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arXiv:2601.02459 · Physical Review Letters

Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains

Christopher Yang, Thomas Scaffidi

cond-mat.stat-mechcond-mat.str-elhep-thquant-phSubmitted Jan 5, 2026• Announced Jan 7, 2026• 0 citations• Revised Apr 2, 2026 (v2)
v1 · Jan 5, 2026v2 · Apr 2, 2026
arXivHTMLPDF

Abstract

The two-dimensional $O(N)$ nonlinear sigma model (NLSM) is asymptotically free for $N>2$: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with $O(N)$ symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at $N = 3$ in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-$1/2$ ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from $N\leq 2$. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.

Journal reference
Phys. Rev. Lett. 137, 051601 (2026)
DOI
10.1103/6tsl-d9kn
Comments
10 pages, 7 figures + Supplemental Materials

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The two-dimensional O(N)O(N)O(N) nonlinear sigma model (NLSM) is asymptotically free for N>2N>2N>2: it exhibits neither a nontrivial fixed point nor spontaneous symmetry-breaking. Here we show that a nontrivial fixed point generically does exist in the complex coupling plane and is described by a complex conformal field theory (CCFT). This CCFT fixed point is generic in the sense that it has a single relevant singlet operator, and is thus expected to arise in any non-Hermitian model with O(N)O(N)O(N) symmetry upon tuning a single complex parameter. We confirm this prediction numerically by locating the CCFT at N=3N = 3N=3 in two non-Hermitian spin-1 antiferromagnetic Heisenberg chains, and in a non-Hermitian spin-1/21/21/2 ladder, finding good agreement between the complex central charge and scaling dimensions and those obtained by analytic continuation of real fixed points from N≤2N\leq 2N≤2. We further construct a realistic Lindbladian for a spin-1 chain whose no-click dynamics are governed by the non-Hermitian Hamiltonian realizing the CCFT. Since the CCFT vacuum is the eigenstate with the smallest decay rate, the system naturally relaxes under dissipative dynamics toward a CFT state, thus providing a route to preparing long-range entangled states through engineered dissipation.
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Motivated by the recent interest in the criticality of open quantum many-body systems, we study nonlinear sigma models with complexified couplings as a general framework for nonunitary field theory. Applying the perturbative renormalization-group analysis to the tenfold symmetric spaces, we demonstrate that fixed points with complex scaling dimensions and critical exponents arise generically, without counterparts in conventional nonlinear sigma models with real couplings. We further clarify the global phase diagrams in the complex-coupling plane and identify both continuous and discontinuous phase transitions. Our work thus identifies nonlinear sigma models as a representative setting for studying complex critical points and elucidates universal aspects of critical phenomena in complexified field theory.
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