Duality between the Deconfined Quantum-Critical Point and the Bosonic Topological Transition

Recently, significant progress has been made in ($2+1$)-dimensional conformal field theories without supersymmetry. In particular, it was realized that different Lagrangians may be related by hidden dualities; i.e., seemingly different field theories may actually be identical in the infrared limit. Among all the proposed dualities, one has attracted particular interest in the field of strongly correlated quantum-matter systems: the one relating the easy-plane noncompact ${\mathrm{CP}}^1$ model (${\mathrm{NCCP}}^1$) and noncompact quantum electrodynamics (QED) with two flavors ($N=2$) of massless two-component Dirac fermions. The easy-plane ${\mathrm{NCCP}}^1$ model is the field theory of the putative deconfined quantum-critical point separating a planar ($XY$) antiferromagnet and a dimerized (valence-bond solid) ground state, while $N=2$ noncompact QED is the theory for the transition between a bosonic symmetry-protected topological phase and a trivial Mott insulator. In this work, we present strong numerical support for the proposed duality. We realize the $N=2$ noncompact QED at a critical point of an interacting fermion model on the bilayer honeycomb lattice and study it using determinant quantum Monte Carlo (QMC) simulations. Using stochastic series expansion QMC simulations, we study a planar version of the $S=1/2$ $J\text{−}Q$ spin Hamiltonian (a quantum $XY$ model with additional multispin couplings) and show that it hosts a continuous transition between the $XY$ magnet and the valence-bond solid. The duality between the two systems, following from a mapping of their phase diagrams extending from their respective critical points, is supported by the good agreement between the critical exponents according to the proposed duality relationships. In the $J\text{−}Q$ model, we find both continuous and first-order transitions, depending on the degree of planar anisotropy, with deconfined quantum criticality surviving only up to moderate strengths of the anisotropy. This explains previous claims of no deconfined quantum criticality in planar two-component spin models, which were in the strong-anisotropy regime, and opens doors to further investigations of the global phase diagram of systems hosting deconfined quantum-critical points.

Popular Summary

A unified theory of our Universe has long been one of the ultimate goals of physics. To achieve this unity, it is important to investigate how seemingly different mathematical models of physical systems can actually be equivalent, or dual. For physicists who study condensed matter, one possible duality that has attracted considerable interest involves physics beyond the standard picture for transitions between states of matter. Here, the first theory involves the transition from a complex insulating state to another trivial insulator, and the second one describes a transition between states of a spin system (a certain type of magnetism). If these descriptions of two seemingly different systems are mathematically equivalent, then demonstrating this equivalence would help us to better understand the physical systems, their states, and phase transitions. To demonstrate this duality, we examined versions of the two models that can be solved using large-scale computer simulations.

Formally, the uncovered duality is between the quantum electrodynamics with two fermionic and bosonic matter fields in two spatial dimensions. The fermionic model is designed to host a phase transition between two insulating states: a so-called symmetry-protected topological insulator (whose surface behaves very differently from its bulk) and a simpler, featureless Mott insulator. The bosonic model hosts a transition between two different crystal-like states of a system of electronic spins (magnetic moments). We demonstrate that the critical points (at which the changes in the types of states take place) have identical properties.

This provides unprecedented evidence for the proposed duality, which, in turn, is an important step toward a unified description of a class of condensed-matter systems.

Figure 1

Schematic phase diagrams of (a) the bilayer honeycomb (BH) model, (b) the easy-plane $J-Q_3$ (EPJQ) model, and (c) the $N=2$ QED theory. In (a), the BH model contains two symmetry-breaking phases, the spin-density wave (SDW) and superconducting (SC) phases, and two symmetric phases, the bosonic symmetry-protected topological (BSPT) and the trivial Mott-insulating phases. In (b), the EPJQ model also contains two symmetry-breaking phases, the Néel antiferromagnetic (AFM) phase and the valence-bond solid (VBS) phase, and two spin-polarized phases induced by an external staggered field. In (c), as was shown in Refs. [22, 27, 33], when tuning the two masses $m$ and $M$, the $N=2$ QED theory also has two symmetry-breaking (SB) phases and two symmetric (SY) phases, one of which is the BSPT state. In all models, the four phases merge at the deconfined quantum-critical point. Phases of the same color can be mapped to each other among the models, following the duality relations proposed in the table on the right-hand side. The double arrows in (a) and (b) indicate the quantum phase transitions investigated numerically in this paper, with the paths chosen to cross the respective critical points in the physically most interesting and computationally most tractable way. In (c), recent lattice QED calculations have focused exclusively on the critical point separating all the phases [34].

Figure 2

Crossing-point analysis of the EPJQ model at $\mathrm{\Delta }=1/2$. (a) $U(q,L)$ versus $q$ in the neighborhood of $q_c$ for several system sizes $L$. (b) The crossing points seen in (a) for system-size pairs $(L,2L)$, analyzed according to the expected finite-size scaling form, Eq. (21). The procedure including error analysis gives $q_c=0.6197(2)$ and $1/{\nu }_{J\text{−}Q}^{xy}+\omega =4.0(2)$. (c) A similar analysis of $U_c^{*}(L)$ based on Eq. (22), giving $\omega =2.3(1)$. (d) Finite-size estimates ${\nu }_{J\text{−}Q}^{xy,*}$ of the correlation-length exponent defined in Eq. (23), using the slopes of the cumulants at the $(L,2L)$ crossing points. Analysis according to Eq. (24) gives ${\nu }_{J\text{−}Q}^{xy}=0.48(2)$ for the exponent in the thermodynamic limit. In all fits, we exclude small system sizes until statistically sound fits are produced.

Figure 3

Extraction of the anomalous dimensions ${\eta }_{J\text{−}Q}^{xy}$ and ${\eta }_{J\text{−}Q}^z$ of the EPJQ model at the estimated critical point $q_c$ for $\mathrm{\Delta }=1/2$. The squares of order parameters are graphed versus $L$ and analyzed with power-law fits (lines on the log-log plots). The off-diagonal spin order parameter square $M_{xy}^2(L)$ in (a) and the dimer order parameter square $D^2(L)$ in (b) give ${\eta }_{J-Q}^{xy}=0.13(3)$. The diagonal spin order parameter square $M_z^2(L)$ in (c) gives ${\eta }_{J\text{−}Q}^z=0.91(3)$.

Figure 4

DQMC results for the BH model close to its phase transition. The derivative with respect to the coupling $V$ of (a) the kinetic energy density and (b) the total ground-state energy density for linear system sizes $L=6$, 9, and 12. Panel (c) shows the zero-frequency susceptibility of the $O(4)$ vector in the pairing channel for $L=6$, 9, 12, 15, 18. The dashed line indicates our estimated critical point $V_c=2.82(1)$. In all cases the error bars are smaller than symbol size.

Figure 5

(a) Excitation gaps computed from the imaginary-time decay of the $O(4)$ vector across the BSPT-Mott transition for the BH model with $L=6$, 9, 12, 15, 18. A gap closing at $V_c=2.82t$ (vertical dashed line) is expected. (b) Finite-size scaling of $L{\mathrm{\Delta }}_{O(4)}$ for $V/t=2.80$, 2.82, 2.85 versus the inverse system size. The behavior at $V/t=2.82$ indicates $z=1$. (c) Data collapse of the data in (a) according to the expected scaling form Eq. (32), yielding ${\nu }_{\mathrm{BH}}=0.53(5)$ and $V_c/t=2.80(1)$.

Figure 6

Analysis of the anomalous dimensions ${\eta }_{\mathrm{BH}}^{\mathrm{\Delta }}$ and ${\eta }_{\mathrm{BH}}^{\rho }$ of the BH model from squared order parameters close to the critical point. Only the $V=2.817$ data points for each quantity are used in the fits, and the three points for $V=2.82$ do not exhibit any deviations from the $V=2.817$ values within the error bars. Panel (a) shows the $O(4)$ order parameter defined in Eq. (33). A power-law fit (straight line on the log-log scale) to the $L\ge 15$ data delivers the exponent ${\eta }_{\mathrm{BH}}^{\mathrm{\Delta }}=0.10(1)$. (b) The pair-density order parameter Eq. (34). Here the power-law fit works well for all system sizes and we obtain ${\eta }_{\mathrm{BH}}^{\rho }=1.00(1)$.

Figure 7

Example of finite-size crossing points $q_c^{*}(L)$,$U_c^{*}(L)$ of the Binder cumulant, here for the EPJQ model with $L=16$ and $L=32$. The data are here fitted to third-order polynomials. The crossing point and derivatives at the crossing point can be determined from the fitted polynomials.

Figure 8

Crossing-point analysis of the extreme, $\mathrm{\Delta }=1$, version of the EPJQ model, Eq. (b1). (a) The magnetic Binder cumulant versus the coupling $Q$ in the neighborhood of the phase transition. (b) Size dependence of the critical point defined as the crossing point of $U$ versus $Q$ for system sizes $L$ and $2L$, with a power-law fit giving $Q_c=1.732(2)$. (c) The cumulant evaluated at the crossing points. Here, it is not clear whether the asymptotic behavior has been reached, and we refrain from carrying out a fit. (d) The finite-size correlation length. Here, as well, we do not present any fit, as the asymptotic behavior is not yet clear.

Figure 9

Comparison of the $XY$ magnetization squared of the $\mathrm{\Delta }=1/2$ and $\mathrm{\Delta }=1$ EPJQ models at their respective critical points. In (a), fits indicating a non-vanishing order parameter in the thermodynamic limit are indicated, with the dashed line corresponding to a purely linear correction and the solid curve to a quadratic form. In (b), a second-order fit is shown. The extrapolated value at $L=\infty$ is 0 to within two error bars.