Deconfined Quantum Criticality, Scaling Violations, and Classical Loop Models

Numerical studies of the transition between Néel and valence bond solid phases in two-dimensional quantum antiferromagnets give strong evidence for the remarkable scenario of deconfined criticality, but display strong violations of finite-size scaling that are not yet understood. We show how to realize the universal physics of the Néel–valence-bond-solid (VBS) transition in a three-dimensional classical loop model (this model includes the subtle interference effect that suppresses hedgehog defects in the Néel order parameter). We use the loop model for simulations of unprecedentedly large systems (up to linear size $L=512$). Our results are compatible with a continuous transition at which both Néel and VBS order parameters are critical, and we do not see conventional signs of first-order behavior. However, we show that the scaling violations are stronger than previously realized and are incompatible with conventional finite-size scaling, even if allowance is made for a weakly or marginally irrelevant scaling variable. In particular, different approaches to determining the anomalous dimensions ${\eta }_{\mathrm{VBS}}$ and ${\eta }_{\mathrm{N}é\mathrm{el}}$ yield very different results. The assumption of conventional finite-size scaling leads to estimates that drift to negative values at large sizes, in violation of the unitarity bounds. In contrast, the decay with distance of critical correlators on scales much smaller than system size is consistent with large positive anomalous dimensions. Barring an unexpected reversal in behavior at still larger sizes, this implies that the transition, if continuous, must show unconventional finite-size scaling, for example, from an additional dangerously irrelevant scaling variable. Another possibility is an anomalously weak first-order transition. By analyzing the renormalization group flows for the noncompact ${\mathrm{C}\mathrm{P}}^{n-1}$ field theory (the $n$-component Abelian Higgs model) between two and four dimensions, we give the simplest scenario by which an anomalously weak first-order transition can arise without fine-tuning of the Hamiltonian.

Popular Summary

Two-dimensional Mott insulators can undergo a quantum phase transition between an antiferromagnetic state and a spontaneously dimerized, nonmagnetic state. This transition, which is a remarkable and paradigmatic example of deconfinement and fractionalization in spin systems, nonetheless remains controversial. There is ongoing debate about whether the proposed “deconfined critical point” exists in such magnets or whether there is instead a very weak first-order transition. Numerical studies have been unable to settle this debate because of perplexing violations of conventional scaling behavior. Here, we shed new light on the deconfined transition by combining state-of-the-art numerical work (on systems of up to 512 lattice spacings) with new analytical tools.

A lynchpin of statistical mechanics is the correspondence between the partition functions of two-dimensional quantum systems and three-dimensional classical systems. However, the unusual features of the deconfined transition rely crucially on the fact that the weights of the spin configurations contributing to the partition function are complex instead of real and positive. This situation makes it impossible to map to a classical three-dimensional spin system. We instead show that there is a surprising correspondence with an isotropic three-dimensional loop model, in which the degrees of freedom are fluctuating strings instead of spins. By simulating this model, we rule out the possibility that the scaling violations at the deconfined transition are simply large “corrections to scaling” of the conventional type. We uncover surprising features in the data that hint that the deconfined critical point may be even more exotic than previously suspected, in a manner that we clarify. We also pin down the topology of the renormalization-group flows in the relevant field theory, showing what would have to occur for the scaling violations to be associated with a very weak first-order transition (which would characteristically exhibit a discontinuous jump in the order parameters).

We expect that our findings will advance our understanding of exotic critical behavior in two-dimensional quantum magnets and field theories.

Figure 1

Structure of L lattice. Left figure: Nodes and links (with associated orientations) in the unit cell of the L lattice. The two sublattices of nodes are indicated. Right figure: One of the four equivalent packings of minimal-length loops on this lattice.

Figure 2

The two possible configurations of a node (${\sigma }_r=\pm 1$).

Figure 3

Néel order parameter $N$ (dashed lines) and VBS order parameter $|\overrightarrow{\phi }|$ (solid lines) as a function of $J$ for several system sizes $L$. Continuous curves are interpolations using the multiple histogram method [43]. Inset: Pseudocritical couplings $J^{*}(L)$ obtained from various observables [$N$ (blue rhombi), $|\overrightarrow{\phi }|$ (down-pointing magenta triangles), crossings of $4N$ & $|\overrightarrow{\phi }|$ (grey stars), $C$ (black squares), ${\mathcal{N}}_s$ (up-pointing green triangles), $B_{\phi }$ (red circles), and $V$ (turquoise asterisks)].

Figure 4

Probability distribution of $\overrightarrow{\phi }=({\phi }_1,{\phi }_2)$ for size $L=64$ and (from left to right) $J=0.0886$, $J=0.091$, and $J=0.096$, corresponding, respectively, to the critical point, the VBS phase in the U(1) regime, and the VBS phase in the ${\mathbb{Z}}_4$ regime. The crossover from U(1) to ${\mathbb{Z}}_4$ symmetry is analyzed further in Sec. 4b.

Figure 5

Spanning number ${\mathcal{N}}_{\mathrm{s}}$ as a function of $J$, for various $L$. Inset: $L$ dependence of critical spanning number ${\mathcal{N}}_{\mathrm{s}}^{*}$, defined (1) as ${\mathcal{N}}_{\mathrm{s}}$ at $J=J_c$ (red dots) and (2) as ${\mathcal{N}}_{\mathrm{s}}$ at the point where the slope $|\mathrm{d}{\mathcal{N}}_{\mathrm{s}}/\mathrm{d}J|$ is maximal (black dots). Continuous curves are power-law fits, ${\mathcal{N}}_{\mathrm{s}}^{*}=A{L}^p+B$, with exponents $p=0.61(5)$ and $p=0.62(3)$, respectively. (Note log-log scale.)

Figure 6

Main panel: Heat capacity $C$ versus $J$. Upper inset: Peak height $C_{\max }$ with constant $C_0=0.9836$ subtracted to account for background. The fit is the power law with exponent $\alpha /\nu =1.52(2)$. Lower inset: Standard deviation in energy at $J=0.08850$. The fit is with exponent $(\alpha /\nu -3)/2=-0.97(4)$.

Figure 7

Correlators $G_{\mathrm{N}é\mathrm{el}}(r,L)$ (top panel) and $G_{\mathrm{VBS}}(r,L)$ (bottom panel) plotted against $r$, for various $L$. Straight lines correspond to the estimates ${\eta }_{\mathrm{N}é\mathrm{el}}=0.259(6)$ and ${\eta }_{\mathrm{VBS}}=0.25(3)$ from Eq. (12).

Figure 8

Néel and VBS correlation functions at separation $r=L/2$, plotted against $L$ on a log-log scale. The slopes give effective finite-size exponent values ${\eta }_{\mathrm{N}é\mathrm{el}}(L)$ and ${\eta }_{\mathrm{VBS}}(L)$, plotted in the left inset. The right inset shows the ratio $G(L/2)/G(L/4)$ versus $L$.

Figure 9

Derivatives of Néel and VBS correlators, $\mathrm{d}{G}_{\mathrm{N}é\mathrm{el}}(r)/\mathrm{d}r$ and $\mathrm{d}{G}_{\mathrm{VBS}}(r)/\mathrm{d}r$. The main panels show raw data and power-law fits (dashed lines) with ${\eta }_{\mathrm{N}é\mathrm{el}}=0.259(6)$ and ${\eta }_{\mathrm{VBS}}=0.25(3)$. The insets show scaling collapse of $L^{2+\eta }{G}^{\prime }$ versus $r/L$, with the same values for ${\eta }_{\mathrm{N}é\mathrm{el},\mathrm{VBS}}$.

Figure 10

Susceptibilities for Néel and VBS order parameters. The inset shows peak heights as a function of $L$.

Figure 11

$U_{\mathrm{VBS}}$ versus $J$ for several system sizes ranging from 32 to 512. Inset: Maximum value of $\mathrm{d}{U}_{\mathrm{VBS}}/\mathrm{d}J$ versus $L$.

Figure 12

Data for $⟨\cos (4\theta )⟩$ versus $J$ in several system sizes. Inset: $⟨\cos (4\theta )⟩$ as a function of the scaling variable $L^{1/{\nu }_4}(J-J_{\mathrm{c}})F_2(J-J_{\mathrm{c}})$.

Figure 13

Probability $P_k$ that the spanning number ${\mathcal{N}}_s$ equals $2k$, for a range of $L$. In the main panel, $J$ is tuned so that $P_0=0.3$ for each $L$. In the inset, $J$ is fixed, $J=0.08850$.

Figure 14

Parameter-free scaling: $P_1(J)$ as a function of the average number ${\mathcal{N}}_s(J)$ of spanning curves. Inset: Size dependence of the maximum of $P_1(J)$ on a double logarithmic scale.

Figure 15

First three panels starting from the top: Energy distribution functions for system sizes $L=320$, 400, and 512, respectively, for various $J$. Bottom panel: Energy distribution for $J=0.08850$ for various $L$.

Figure 16

$V_{\max }$ is shown as a function of system size on a double logarithmic scale with the background contribution subtracted. Inset: Raw data for $V$ plotted as a function of $J$ on a semilogarithmic scale for several values of $L$.

Figure 17

Size dependence of the effective correlation-length exponent ${\nu }_{\mathrm{eff}}$ obtained from finite-size scaling of various quantities.

Figure 18

Estimates of the correlation length obtained from (1) the exponential decay of the spanning number ${\mathcal{N}}_s$ in the VBS phase, $J>J_c$ (red points) and (2) the coefficient of the linear growth of ${\mathcal{N}}_s$ in the Néel phase, $J<J_c$ (black points). Note: The overall normalization is arbitrary in the latter case.

Figure 19

Topology of the sheet of RG fixed points for dimensionalities around three (the limiting cases $d=2$ and $d=4$ are not shown).