Quasiparticle tunnelling in two coupled chiral SYK model

TL;DR

This work analyzes quasiparticle tunnelling between two coupled chiral SYK systems in 1+1 dimensions, showing that a weak bilinear intersystem coupling does not induce a mass gap or thermal phase transition. Using perturbative Dyson–Schwinger equations, the authors obtain analytic finite-temperature two-point functions expressed via complete elliptic integrals and demonstrate that the leading free-energy correction is temperature-independent, leaving the entropy as in the decoupled case. Real-time and spectral analyses reveal massless collective bosonic modes propagating between the subsystems at zero temperature, i.e., gapless tunnelling between chiral edges with velocities $u_{\pm}=1\pm J/(2\pi)$. The results reveal a sharp dimensional distinction from the 0+1D case, highlighting robust gapless edge dynamics against explicit scaling-symmetry breaking and informing the holographic interpretation of chiral edge theories in topological phases.

Abstract

The chiral SYK model is an 1+1 dimensional generalisation of the Sachdev-Ye-Kitaev model with chiral Majorana fermions and homogeneous random interactions. In the large N limit, the model admits an exact solution of the two-point function due to its scaling symmetry and exhibits a quantised thermal Hall conductance consistent with that of a 2+1 dimensional gapped topological system. We study two chiral SYK systems coupled by a relevant quadratic interaction that explicitly breaks scaling and time reversal symmetry. Working in the regime of weak intersystem coupling, we solve the Dyson-Schwinger equations perturbatively and obtain analytic expressions for two-point functions at finite temperature. Unlike the coupled SYK model in 0+1 dimensions, the 1+1 dimensional chiral system does not develop a mass gap, and no thermal phase transition is observed. We show that the leading correction to the thermodynamic free energy is temperature-independent, implying that the entropy density remains identical to that of two uncoupled chiral SYK systems. A real-time analysis of the retarded correlator reveals the emergence of massless collective bosonic modes propagating between the two subsystems at zero temperature, signalling quasiparticle tunnelling without gap generation. Our results demonstrate a sharp qualitative distinction between relevant deformations of SYK models in zero and one spatial dimensions, and highlight the robustness of gapless chiral edge dynamics against explicit scaling symmetry-breaking interactions.

Figures (6)

• Figure 1: The spectral density of states of the chiral SYK model for fixed momentum $k >0$.
• Figure 2: The unit circle contour is deformed to avoid the poles and the branch cuts. With no poles or cuts inside, the contour reduces to the four integral, $-I_1-I_2-I_3-I_4$.
• Figure 3: Countably infinite branch cuts of the integrand of $I_a$. Depending on the imaginary part of $t_1$, the cut shown on the Re $x_1$ axis will lie inside or outside of the rectangular contour $C$.
• Figure 4: The real and imaginary parts of the two-point functions are shown for parameter values $\beta = 50$, $J = 3$ and $\mu = 0.01$. For fixed $x$, the correlators follow same profile with different strengths.
• Figure 5: (a) The circular contour $|z'|=1$ deformed to exclude the poles situated on real $\tilde{z}$ axis. (b) The infinite square contour $\mathcal{C}'$ reduces to the $x$ integral. The two poles shown on the real $x$ axis lie infinitesimally above for Im$(t) = \epsilon$ and infinitesimally below for Im$(t) = -\epsilon$.