Stiefel Liquid: Non-Lagrangian Quantum Criticality

• Stiefel liquids are critical quantum states in 2+1 dimensions defined by non-linear sigma models on SO(N)/SO(4) with quantized WZW terms.
• They exhibit a cascade of intertwined orders governed by strict quantum anomaly matching that constrains lattice realizability.
• Their geometric interpretation on Stiefel manifolds inspires novel optimization flows and insights into quantum channel parametrizations.

A Stiefel Liquid is a class of critical quantum states in $2+1$ dimensions, characterized by non-linear sigma models with target space $SO(N)/SO(4)$ and supplemented by quantized Wess-Zumino-Witten (WZW) terms. These states exhibit emergent large symmetry groups, a cascade structure of intertwined orders, and nontrivial quantum anomalies. The paradigm notably includes certain well-known critical points such as the deconfined quantum critical point and the $U(1)$ Dirac spin liquid as special cases ($N=5$ and $N=6$ respectively); for $N>6$, Stiefel liquids are conjectured to be non-Lagrangian, i.e., they flow to infrared fixed points not described by any renormalizable continuum Lagrangian and hence go beyond conventional parton gauge mean-field theory frameworks. The realization of Stiefel liquids in microscopic spin systems is controlled by symmetry embedding and quantum anomaly matching conditions, providing precise constraints on their emergibility in lattice models.

1. Field-Theoretical Definition and Structure

Stiefel liquids are constructed from $2+1$ dimensional non-linear sigma models on a coset target space $SO(N)/SO(4)$, equipped with a topological WZW term. The action is

$$S[n] = \frac{1}{2g} \int d^{3}x\, \operatorname{Tr}(\partial_{\mu} n^T \partial^{\mu} n) + k\;\text{WZW},$$

where $n$ is an $N \times (N-4)$ matrix field subject to orthogonality constraints, and $k$ quantizes the WZW coupling. This construction leads to ordered parameter fields transforming under product symmetry groups $SO(N)\times SO(N-4)$. For $N>6$, the infrared (IR) theories resulting from these models lack known renormalizable Lagrangian descriptions—hence "non-Lagrangian quantum criticality." The ordered parameter field $n$ exhibits a cascade structure in its operator product expansion (OPE), with components in various irreducible representations:

$$n \times n = (S,S)^+ \oplus (S,T)^+ \oplus (T,S)^+ \oplus (T,T)^+ \oplus (A,A)^+ \oplus (A,S)^- \oplus (S,A)^- \oplus (T,A)^- \oplus (A,T)^-,$$

where $S, A, T$ denote singlet, antisymmetric, and symmetric traceless representations respectively, and the superscript indicates spin parity. The complex structure and cascade of intertwined orders reflect the underlying manifold geometry and topological term.

2. Emergent Symmetry and Quantum Anomaly Constraints

The emergent IR symmetry group in Stiefel liquids generically grows large, often of the product form $SO(N)\times SO(N-4)$, sometimes further enriched by discrete symmetries. The key physical question is the correspondence between microscopic (UV) lattice symmetries and IR symmetry realization, formalized by a group homomorphism $\varphi: G_{\text{UV}} \to G_{\text{IR}}$. Physical legitimacy of such symmetry embeddings is dictated by quantum anomaly matching: the IR theory must have topological response functions $w[G_{\text{IR}}]$ whose pullback via $\varphi$ precisely matches the lattice anomaly $w[G_{\text{UV}}]$ imposed by generalized Lieb-Schultz-Mattis (LSM) constraints:

$w[G_{\text{UV}}] = \varphi^{*}(w[G_{\text{IR}}]).$

This requirement is stringent as quantum anomalies are invariant under renormalization group flow, thereby providing a robust criterion for "emergibility"—a low-energy theory is realizable as a lattice Hamiltonian only if its anomalies match exactly those mandated by the UV symmetries and constraints. The matching encompasses both on-site symmetries (e.g., spin rotation $SO(3)$, time reversal) and lattice point group symmetries, with possible non-uniqueness in embedding choices. Only homomorphisms satisfying anomaly matching and IR dynamic stability are permitted. Perturbations allowed by UV symmetry must remain irrelevant at the IR fixed point to ensure stability.

3. Non-Lagrangian Quantum Criticality and Intertwined Orders

For $N>6$, Stiefel liquids are conjectured to realize non-Lagrangian quantum phase transitions where the IR fixed point cannot be described by any known local, renormalizable Lagrangian. These states escape classification by parton mean-field theory (e.g., gauge fractionalization frameworks), presenting an intrinsic obstacle to conventional mean-field constructions. The absence of such a construction means deciding physical realizability from UV (lattice) models is only possible through quantum anomaly matching and dynamic stability. The intertwined order arises from non-coplanar magnetism and valence bond solid (VBS) orderings in frustrated quantum spin systems such as the triangular and kagome lattices, providing a physical basis for the criticality associated with Stiefel liquids.

4. Conformal Bootstrap Analysis of Scaling Dimensions and Stability

Direct analysis of non-Lagrangian fixed points is intractable by perturbative or Lagrangian approaches. Instead, the conformal bootstrap provides rigorous bounds by leveraging crossing symmetry and unitarity in operator correlation functions—without reference to a microscopic construction. For the $N=7$ Stiefel liquid, four-point functions of the order parameter field $n$ were bootstrapped under lattice symmetry constraints. The stability of the critical Stiefel phase on different lattices translates to lower bounds on the scaling dimension $\Delta_n$ of the fundamental order parameter:

• Triangular lattice: Requires $\Delta_{(T,S)} > 3 \implies \Delta_n > 1.003$.
• Kagome lattice: Requires $\Delta_{(T,S)} > 3$ and $\Delta_{(T,T)} > 3 \implies \Delta_n > 1.123$.

These bounds ensure that dangerous, potentially relevant singlet operators remain irrelevant, securing phase stability under lattice symmetries. The bootstrap results are consistent with those for related theories (e.g., $U(1)$ Dirac spin liquid, which exhibits similar lower bounds for monopole operators). There is no evidence, in the bootstrap kinks, for physical realization of Stiefel liquids via enhanced symmetry sectors as in standard $O(N)$ kinks; the actual operator spectrum is controlled by the stability constraints.

Lattice	Stability Condition	Lower Bound for $\Delta_n$
Triangular	$\Delta_{(T,S)} > 3$	$>1.003$
Kagome	$\Delta_{(T,S)}, \Delta_{(T,T)} > 3$	$>1.123$

5. Optimization Flows and Geometric Liquid Behavior

In the domain of optimization and geometry, "Stiefel liquid" is evocative of continuous flows on the Stiefel manifold—the set of orthonormal matrices. A dynamical system

$$\frac{dX}{dt} = -[\psi(X)X + \lambda \nabla \mathcal{N}(X)],$$

with $\psi(X) = 2\:\text{skew}(\nabla f(X) X^T)$ and $\mathcal{N}(X) = \frac{1}{4} \|X^T X - I_p\|$ (Euclidean norm), describes a flow that asymptotically lands on the Stiefel manifold. The vector field decomposes into a Riemannian gradient component (tangential to the manifold) and a normal correction driving the iterates onto the manifold. The flow is globally convergent—any full-rank starting point asymptotically approaches the manifold and converges to a critical point. Isolated local minima are asymptotically stable. This geometric "liquid" viewpoint allows unconstrained evolution ("liquefaction") before successful "freezing" onto the constraint set. Such flows inspire efficient algorithms for manifold-constrained optimization, avoiding expensive retraction steps common in iterative manifold methods.

6. Quantum Channels, Stiefel Manifolds, and Liquid Analogies

Quantum channels, represented as completely positive trace-preserving (CPTP) maps, can be parameterized through block matrices of Kraus operators satisfying $K^\dagger K = I$, thus residing on a complex Stiefel manifold $V_k(\mathbb{C}^n)$. The set of channels is homeomorphic to the quotient $V_n(\mathbb{C}^{nm})/U(nm)$, as established rigorously. The Riemannian metric on the Stiefel manifold induces a metric on quantum channels:

$$D(\Phi_1, \Phi_2) = \inf \{d(S_1, S_2): S_j \in h^{-1}(\Phi_j) \}$$

where $h$ is the Kraus to channel map. This geometric structure supports optimization over quantum channels, including convex functionals for state preparation, gate synthesis, and thermodynamic objectives. Convexity ensures absence of "kinematic traps" (suboptimal local minima), facilitating efficient quantum control. While "Stiefel liquid" is not explicit in this context, the geometric landscape of quantum channels (as equivalence classes on Stiefel manifolds) supports fluid-like interpretations—potentially relevant for statistical approaches or modeling collective channel behavior.

7. Implications, Realizability, and Further Directions

Stiefel liquids represent a significant extension of quantum criticality theory, encompassing non-Lagrangian fixed points determinable only by anomaly matching and stability analyses, with deep implications for frustrated spin systems and intertwined quantum orders. Enhanced emergent symmetry and robust anomaly matching are necessary (but not sufficient) for physical realizability in lattices. Bootstrap and manifold flow techniques offer rigorous tools for bounding operator spectra and analyzing stability. The geometric and optimization aspects of Stiefel manifolds provide complementary insights, with possible ramifications for quantum information and control. Future research may further elucidate liquid-like collective behaviors in channel spaces, manifold optimization, and the landscape of exotic quantum critical states beyond conventional Lagrangian descriptions.