Universal Signatures of Quantum Critical Points from Finite-Size Torus Spectra: A Window into the Operator Content of Higher-Dimensional Conformal Field Theories

The low-energy spectra of many body systems on a torus, of finite size $L$, are well understood in magnetically ordered and gapped topological phases. However, the spectra at quantum critical points separating such phases are largely unexplored for $(2+1)\mathrm{D}$ systems. Using a combination of analytical and numerical techniques, we accurately calculate and analyze the low-energy torus spectrum at an Ising critical point which provides a universal fingerprint of the underlying quantum field theory, with the energy levels given by universal numbers times $1/L$. We highlight the implications of a neighboring topological phase on the spectrum by studying the Ising* transition (i.e. the transition between a $Z_2$ topological phase and a trivial paramagnet), in the example of the toric code in a longitudinal field, and advocate a phenomenological picture that provides qualitative insight into the operator content of the critical field theory.

Figure 1

The two torus geometries with fourfold and sixfold rotation symmetry and their momentum-space grid in the vicinity of the $\mathrm{\Gamma }=(0,0)$ point. In the center of the lower row we display the Wigner-Seitz cell of the torus, highlighting the sixfold symmetry. The momentum space variable $\kappa$ is defined as $\kappa =(L/2\pi )|\mathbf{k}|{\tau }_2$ with $\tau ={\tau }_1+i{\tau }_2$, $L=|{\omega }_1|=|{\omega }_2|$ and $\mathbf{k}$ a momentum of the finite-size cluster.

Figure 2

Normalized low-energy torus spectrum for the Ising QFT for the modular parameters $\tau =i$ and $\tau =1/2+\sqrt{3}/2i$ obtained with ED (large symbols) and QMC simulations (small red filled circles). Filled (empty) symbols denote ${\mathbb{Z}}_2$ even (odd) levels. Linear fits in $1/N$ for levels with $\kappa =0$ ($\kappa =1$) are shown by blue solid (green dashed) lines (cf. color coding in Fig. 1) and the values of the fields after extrapolation to the thermodynamic limit $1/N\to 0$ are given in parentheses. The normalization constant ${\mathrm{\Delta }}_0$ is chosen such that the first ${\mathbb{Z}}_2$ odd level extrapolates to one. We observe a universal torus spectrum for the lattices with the same type of IR cutoff (same $\tau$).

Figure 3

Universal torus spectra for the Ising QFT for the modular parameters $\tau =i$ (left panel) and $\tau =1/2+\sqrt{3}/2i$ (right panel). Full symbols denote numerical results obtained by $\mathrm{E}\mathrm{D}+\mathrm{Q}\mathrm{M}\mathrm{C}$ (the lowest $Z_2$ odd levels), while empty symbols denote the $\epsilon$-expansion results. The dashed line shows a dispersion according to the speed of light.

Figure 4

Universal torus spectra for the Ising* QFT and the modular parameter $\tau =i$. The labels A-P etc. denote the boundary conditions along the two directions of the torus, where P (A) means periodic (antiperiodic). Left: Normalized low-energy spectrum from ED with the same normalization constant ${\mathrm{\Delta }}_0$ as in Fig. 2. The levels in the P-P sector are the ${ϵ}_T(+\mathrm{\Delta }\kappa )$ levels from the TFI spectrum. A very remarkable feature is the presence of the four very low-lying levels which govern the fourfold degenerate ground state manifold in the deconfined phase. See Fig. 2 for further details. Right: Full symbols denote numerical results obtained by ED, while empty symbols denote $\epsilon$-expansion results. The dashed line shows a dispersion with the speed of light. The inset is an enlargement of the four lowest levels. See Fig. 3 for further details.