Operator fusion from wavefunction overlaps: Universal finite-size corrections and application to Haagerup model

Abstract

Given a critical quantum spin chain described by a conformal field theory (CFT) at long distances, it is crucial to understand the universal conformal data. One most important ingredient is the operator product expansion (OPE) coefficients, which describe how operators fuse into each other. It has been proposed in [Zou, Vidal, Phys. Rev. B 105, 125125] that the OPE coefficients can be computed from overlaps of low-energy wavefunctions of the spin chain. In this work, we establish that all conformal data including central charge, conformal dimensions, and OPE coefficients are encoded in the wavefunction overlaps, with universal finite-size corrections that depend on the operator content of the cyclic orbifold CFT. Thus this method allows us to numerically compute all the conformal data based solely on the low-energy eigenstates. The predictions are verified in the Ising and XXZ model. As an application, we study the recently proposed Haagerup model built from the Haagerup fusion category. We find that the CFT has central charge $c \approx 2.1$ and the lowest spin-$1$ operator in the twisted sector has scaling dimension $1 < Δ_J \leq 1.4$.

Figures (12)

• Figure 1: The path integral for the wavefunction overlap $A_{\alpha\beta\gamma}=\langle \phi^{3*}_{\gamma}|\phi^{1}_{\alpha}\phi^{2}_{\beta}\rangle$. The geometry is a three-sided cylinder $\Sigma$ with circumference $L_1+L_2=L_3$.
• Figure 2: Illustration of the conformal transformations that map three-sided cylinder to the complex plane. (a) Eq. \ref{['eq:conf_trans1']} for the case of $L_1=L_2$ and (b) Eq. \ref{['eqn:third']}, for the case of $L_2=2L_1$. Colored lines represent equal time slices with equal real part of $z$. Darker color corresponds to larger $\tau$. Red lines correspond to equal time slices on cylinders $1$ and $2$, and the blue lines correspond to equal time slices on cylinder $3$. For better illustration purposes we show both $w$ and $1/w$ for the two conformal transformations.
• Figure 3: Mapping the path integral for the wavefunction overlap $A_{\alpha\beta\gamma}$ to the cyclic orbifold path integral. (a) The three-sided cylinder is conformally equivalent to the double-sheeted Riemann sphere with branch cut on $[1,\infty)$. The conformal transformation is achieved by $z'=e^{2\pi z/L_1}$. (b) The corresponding path integral of the cyclic orbifold for $A_{\alpha\beta\gamma}$. The insertions of branch-point twist operators at $\tau=0$ and $\tau=+\infty$ correspond to the branch cut in (a).
• Figure 4: Finite-size corrections of wavefunction overlap in the critical Ising model, for the four channels: $\sigma\times\epsilon\rightarrow\sigma$, $\sigma\times\sigma\rightarrow \epsilon, \epsilon\times\epsilon\rightarrow \mathbbm{1}, \mathbbm{1}\times\epsilon\rightarrow \mathbbm{1}$ under discussion. The powers of leading finite-size corrections are $p=\frac{1}{2},\frac{3}{2},2,\frac{1}{2}$, respectively, from the orbifold CFT. In the numerical computation we choose the system size $N_1\in[100,500)$, and we see the discrepancy between the theory and numerical value is less than $1\%$.
• Figure 5: Finite-size correction of wavefunction overlap of the XXZ model at $\Delta=0$, for the three channels under discussion. The system size is chosen in the range $N_1\in[100,500)$. The discrepancy between the theory and numerical value is less than $0.1\%$.