Torus spectroscopy of the Gross-Neveu-Yukawa quantum field theory: Free Dirac versus chiral Ising fixed point

We establish the universal torus low-energy spectra at the free Dirac fixed point and at the strongly coupled chiral Ising fixed point and their subtle crossover behavior in the Gross-Neuveu-Yukawa field theory with $n_{\text{D}}=4$ component Dirac spinors in $D=(2+1)$ dimensions. These fixed points and the field theories are directly relevant for the long-wavelength physics of certain interacting Dirac systems, such as repulsive spinless fermions on the honeycomb lattice or $\pi$-flux square lattice. The torus energy spectrum has been shown previously to serve as a characteristic fingerprint of relativistic fixed points and is a powerful tool to discriminate quantum critical behavior in numerical simulations. Here, we use a combination of exact diagonalization and quantum Monte Carlo simulations of strongly interacting fermionic lattice models, to compute the critical torus energy spectrum on finite-size clusters with periodic boundaries and extrapolate them to the thermodynamic limit. Additionally, we compute the torus energy spectrum analytically using the perturbative expansion in $\epsilon =4-D$, which is in good agreement with the numerical results, thereby validating the presence of the chiral Ising fixed point in the lattice models at hand. We show that the strong interaction between the spinor field and the scalar order-parameter field strongly influences the critical torus energy spectrum and we observe prominent multiplicity features related to an emergent symmetry predicted from the quantum field theory. Building on these results we are able to address the subtle crossover physics of the low-energy spectrum flowing from the chiral Ising fixed point to the Dirac fixed point, and analyze earlier flawed attempts to extract Fermi velocity renormalizations from the low-energy spectrum.

Figure 1

Illustration of (a) the honeycomb lattice, (b) the $\pi$-flux square lattice, and the two corresponding torus geometries with (c) sixfold, and (d) fourfold rotational symmetry. The two sublattices $A$ and $B$ are indicated by black and gray points respectively. The lattice constant $a$ is given by the distance between nearest neighbors, the lattice vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are indicated by the blue arrows, and the unit cell is traced by the grey dotted lines. Finite clusters with $N_s=2L^2$ sites span $L$ unit cells in the direction of ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ respectively, i.e., $|{\omega }_1|=|{\mathbf{a}}_1|L$, $|{\omega }_2|=|{\mathbf{a}}_2|L$. In (b) we have chosen the specific gauge ${\theta }_{ij}=\pi /4$, ${\theta }_{ji}=-{\theta }_{ij}$, where the tail (head) of the arrows indicates site $i$ ($j$).

Figure 2

Tight-binding dispersion relation (upper band ${\varepsilon }^0\left(\mathbf{k}\right)\ge 0)$ on the (a) honeycomb lattice, (b) $\pi$-flux square lattice. Dirac points are found at two distinct points in the BZ labeled (a) $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ and (b) $\mathbf{X}$ and ${\mathbf{X}}^{\prime }$, respectively. The grey line traces the BZ boundary.

Figure 3

Illustration of the different finite-size cluster families for the (a) honeycomb, (b) square lattice. Shown is a zoom into the momentum space around one of the Dirac points $\mathbf{K}/\mathbf{X}$. The closest momentum points are plotted for clusters of different linear size $L$, and, as a guide to the eye, we have connected them by lines for the largest $L$ shown. The Dirac points are (not) part of the momentum space for cluster families H/S (H'/S'/S”). See text for details.

Figure 4

Critical torus low-energy spectrum for (a) the free, massless Dirac CFT and (b) the $N=4$-component chiral Ising CFT, as a function of the reduced momentum $\kappa$ on honeycomb tori that contain the Dirac points (family H). Different symbols and colors indicate the fermion number sectors $n_f$ relative to half filling. The numbers in the plots indicate the degeneracy of the levels, while the parenthesis in (b) indicate nearly degenerate levels as described in the main text. The spectrum in (a) is analytically exact, the spectrum in the right panel of (b) is based on extrapolated numerical data from the microscopic lattice model $H_{\mathrm{h}}$. The left panel in (b) shows results from the $\epsilon$ expansion for $\tau =\text{exp}(\mathrm{i}\pi /3)$ and $\kappa =0$ as a comparison. The black, dashed lines show a linear dispersion according to the Fermi velocity $v_{\text{F}}^0=3/2$ and $v_{\text{F}}^c$ in the panels (a) and (b), respectively. Here, $v_{\text{F}}^c$ has been estimated as described in Sec. 3d. The hatching in (b) indicates that we have not computed energy levels within this higher energy regime.

Figure 5

Sketch of crossover effects in the torus spectrum near the chiral Ising CFT. In (a) we show the universal numbers ${\xi }_i$ for two different levels within the SM phase $V<V_c$ and at the QCP, $V=V_c$. (b) shows the scaled energy gaps ${\mathrm{\Delta }}_i\times L\propto v_{\text{F}}\left(V\right){\xi }_i$ in the thermodynamic limit $L\to \infty$, assuming a linear Fermi velocity renormalization $v_{\text{F}}\left(V\right)\propto V$, while (c) illustrates the scaled energy gaps measured from finite size systems with linear size $L$.

Figure 6

Renormalized Fermi velocity $v_{\text{F}}\left(V\right)$ of the honeycomb lattice model $H_{\mathrm{h}}$, based on the energy gap to the single-particle excitation $n_f=\pm 1$ at the momentum closest to the Dirac points $\kappa =1$, as obtained after an extrapolation to the TDL. The red line indicates a linear regression of the data points within the shaded region and is an estimator for $v_{\text{F}}\left(V\right)/v_{\text{F}}^0$ within the SM phase. Based on the data we assume a continuous behavior of $v_{\text{F}}\left(V\right)$ up to the critical point $V_c$ to estimate a critical velocity $v_{\text{F}}^c$ which is approximately $35\%$ larger than $v_{\text{F}}^0$. The finite-size results which give the data points in this plot after finite-size extrapolation are shown in Fig. 11.

Figure 7

Unwarping of the energy spectrum for clusters that do not contain the Dirac points in order to reduce finite-size effects. The finite-size spectrum for all interaction strengths $V$ is multiplied by a ($N_s$-dependent) constant such that the noninteracting fermion dispersion aligns with the (effective) linear Dirac cone with velocity $v_{\text{F}}^0$ at $\mathbf{k}={\mathbf{k}}_{\text{min}}$. See the main text for further details.

Figure 8

Low-energy spectra of the model Eq. (5) as a function of $V$ for clusters of size (a) $N_s=24$, (b) $N_s=32$. The left panels show the half-filled sector $n_f=0$; the right panels show the sectors with one additional fermion/hole $n_f=\pm 1$. The black vertical line indicates the critical point $V_c\approx 1.355$ [4, 33]. The cluster in (a) has the Dirac point in its momentum space (family H) and the fermionic excitation (right panel) is gapless for $V=0$. The cluster in (b) does not feature the Dirac point (family H') and the fermionic excitation is always gapped. This influences the spectrum at criticality such that different families of clusters have to be distinguished in the extrapolation to the TDL. We have also, in the corresponding colors, indicated the labels for the most important low-energy spectral levels: 1, ${\sigma }_T$, ${\epsilon }_T$, ${\psi }_T$, Chern, Kekulé (see text). Empty (filled) symbols for $n_f=0$ represent even (odd) levels under particle-hole inversion.

Figure 9

Critical torus spectrum of the $N=4$ chiral Ising theory for the honeycomb and square models. Shown are the most prominent low-energy levels after finite-size extrapolation [cf. Fig. 10, Fig. 13]. The left panel shows the extrapolated levels from tori that contain the Dirac points, while the right panel shows those that do not contain the Dirac points. The spectrum has been normalized by the critical Fermi velocities $v_{\text{F}}^c$ for the different geometries [cf. Fig. 11, Fig. 14]. Full symbols show levels extrapolated from QMC data; empty symbols denote levels extrapolated from ED data alone.

Figure 10

Extrapolation of the critical energy spectrum on the honeycomb lattice. We use second or first order polynomial functions (depending on the number of available data points) in $1/\sqrt{N_s}$ to extrapolate the gaps to $N_s\to \infty$. Open symbols denote ED data, while full symbols show QMC data, and we use the same labeling for the levels as in Fig. 8. Dashed (dotted) lines show extrapolations of QMC (ED) data. (a) shows clusters that contain the Dirac points, (b) shows clusters that do not. In (b) we extrapolate $Lmod3=1$ and $Lmod3=2$ clusters separately, where possible. We use clusters of linear size up to (a) $L=15$, (b) $L=14$ for the extrapolation. Note that not all gaps could be obtained on the largest clusters with QMC due to small overlap of some excitations with the ground state for the chosen operators, which resulted in noisy estimators and prevented us to reliably extract gaps for the largest systems.

Figure 11

Renormalization of the Fermi velocity in the SM phase for the honeycomb lattice model. (a) Energy gaps ${\widehat{\mathrm{\Delta }}}_i\times L$ of low-lying levels for clusters without Dirac points (family H') as a function of the interaction strength $V$. The gaps are normalized to one at $V=0$ for easier comparison. Blue square (circle) symbols show finite-size data from QMC (ED), while the black diamonds are the finite-size extrapolated values ($L\to \infty$) for each $V$. The extrapolated data for the fermion excitation $n_f=1$ (center panel) is fitted by a linear function (full red line), showing the Fermi velocity renormalization. This fit is included in the other panels by a dashed line as a comparison. The shaded region indicates the fitting window. (b) Energy gap of the single fermion excitation at ${\mathbf{k}}_{\text{min}}$ for clusters that contain the Dirac points (family H) together with a fit describing the $v_{\text{F}}$ renormalization. Note that the gaps for levels shown in (a) would vanish in the SM phase for these clusters. See text for further discussion.

Figure 12

Real space illustration of the point group symmetry of the honeycomb lattice (a), as well as the operator representations used for the ${\sigma }_T$ (b), Kekule (c), and Chern level (d). In panel (a) the blue (orange) dotted lines denote the axes of vertical (horizontal) mirror reflection, while the fixed point of lattice rotations is the center of the hexagon. Panel (b) shows the antisymmetric sublattice modulation of Eq. (A2) with the standard two-site unit cell of the honeycomb lattice. Panel (c) depicts the three distinct Kekule patterns $K_1$, $K_2$, and $K_3$ of the honeycomb lattice, and panel (d) depicts the current pattern of the bonds $N_1$, $N_2$, and $N_3$ used in Eq. (A11).

Figure 13

Extrapolation of the critical torus spectrum on the $\pi$-flux square lattice for clusters that contain the Dirac points (a), or not (b). In (b) there are two classes of lattices, denoted S' (empty symbols) and S” (half-filled symbols). The label for the states (see legend on the left) is indicated by the color and shape of the symbols in the right panel. See Fig. 10 for a comparison.

Figure 14

Renormalization of the Fermi velocity for the $\pi$-flux square lattice model. Here we use the not-extrapolated $N_s=34$ site data for the extrapolations in (a) [$N_s=32$ in (b)]. The shaded region indicates the fitting window. See Fig. 11 for a comparison.