic orders and valence-bond-solid orders.