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.
Highlights
The richness of quantum phases and phase transitions never ceases to surprise us
We focus on critical quantum states that are effectively described by some conformal field theories (CFTs) at low energies
(1) We propose a class of exotic ð2 þ 1ÞD quantum many-body states dubbed Stiefel liquids, each indexed by two integers, N ≥ 5 and k ≠ 0
Summary
The richness of quantum phases and phase transitions never ceases to surprise us. Over the years many interesting many-body states have been discovered or proposed in various systems, such as different symmetry-breaking orders, topological orders, and even exotic quantum criticality. If we can extend the NLSM construction to the Uð1Þ DSL, we may “extrapolate” the two theories to obtain an entire series of theories, some of which could possibly go beyond any mean field plus weak fluctuations description With these motivations, we study a special type of ð2 þ 1ÞD quantum many-body states, each labeled by two integers ðN; kÞ, with N ≥ 5 and k ≠ 0. (to obtain nonuniversal details of a manybody system, its Hamiltonian, Lagrangian or wave function is needed.) such an intrinsic characterization has been (partly) achieved in various systems, such as CFTs in ð1 þ 1ÞD [40], a large class of gapped phases in various dimensions [41,42,43,44,45,46,47,48,49,50,51,52,53], and symmetry-enriched Uð1Þ quantum spin liquids in ð3 þ 1ÞD [54,55,56,57,58,59]. The present work can be viewed as a small step toward this ambitious goal for more complicated critical states of matter
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