Shannon-Rényi entropies and participation spectra across three-dimensional $O\left(3\right)$ criticality

Universal features in the scalings of Shannon-Rényi entropies of many-body ground states are studied for interacting spin-$\frac{1}{2}$ systems across (2+1)-dimensional $O\left(3\right)$ critical points, using quantum Monte Carlo simulations on dimerized and plaquettized Heisenberg models on the square lattice. Considering both full systems and line-shaped subsystems, SU(2) symmetry breaking on the Néel ordered side of the transition is characterized by the presence of a logarithmic term in the scaling of Shannon-Rényi entropies, which is absent in the disordered gapped phase. Such a difference in the scalings allows to capture the quantum critical point using Shannon-Rényi entropies for line-shaped subsystems of length $L$ embedded in $L\times L$ tori, as the smaller subsystem entropies are numerically accessible to much higher precision than for the full system. Most interestingly, at the quantum phase transition an additive subleading constant $b_{\infty }^{*\mathrm{line}}=0.41\left(1\right)$ emerges in the critical scaling of the line Shannon-Rényi entropy $S_{\infty }^{\mathrm{line}}$. This number appears to be universal for 3d $O\left(3\right)$ criticality, as confirmed for the finite-temperature transition in the 3d antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model. Additionally, the phases and phase transition can be detected in several features of the participation spectrum, consisting of the diagonal elements of the reduced density matrix of the line subsystem. In particular, the Néel ordering transition can be simply understood in the $\left\{S^z\right\}$ basis by a confinement mechanism of ferromagnetic domain walls.

Figure 1

Plaquettized (left) and dimerized (right) lattices. The thick (red) lines correspond to strong bonds with coupling $J_1$ and we refer to them as plaquettes/dimers. The dotted bonds are the weak interplaquette/interdimer couplings $J_2\le J_1$. Periodic boundary conditions are implicit.

Figure 2

$S_{\infty }^{\square }$ for different values of the plaquette coupling strength $J_2$. The result for the limit $J_2=0$ is given by $S_{\infty }^{\square }=\frac{ln3}{4}N$ (see Appendix pp2). An emerging logarithmic scaling term for $J_2>J_c$ can be guessed. Lines are guides to the eye.

Figure 3

Prefactor $l_{\infty }^{\square }$ of the logarithmic scaling term of $S_{\infty }^{\square }$ extracted from fits over different size windows for the plaquettized square lattice. The fits include system sizes $N\in [N_{\min },N_{\max }]$ corresponding to the data points in Fig. 2 (e.g., $N=16,36,64,100,144$ for $N_{\min }=16$ and $N_{\max }=144$). $N_{\max }$ is the maximal accessible size for which the entropy is smaller than $\approx 20$ (see Fig. 2). We used $N_{\max }=144$ for $J_2>0.3$ and were able to push calculations up to $N_{\max }=196$ around the critical point and even to $N_{\max }=256$ for $J_2=1$. For $J_2<J_c$, the fits are difficult because of greater error bars for large entropies and large finite size effects. The behavior is nevertheless consistent with a vanishing $l_{\infty }^{\square }$ in the disordered phase. For $J_2>J_c$, a plateau emerges and $l_{\infty }^{\square }$ is found to assume approximately the same nonzero value in the whole ordered phase. The inset shows the subleading constant term ${\tilde{b}}_{\infty }^{\square }$ obtained from fits excluding a logarithmic scaling term, in the relevant low-$J_2$ phase.

Figure 4

SR entropies for the line-shaped subsystem across the transitions in plaquettized ($S_{\infty }^{\mathrm{line},\square }$, top) and dimerized ($S_{\infty }^{\mathrm{line},|}$, bottom) transitions.

Figure 5

Linear scaling prefactors $a_{\infty }^{\mathrm{line}}$ of the subsystem entropy $S_{\infty }^{\mathrm{line}}$ across the transitions in the plaquettized (top) and dimerized (bottom) models. We show both fits to the forms ${\tilde{a}}_{\infty }^{\mathrm{line}}L+{\tilde{b}}_{\infty }^{\mathrm{line}}$ [Eq. (7), valid for $J_2<J_c$, bold in valid regime, pale for $J_2>J_c$] and $a_{\infty }^{\mathrm{line}}L+l_{\infty }^{\mathrm{line}}lnL+b_{\infty }^{\mathrm{line}}$ [Eq. (6), valid for $J_2>J_c$, bold in valid regime, pale for $J_2<J_c$]. As $b_{\infty }^{\mathrm{line}}=0$ and $l_{\infty }^{\mathrm{line}}=0$ in the quantum disordered phase when $J_2<J_c$, the fits forcing $l_{\infty }^{\mathrm{line}}=0$ are slightly better. In the ordered phase, the fit to Eq. (7) does not work because of the existence of the logarithmic scaling term and fit quality factors of $Q\approx 0$ (see e.g., Ref. [23]) were obtained here.

Figure 6

Logarithmic scaling term $l_{\infty }^{\mathrm{line}}$ of the line SR entropy $S_{\infty }^{\mathrm{line}}$ across the transitions in the plaquettized (top) and dimerized (bottom) models, as obtained from fits to Eq. (6). We show fits over different system size windows. The logarithmic term vanishes in the quantum disordered phase, while in the ordered phase it assumes a nonzero, almost constant value, which is similar for both models for a given fitting size window.

Figure 7

Constant scaling term ${\tilde{b}}_{\infty }^{\mathrm{line}}$ of the subsystem entropy $S_{\infty }^{\mathrm{line}}$ across the transitions in the plaquettized (top) and dimerized (bottom) models, as obtained by a fit to Eq. (7). This form is clearly not valid in the ordered phase ($J_2>J_c$), where a logarithmic scaling term of $l_{\infty }^{\mathrm{line}}>0$ is found. Fit qualities drop to zero for $J_2>J_c$ and ${\tilde{b}}_{\infty }^{\mathrm{line}}$ is therefore shown in pale colors. In the disordered phase $J_2<J_c$, ${\tilde{b}}_{\infty }^{\mathrm{line}}=b_{\infty }^{\mathrm{line}}$ is found to be 0 (bold). The lines cross at the critical point at $b_{\infty }^{*,\mathrm{line},\square }=0.412\left(6\right)$ (plaquettized model) and $b_{\infty }^{*,\mathrm{line},|}=0.41\left(1\right)$ (dimerized model).

Figure 8

Spin stiffness multiplied by system size ${\rho }_{s}L$ as a function of temperature $T$ for different linear system sizes $L$ in the 3d $S=1/2$ antiferromagnetic Heisenberg model on the simple cubic lattice. The inset shows our best estimate for the crossing point $T^{*}$ of the spin stiffness for sizes $L$ and $2L$ as obtained from a cubic fit to our data including a bootstrap analysis for the error bars (black points). The red line corresponds to $T^{*}$ calculated from our best fit of our data to a universal function $f(L,T)=(1+c/L^{\omega })k(L^{1/\nu }(T-T_c)+d/L^{\varphi })$ (cf. Ref. [16]), including empirical scaling corrections. We approximated $k$ by a third order polynomial. For the critical temperature, we obtain $T_c=0.94408\pm 0.00002$ using ${10}^4$ bootstrap samples (1$\sigma$ errorbar indicated by the narrow rectangle).

Figure 9

Fit result for the subleading constant $b_{\infty }^{\mathrm{line},3\mathrm{d}}$ in the scaling of the SR entropy $S_{\infty }^{\mathrm{line},3\mathrm{d}}$ with system size close to the critical temperature in the 3d Heisenberg model. We only performed fits to the form in Eq. (7), which is strictly only valid in the absence of logarithmic scaling terms (i.e., in the disordered phase at $T\ge T_c$). At the critical point, $b_{\infty }^{\mathrm{line},3\mathrm{d}}$ converges well with system size and the estimate for the fit window with the largest sizes is given by $b_{\infty }^{*,\mathrm{line},3\mathrm{d}}=0.41\left(1\right)$. The insets show the scaling of $S_{\infty }^{\mathrm{line},3\mathrm{d}}$ as a function of $L$ in the ordered phase $T<T_c$ with a clear sign of the logarithmic scaling correction [$l_{\infty }^{\mathrm{line},3\mathrm{d}}=0.8\left(3\right)$] and in the paramagnetic phase $T>T_c$, where the scaling is purely linear with a vanishing constant $b_{\infty }^{\mathrm{line},3\mathrm{d}}=0.003\left(9\right)$.

Figure 10

Schematic picture for a basis state having two domain walls $\downarrow \downarrow$ and $\uparrow \uparrow$ separating the two Néel configurations ${\mathrm{N}}_{\mathrm{A}}$ and ${\mathrm{N}}_{\mathrm{B}}$.

Figure 11

Development of the density of states in the participation spectrum of ${\rho }_B$ across the transition on the plaquettized (left) and dimerized (right) square lattice. Here, $L=20$.

Figure 12

Density of states for different classes of basis states, distinguished by the number of domain walls $n_{\mathrm{dw}}$ in the Heisenberg limit $J_2=1$. Basis states with different numbers of domain walls are clearly separated in pseudoenergy $\omega$, while the bands come closer with system size and have a small overlap. Note that the density of states is normalized for each state band. The fine line displays the overall density of states scaled by a factor for visibility.

Figure 13

Domain wall resolved density of states for the plaquettized model in the gapped phase ($J_2=0.1$ and $L=24$). The number of domain walls is not a good identifier of the state packets in this case. In the gapped phase, even in the thermodynamic limit there are pronounced gaps between the bands of states which in this case are characterized by the number of strong domain walls $n_{\mathrm{strong}}$ [indicated by bold face (red) numbers].

Figure 14

Development of participation spectrum as a function of $|S^z|$ (indicated by the numbers on top of each state column) across the transition in the plaquettized model for $L=16$.

Figure 15

The band of states with four domain walls in the homogeneous Heisenberg limit is formed from isolated plaquette states with 0, 1, 2, 3, and 4 strong domain walls ($L=16$). Bold face (red) numbers indicate the number of strong bonds for the state packet.

Figure 16

Second derivative of $S_{\infty }^{\mathrm{line}}$ for the plaquettized (left) and dimerized (right) lattices for different lattice sizes. The sign change shows a clear inflection point right at the quantum phase transition, which corresponds to a pronounced minimum of the first derivative of $S_{\infty }^{\mathrm{line}}$.

Figure 17

Dependence of the width $\delta /L$ of state packets as a function of system size $L$ for fixed $(n_{\mathrm{strong}},n_{\mathrm{dw}},|S^z\left|\right)$ (top panel) and $n_{\mathrm{dw}}$ (bottom panel, including states from all $S^z$ sectors) across the transition in the plaquettized (top) or dimerized (bottom) model. Lower numbers of domain walls $n_{\mathrm{dw}}$ correspond to low pseudoenergy part of the participation spectrum. The crossing of $\delta /L$ for different sizes close to the critical point indicated by a vertical line is most prominent for the subsystem basis state packets with low pseudoenergy (e.g., for the packet with two domain walls for both panels) but clearly exists with considerable larger size effects (such as drift crossing) in higher parts of the spectrum.

Figure 18

Width $\delta /L$ of state packet (1, 2, 1) as a function of inverse system size $1/L$ for different values of $J_2$ in the case of the transition in the plaquettized model. In the limit of large system sizes, $\delta /L$ vanishes in the quantum disordered phase, while it goes to a constant in the ordered phase. (Inset) Zoom for small values of $J_2$ in the gapped phase. The linear behavior in $1/L$ suggests that $\delta /L$ vanishes like $1/L$.

Figure 19

Pseudoenergies of the line subsystem as a function of domain wall distance $d$ (expressed in terms of the chord distance on the ring) for all states with two domain walls in the $S^z=0$ sector for the plaquettized (top) and dimerized (bottom) model, for different values of $J_2$ (equidistant in steps of $0.05$). In order to compare the curves, we subtracted the pseudoenergy of the state with the minimal distance $d=2$ between domain walls (which is always the lowest).