NCFT₁ in the Complex SYK Model

• NCFT₁ is a one-dimensional field theory with an approximate infrared conformal symmetry, characterized by the asymmetry parameter 𝓔.
• The framework leverages the complex SYK model, using Schwinger-Dyson equations and ladder diagram techniques to compute two- and four-point functions.
• The operator product expansion reveals that spectral data and OPE coefficients vary continuously with μ and 𝓔, indicating tunable non-symmetric IR physics.

Nearly Conformal Field Theory (NCFT$_{1}$) refers to a class of one-dimensional field theories characterized by an infrared (IR) regime that closely approximates, but does not exactly preserve, conformal invariance. In the context of the complex Sachdev-Ye-Kitaev (cSYK) model, introducing a finite chemical potential $μ$ yields a continuous, one-parameter family of distinct NCFT$_{1}$s, each parameterized by an asymmetry variable $\mathscr{E}$ that encodes the effect of $μ$ on correlation functions and operator dynamics. This framework allows for explicit computation of correlation functions, spectral data, and operator product expansion (OPE) coefficients, extending the structure known from the standard (Majorana) SYK model to a broader, non-symmetric setting (Akyuz et al., 7 Jan 2026).

1. The Complex SYK Model and Infrared NCFT$_{1}$ Regime

The cSYK model consists of $N$ complex fermions $\psi^i$, interacting via random all-to-all $q$-body couplings and subjected to a chemical potential $μ$. In the large $N$ limit, the system is governed by Schwinger–Dyson equations for the imaginary-time two-point function $$G(\tau) = \langle T\, \psi^i(\tau)\, \bar{\psi}_i(0) \rangle$$, with the action of $μ$ manifest as an explicit symmetry-breaking term: $$(\partial_\tau - μ)G(\tau) + \int_0^\beta d\tau' \Sigma(\tau-\tau') G(\tau') = \delta(\tau), \quad \Sigma(\tau) = J^2\, G(\tau)^{q/2} G(\beta-\tau)^{q/2-1}.$$ In the strong-coupling, low-temperature regime ($\beta J \gg 1$), and for times away from endpoints, the conformal symmetry emerges approximately. The presence of $μ$ induces a particle-hole asymmetry, summarizable in a single parameter $\mathscr{E}$ (or equivalently $\theta$) tied via

$$e^{2\pi\mathscr{E}} = \frac{\sin(\pi\Delta+\theta)}{\sin(\pi\Delta-\theta)},\quad \Delta = 1/q.$$

The conformal solution for the two-point function is

$$G_c(\tau) = b^\Delta\, \mathrm{sgn}\, \tau\, e^{2\pi\mathscr{E} (|\tau|/\beta - 1/2) \mathrm{sgn} \tau} \left[\frac{\pi}{\beta J}\, \sin\left(\frac{\pi |\tau|}{\beta}\right)\right]^{-2\Delta},$$

where the normalization $b$ is determined by the SD equations. In the zero-temperature limit, this reduces to

$$G_c(\tau) = b^\Delta\, \mathrm{sgn}\, \tau\, e^{-\pi\mathscr{E}\, \mathrm{sgn}\,\tau}\, |J\tau|^{-2\Delta}.$$

The family of IR fixed points parameterized by $\mathscr{E}$ are inequivalent, each defining a distinct NCFT$_{1}$ (Akyuz et al., 7 Jan 2026).

2. Four-Point Function and Ladder Kernel in NCFT$_{1}$

The leading connected four-point function at $O(1/N)$,

$$\frac{1}{N} \mathcal{F}(\tau_1,\tau_2;\tau_3,\tau_4) = \langle \psi^i(\tau_1) \bar{\psi}_i(\tau_2)\psi^j(\tau_3)\bar{\psi}_j(\tau_4) \rangle_\mathrm{conn},$$

can be computed via a sum over ladder diagrams. The conformal (IR) limit enables the replacement $G \to G_c$ in diagrammatic computations, leading to a kernel $K_c$ acting on bilocal test functions. A key simplification is the diagonalization of $K_c$ in an $\mathrm{SL}(2)$ basis, with temperature-dependent cross-ratio

$$\chi = \frac{\sin(\pi\tau_{12}/\beta)\sin(\pi\tau_{34}/\beta)}{\sin(\pi\tau_{13}/\beta)\sin(\pi\tau_{24}/\beta)}.$$

At $μ = 0$, kernel eigenmodes organize into an antisymmetric sector ($h$ even), a symmetric sector ($h$ odd), and a continuous principal series. For general $μ$ (finite $\mathscr{E}$), the kernel is promoted to a $2\times 2$ non-symmetric matrix, with eigenvalues $k_1(h),\,k_2(h)$ that may become complex-valued at strong asymmetry. The four-point function,

$$\mathcal{F}_c / [G_c(\tau_{12}) G_c(\tau_{34})] = \alpha_0^{aa} \sum_{n=1}^\infty \frac{k_c^a(2n)}{1-k_c^a(2n)} \Psi_{2n}^{aa}(\chi) + \alpha_0^{ss} \sum_{n=1}^\infty \frac{k_c^s(2n+1)}{1-k_c^s(2n+1)} \Psi_{2n+1}^{ss}(\chi),$$

is expressed as a sum over kernel eigenmodes; the explicit eigenvalues and conformal blocks encode the spectrum and operator content of the NCFT$_{1}$ (Akyuz et al., 7 Jan 2026).

3. Operator Product Expansion and Structure Constants

The OPE of two fermions in a one-dimensional CFT is schematically

$$\psi^i(\tau_1) \bar{\psi}_i(\tau_2) = G_c(\tau_{12}) + \frac{G_c(\tau_{12})}{N} \sum_{h_m} C_h^{(a)} |\tau_{12}|^{-h} C_h^a(\tau_{12}, \partial_2)\, \mathcal{O}_h(\tau_2) + \dots$$

where $\mathcal{O}_h$ denotes bilinear primaries of scaling dimension $h = h_m$. The spectrum organizes as $h_0=1$ (corresponding to the $U(1)$ current), $h_1=2$ (Hamiltonian), and $h_{m\ge 2}$ (genuinely interacting primaries). Inserting this OPE into the four-point function, and using operator two-point functions

$$\langle \mathcal{O}_h(\tau)\mathcal{O}_h(0)\rangle \propto N\,|\tau|^{-2h},$$

allows extraction of OPE coefficients $c^a_h$, $c^s_h$: $$\begin{aligned} (c^a_h)^2 &= \frac{1}{b(q-1)}\frac{h-1/2}{2\pi\tan(\pi h/2)}\frac{[\Gamma(h)]^2}{\Gamma(2h)}\,\frac{1}{k_1'(h)}, &\quad h=h_{2m+1},\ (c^s_h)^2 &= -\frac{1}{b}\frac{h-1/2}{2\pi\cot(\pi h/2)}\frac{[\Gamma(h)]^2}{\Gamma(2h)}\,\frac{1}{k_2'(h)}, &\quad h=h_{2m}. \end{aligned}$$ The set of dimensions $\{h_m(\mathscr{E})\}$ is uniquely determined by solving $k_i(h_m) = 1$, and the OPE coefficients vary continuously as functions of both microscopic parameters and the asymmetry parameter $\mathscr{E}$ (Akyuz et al., 7 Jan 2026).

4. Parametrization by Chemical Potential and NCFT$_{1}$ Line

The introduction of a chemical potential $μ$ results in a monotonically varying asymmetry parameter $\mathscr{E}$. For each value of $\mathscr{E}$, the IR two-point function specifies a distinct NCFT$_{1}$ theory. While the scaling dimension $\Delta = 1/q$ is fixed by the interaction order, the overall normalization $b(\mathscr{E})$ of the two-point function

$$b = \frac{(1-2\Delta)\sin(\pi\Delta+\theta)\sin(\pi\Delta-\theta)}{\pi\sin(2\pi\Delta)}$$

varies with the asymmetry. For small $μ$, the kernel eigenvalues $k_{1,2}(h;\mathscr{E})$ remain real, while for sufficiently large $μ$ complex conjugate pairs may appear, signaling potential PT-symmetry transitions in operator content. Operator dimensions $\{h_m(\mathscr{E})\}$ and OPE coefficients $\{c_h\}$ traverse a continuous line as functions of $μ$ and $\mathscr{E}$. For $μ=0$, standard symmetric Majorana-SYK NCFT$_{1}$ is recovered, but for $μ\neq 0$ new non-symmetric structures, including mixed OPE coefficients $c^a c^s\neq 0$, are realized (Akyuz et al., 7 Jan 2026).

5. Spectrum and Correlation Structures Across the NCFT$_{1}$ Family

Each point along the one-parameter family of NCFT$_{1}$s is distinguished by its specific spectrum of primary bilinear operator dimensions and associated OPE structure constants. Operator scaling dimensions shift continuously with $\mathscr{E}$, and the two-point function normalization encapsulated in $b(\mathscr{E})$ encodes the response to particle-hole asymmetry. The four-point function, governed by the sum over conformal blocks, reflects the ladder kernel’s sectoral (antisymmetric/symmetric) decomposition and the cross-ratio dependence. As $\mathscr{E}$ approaches certain thresholds, the structure of kernel eigenvalues and hence the analytic structure of correlation functions can change qualitatively. The table below summarizes the principal features:

Parameter	Effect on NCFT$_{1}$	Notable Consequence
$\mathscr{E}$	Labels line of theories	Continuously tunable operator dimensions
$Δ=1/q$	Fermion scaling dimension	Fixed for given $q$, controls power-law decay
$b(\mathscr{E})$	2-pt function normalization	Encodes asymmetry, affects all higher correlators
$k_{1,2}(h;\mathscr{E})$	Kernel eigenvalues	Real/complex pair structure affects OPE spectrum

The continuous line of NCFT$_{1}$s thus encompasses a wide range of correlation structures, including possible transitions as $μ$ is tuned (Akyuz et al., 7 Jan 2026).

6. Relations to Other SYK Family Models and Outlook

The NCFT$_{1}$ structure arises naturally in the large $N$ limit of the cSYK model, generalizing the Majorana-SYK paradigm to settings with explicit particle-hole asymmetry. The explicit dependence of spectral data and OPE coefficients on $μ$ and $\mathscr{E}$ enables controlled exploration of IR physics with tunable non-symmetry. Connections to the spectral properties of random Hamiltonians, PT-symmetry breaking, and the dynamics of bilinear operators are immediately apparent. For $μ=0$, all results smoothly reduce to those previously derived for the symmetric (Majorana) SYK regime.

Further investigation may clarify physical implications, such as the behavior of transport, chaos, and entanglement within distinct NCFT$_{1}$s, as well as the ultimate universality (or lack thereof) across the continuous NCFT$_{1}$ line (Akyuz et al., 7 Jan 2026).