Surface Topological Quantum Criticality II: Conformal manifolds, Isolated fixed points and Entanglement

TL;DR

This work demonstrates that conformal manifolds naturally arise in 2+1D surface topological quantum critical points under a Yukawa-coupled, multi-flavor framework in the large-$N_c$ limit, with the manifold geometry taking the form $S^{N-1}$. Finite-$N_c$ quantum fluctuations break the manifolds into a discrete set of infrared-stable Wilson-Fisher fixed points, whose universality classes are organized by discrete symmetries and distinguished by maximally entangled flavor-space operators. Entanglement entropy in flavor space, $S_F$, tracks these distinctions and correlates with RG flow directions, offering a quantitative handle on universality within high-dimensional interaction spaces. The paper provides explicit analyses for $ obreak N=2$ and $ obreak N=3$ flavors, revealing ring ($S^1$) and spherical ($S^2$) conformal manifolds that decompose into multiple fixed points, some of which correspond to SUSY-like unstable points and others to IR-stable, maximally entangled states relevant for sTQCP physics. These results suggest a powerful approach to classifying surface topological quantum phase transitions in systems with rich interaction manifolds, and raise open questions about connections to entanglement-based measures and the $F$-theorem in 2+1D CFTs.

Abstract

In this article, we propose the realization of conformal manifolds, which are smooth manifolds of scale-conformal invariant interacting Hamiltonians in two-dimensional quantum many-body systems. Such phenomena can occur in various interacting systems, including topological surfaces or 2D bulks. Building on previous observations, we demonstrate that a conformal manifold can emerge as an exact solution when the number of fermion colors, \(N_c\), becomes infinite. We identify distinct exact marginal deformation operators uniquely associated with the conformal manifolds. By considering \(N_c\) as finite but large, we show that quantum fluctuations induce a fermion field renormalization that results in mildly infrared relevant or irrelevant renormalization-group (RG) flow within a conformal manifold, producing standard isolated infrared stable Wilson-Fisher fixed points. These can be grouped with ultraviolet stable fixed points into a discrete manifold due to the spontaneous symmetry breaking of an emergent \(SO(\mathcal{N})\) dynamical symmetry in the RG flow as \(N_c \rightarrow \infty\). Additionally, we find a correlation between the direction of the RG flow within the manifold and an increase in EPR-like entanglement entropy. The infrared-stable Wilson-Fisher fixed points, induced by quantum fluctuations, are linked to theories on the conformal manifold where interaction operators are maximally entangled in flavor space. Our studies provide an effective framework for addressing topological quantum critical points with high-dimensional interaction parameter spaces, potentially housing many fixed points of various stabilities. They also highlight the central role of entangled conformal operators and their entropy in shaping universality classes of surface topological quantum phase transitions. We conclude with open questions and possible future directions.

Figures (6)

• Figure 1: Phase boundary (Eq. (\ref{['phsebndrmnfldN2']})) separating the gapless phase from the gapped superconducting phase for the $\mathcal{N}=2$ TI surface in the parameter space spanned by $(V_0, V_x, V_z)$ (Eq. (\ref{['Vdef']})). The red ring denotes the conformal manifold (Eq. (\ref{['4fN2fp']})) embedded on the phase boundary dictating its universality class. Red points denote isolated fixed points. Image from Ref.saran(2025)
• Figure 2: Feynman diagrams that contribute to the RGE in Eq. (\ref{['YkwaRGENcfntegen1']},\ref{['YkwaRGENcfntegen2']}). a) represents the leading order diagram in the large $N_c$-expansion that remains finite in the limit of $N_c \rightarrow \infty$. b) and c) are subleading diagrams which is of the first order in the $\dfrac{1}{N_c}$-expansion, while d) and e) represent two-loop diagrams that are also of the first order in the $\dfrac{1}{N_c}$-expansion and subleading in the $\epsilon$-expansion. Solid lines represent fermion propagators while dotted lines represent propagators of complex bosons.
• Figure 3: a) One-loop vertex renormalization diagram, which turns out to be exactly zero. b) Two-loop vertex renormalization, which is of order $O(1/N^2_c)$.
• Figure 4: Phase boundary of the gapless phase of $\mathcal{N}=2$ TI surface found using mean-field approximation (Eq. (\ref{['phsebndrmnfldYkwa']})) of the Yukawa theory in Eq. (\ref{['YkwEFTNc']}). The parameter space is spanned by the Yukawa couplings $(g_1,g_2)$ and the boson mass $M_B$. The red ring denotes the $S^1$ conformal manifold (Eq. (\ref{['fxdpntsYkwalrgeNc']})), which dictates the universality class of the phase boundary in the $N_c \rightarrow \infty$ limit.
• Figure 5: (a) Renormalization group (RG) flow in the plane of the Yukawa couplings $(g_1,g_2)$ for $\mathcal{N}=2$ in the limit of $N_c \rightarrow \infty$ (Eqs. (\ref{['YkwaRGENcfnte2N1']},\ref{['YkwaRGENcfnte2N2']})). The red-colored ring describes the conformal manifold (Eq. (\ref{['cnfmlmnfld_N2']})), i.e., a smooth manifold of scale-invariant fixed-point Hamiltonians, while the black dot is the isolated free-fermion fixed point. The directions of flow are towards the IR scales. (b) RG flow studied at first order in $1/N_c$ and at two-loop level. The conformal manifold in (a) breaks into isolated fixed points. Red dots represent the IR stable universality class (class I, Eq. (\ref{['IRstbleN2']})), while the green dots describe the IR unstable set of fixed points(class II, Eq. (\ref{['IRunstbleN2']}). We set $\epsilon = 0.95$ and $N_c = 10$. (c) Entanglement entropy $S_F(\phi)$ of the fixed point interaction operator evaluated along the $S^1$ manifold (see panel (a) ), plotted as a function of the angular coordinate $\phi$ (Eq. (\ref{['EentrpyN2']})) in the limit of $N_c \rightarrow \infty$. The red circles and the green squares correspond to the isolated fixed points from (b), appearing at first order in $1/N_c$.