Conformal Data and Renormalization Group Flow in Critical Quantum Spin Chains Using Periodic Uniform Matrix Product States

We establish that a Bloch-state ansatz based on periodic uniform matrix product states (PUMPS), originally designed to capture single-quasiparticle excitations in gapped systems, is in fact capable of accurately approximating all low-energy eigenstates of critical quantum spin chains on the circle. When combined with the methods of [Milsted and Vidal, Phys. Rev. B 96, 245105] based on the Koo-Saleur formula, PUMPS Bloch states can then be used to identify each low-energy eigenstate of a chain made of up to hundreds of spins with its corresponding scaling operator in the emergent conformal field theory (CFT). This enables the following two tasks that we demonstrate using the quantum Ising model and a recently proposed generalization thereof due to O’Brien and Fendley [Phys. Rev. Lett. 120, 206403]. (i) From the spectrum of low energies and momenta we extract conformal data (specifying the emergent CFT) with unprecedented numerical accuracy. (ii) By changing the lattice size, we investigate nonperturbatively the renormalization group flow of the low-energy spectrum between two CFTs. In our example, where the flow is from the tricritical Ising CFT to the Ising CFT, we obtain excellent agreement with an analytical result [Klassen and Melzer, Nucl. Phys. B370, 511] conjectured to describe the flow of the first spectral gap directly in the continuum.

Figure 1

Top: Scaling operator spectra of (a) the Ising and (b) the TCI CFTs (with a selection of operators labeled). Bottom: Approximate scaling dimensions and conformal-tower identification for the OF model at $\lambda =0.4$ with (c) $N=256$, $D=52$, and (d) $N=32$, $D=32$, corresponding to points of Fig. 2. We label a selection of states according to a numerical identification of the corresponding CFT operators [12, 29]. Note: We displace data points slightly along the $x$ axis to show degeneracies.

Figure 2

Spectral RG flow (crosses) of the first five energy levels (as apparent scaling dimensions $\mathrm{\Delta }$) at momentum zero, excluding $\mathrm{\Delta }=0$, extracted from the OF model with $\lambda =0.4$, using PUMPS with $D\le 52$. For comparison, we also plot the exact scaling dimensions of the Ising and TCI CFTs (dots, diamonds). The crossover between the two highest levels plotted, which we confirm by tracking conformal tower membership using $H_n$ matrix elements, is consistent with these states belonging to different Kramers-Wannier self-duality sectors.

Figure 3

Connection of the spectral RG flow of Fig. 2 (left) to the “flow” of OF model energy levels as a function of $\lambda$ at fixed system size $N=32$, computed using PUMPS with $D=28$. Note how the apparent scaling dimensions agree with the TCI CFT values at the TCI point ${\lambda }_{\mathrm{TCI}}\approx 0.428$.

Figure 4

Flow of the first spectral gap from Fig. 2 compared [29] with the integrable field theory result of Ref. [31], conjectured to describe the equivalent flow in the continuum.