Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in the spin-1 Heisenberg chain

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 $\textit{complex}$ 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 a non-Hermitian spin-1 antiferromagnetic Heisenberg chain, 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.

Figures (5)

• Figure 1: Schematic RG flow (from UV to IR) of the 2D $O(N>2)$ nonlinear $\sigma$-model (NLSM) with a complex coupling parameter $g$. On the real axis, the flow goes to strong coupling. The CCFTs (red dots) appear at complex-conjugate positions in the complex-$g$ plane, around which the flow forms an outward spiral which eventually also flows to strong coupling.
• Figure 2: (a) Schematic two-parameter RG flow for the $O(N>2)$ NLSM. The parameter $|g-g_c|$ is the distance from the critical point $g_c$ in the complex plane of the coupling constant $g$, and thus corresponds to the energy operator $\epsilon$, which is relevant. The parameter $g_{lc}$ controls the strength of the "loop crossings" and corresponds to the 4-leg watermelon operator, with scaling dimension $\Delta_{lc} \equiv \Delta_{l=4}$, which is irrelevant at the CCFT. Contrary to the loop model, for which $g_{lc}=0$ due to an additional microscopic, non-invertible symmetry, the NLSM generically has $g_{lc}\neq 0$ in the UV. It therefore only flows toward $g_{lc}=0$ in the IR, realizing the same non-invertible symmetry as the loop model, but now as an emergent symmetry that appears only at the CCFT fixed point. (b) Scaling dimension of loop crossings $\Delta_{lc}$, which are irrelevant for all $N>2$. The dashed line indicates the asymptote $\mathrm{Re}~\Delta_{lc} (N \to \infty) = 2.5$.
• Figure 3: Spectrum of the Heisenberg spin-1 chain at the finite-size complex fixed point [Eq. (\ref{['eq:J2Kvals']})] for a length $L = 14$ chain. (a) Real part of the extracted scaling dimensions, obtained as rescaled energy gaps $\Delta E / (2 \pi v / L)$, where $\Delta E = E - E_g$ and $E_g$ is the ground state energy and $v = 0.796-1.03i$ is the complex velocity. Different colors and symbols indicate states with distinct $O(3)$ total spin number $S$. Letters label states with known $O(N)$ CCFT predictions, as summarized in panel (d). Narrow horizontal lines denote the predicted real parts of the scaling dimensions. States labeled by primed letters correspond to descendants of the primaries labeled by the corresponding unprimed letters. On the right, we zoom in on the orange box enclosing states at $k = 0$ with nearly-marginal scaling dimensions, of which D0, D2, and D4 arise from the $\ell = 4$ watermelon operator. (b) Energy spectrum in the complex plane without rescaling by $2\pi v / L$, with the dashed line parameterized by $E_g + vx$, where $x \in \mathbb{R}^+$ showing the direction of the conformal towers. Note that we shift the eigenenergies by an imaginary constant $\delta$ (see the discussion following Eq. (\ref{['eq:lindblad']})). (c) Same as (b), but only displaying states with momenta $k =0$ and $\pi$. On this panel, the clusters of nearly-degenerate states corresponding to the $\ell=1,2,3,4$ watermelon operators (labeled by A, B, C, D) become obvious. (d) Table of identified CCFT states, comparing the predicted and numerically calculated scaling dimensions $\Delta$. WM stands for watermelon operators.
• Figure 4: Realization of a CCFT in a spin-1 Heisenberg chain. Non-Hermiticity can be engineered by introducing jump operators $\boldsymbol S_i - \boldsymbol S_{i+2}$ and $1-P_0$, where $P_0 = (1/3) [(\boldsymbol{S}_i \cdot \boldsymbol S_{i+1})^2-1]$ is a projector onto the singlet sector. After post-selection on no-click trajectories, the dynamics are governed by a non-Hermitian Hamiltonian with a complex $J_2$ and $K$, which can be tuned to a CCFT.
• Figure 5: (a)–(b) Real and imaginary parts of the complex entanglement entropy $S_l$ for different subsystem sizes $l$ at the CCFT. The dashed line indicates the fit to the logarithmic scaling expected at criticality [see Eq. (\ref{['eq:logsl']})], which estimates the central charge to be $c = 1.529 - 0.161i$. (c)–(d) Real and imaginary parts of the ground-state energy density $E_g / L$ and fit to $E_g / L = e_\infty - \pi v c / (6 L^2)$, where $e_\infty$ is a fitting parameter, $c$ is determined from the entanglement entropy [see panels (a)–(b)], and $v$ is obtained from the energy spectrum [see Eq. (\ref{['eq:velocity']})].