Confinement transition in the ${\mathrm{QED}}_3$-Gross-Neveu-XY universality class

The coupling between fermionic matter and gauge fields plays a fundamental role in our understanding of nature, while at the same time posing a challenging problem for theoretical modeling. In this situation, controlled information can be gained by combining different complementary approaches. Here, we study a confinement transition in a system of $N_{\mathrm{f}}$ flavors of interacting Dirac fermions charged under a U(1) gauge field in $2+1$ dimensions. Using quantum Monte Carlo simulations, we investigate a lattice model that exhibits a continuous transition at zero temperature between a gapless deconfined phase, described by three-dimensional quantum electrodynamics, and a gapped confined phase, in which the system develops valence-bond-solid order. We argue that the quantum critical point is in the universality class of the ${\mathrm{QED}}_3$-Gross-Neveu-XY model. We study this field theory within a $1/N_{\mathrm{f}}$ expansion in fixed dimension as well as a renormalization group analysis in $4-\epsilon$ space-time dimensions. The consistency between numerical and analytical results is revealed from large to intermediate flavor number.

Figure 1

(a) Illustration of the lattice model in Eqs. (1) and (2). Blue balls represent fermions and green balls represent the U(1) gauge field, coupled to the nearest-neighbor fermion hopping. The flux term is represented by blue dashed circles in each plaquette. The gauge fields fluctuate from ${\varphi }_{ij}\left({\tau }_1\right)$ at imaginary time slice ${\tau }_1$ to ${\varphi }_{ij}\left({\tau }_2\right)$ at time slice ${\tau }_2$. (b) Zero-temperature phase diagram as a function of the coupling $J$, parametrizing the strength of the gauge field fluctuations, for different fermion flavor numbers $N_{\mathrm{f}}$. In this work, we study the transition between the U1D phase and the confined VBS phase. (c) Illustration of the confinement transition from U1D to VBS. In the U1D phase, the fermions form Dirac cones and interact via a U(1) gauge field, representing a lattice realization of ${\mathrm{QED}}_3$. In the VBS phase, the fermions form gapped spin singlets. We argue that this transition is in the ${\mathrm{QED}}_3$-Gross-Neveu-XY universality class.

Figure 2

Correlation ratio of VBS order parameter across the U1D-VBS phase transition with linear system size up to $L=20$, for (a) $N_{\mathrm{f}}=4$, (b) $N_{\mathrm{f}}=6$, and (c) $N_{\mathrm{f}}=8$. Correlation ratios for $N_{\mathrm{f}}=10$ ($J_c\approx 2.9$) and $N_{\mathrm{f}}=12$ ($J_c\approx 3.2$) are not shown here.

Figure 3

Real-space decay of spin (left column) and dimer (right column) correlation function for $N_{\mathrm{f}}=4,6,8,10,12$, at the U1D-VBS transition point. The scaling dimensions are measured by fitting the slopes in the log-log plot.

Figure 4

Diagrams that contribute to the scaling dimensions ${\mathrm{\Delta }}_{\mathrm{AFM}}$ (a) and ${\mathrm{\Delta }}_{\mathrm{VBS}}$ (b) at $\mathcal{O}(1/N_{\mathrm{f}})$ from gauge fluctuations. Solid (wiggly) lines correspond to fermion-field (gauge-field) propagators at $\mathcal{O}(1/N_{\mathrm{f}}^0)$. The cross ($\times$) denotes ${\mu }^z\otimes {\sigma }^{\alpha }\otimes {\mathbb{1}}_{N_{\mathrm{f}}/2}$ insertions and the square ($\blacksquare$) denotes ${\mu }^a\otimes {\mathbb{1}}_2\otimes {\mathbb{1}}_{N_{\mathrm{f}}/2}$ insertions. Aslamazov-Larkin diagrams (not shown) vanish due to $tr\left({\mu }^z\right)=tr\left({\mu }^a\right)=0$.

Figure 5

Diagrams that contribute to the scaling dimensions ${\mathrm{\Delta }}_{\mathrm{AFM}}$ (a) and ${\mathrm{\Delta }}_{\mathrm{VBS}}$ (b) at $\mathcal{O}(1/N_{\mathrm{f}})$ from scalar fluctuations. Dashed lines correspond to scalar-field propagators at $\mathcal{O}(1/N_{\mathrm{f}}^0)$.

Figure 6

Dyson equation for the scalar-field two-point correlator at $\mathcal{O}(1/N_{\mathrm{f}})$. Thick dashed line corresponds to scalar-field propagator at $\mathcal{O}(1/N_{\mathrm{f}})$.

Figure 7

Schematic RG flow in the space spanned by tuning parameter $r$ and monopole fugacity $z$ for large $N_{\mathrm{f}}$. The theory defined by Eq. (5) describes the line $z=0$. Both in the U1D phase, described by the infrared stable fixed point denoted as gapless ${\mathrm{QED}}_3$, and at the ${\mathrm{QED}}_3\text{−}\mathrm{GN}\text{−}\mathrm{XY}$ quantum critical point, monopole operators are (dangerously) irrelevant. However, when the fermions acquire a mass gap, the fugacity $z$ grows towards the infrared, indicating a proliferation of monopoles and confinement. The corresponding fixed point, denoted as gapped ${\mathrm{QED}}_3$, is unstable towards the formation of VBS order.

Figure 8

Data collapses of VBS correlation ratios according to Eq. (43) within the three different scenarios (first three columns) and using fitted values (last column) for $N_{\mathrm{f}}=4$ (top row), $N_{\mathrm{f}}=6$ (middle row), and $N_{\mathrm{f}}=8$ (bottom row). (a),(e),(i) Scenario 1, assuming standard XY universality. (b),(f),(j) Scenario 2, assuming pure Gross-Neveu-XY universality. (c),(g),(k) Scenario 3, assuming ${\mathrm{QED}}_3\text{−}\mathrm{GN}\text{−}\mathrm{XY}$ universality. (d),(h),(l) Using fitted values as denoted in the insets.

Figure 9

Scaling dimensions ${\mathrm{\Delta }}_{\text{AFM}}$ and ${\mathrm{\Delta }}_{\text{VBS}}$ as extracted from the QMC data of Fig. 3, in comparison to the predictions of the Gross-Neveu-XY and ${\mathrm{QED}}_3\text{−}\mathrm{GN}\text{−}\mathrm{XY}$ models, respectively. The fact that ${\mathrm{\Delta }}_{\text{VBS}}$ is above 1 is consistent only with the ${\mathrm{QED}}_3\text{−}\mathrm{GN}\text{−}\mathrm{XY}$ universality class (Scenario 3).