Confinement transition in the QED₃-Gross-Neveu-XY universality class

The coupling between fermionic matter and gauge fields is of fundamental importance in both high-energy and condensed-matter physics. In the latter context, gauge fields can emerge as a consequence of fractionalization in quantum materials. Prominent examples include spin liquids balents2010spin ; XGWen2019 and deconfined quantum critical points PhysRevB.70.144407 ; PhysRevLett.98.227202 ; PhysRevX.7.031052 ; Zhou2019Quantum ; Ma2018 ; JGuo2019 in frustrated magnets. As such states are characterized by topological order, it is often difficult to identify in an experiment or a simulation the relevant low-energy excitations and their characteristic properties.

In many cases, however, it is possible to tune the system by nonthermal external parameters, such as pressure or magnetic field, through a zero-temperature transition between an exotic phase with topological order and a conventionally ordered phase. If this quantum phase transition is continuous, the presence of fractionalized low-energy excitations on the topologically ordered side of the transition leaves characteristic fingerprints on the pertinent quantum critical behavior. These are in principle measurable and can be used to identify the topological order. To take advantage of this fact, it is decisive to gain a comprehensive understanding of the quantum universality classes that involve fluctuating gauge fields.

Recent work on a system of fermions coupled to a discrete $\mathbbm{Z}_{2}$ gauge field in 2+1 dimensions has demonstrated the existence of such a quantum transition between a deconfined phase with gapless fermionic excitations at weak coupling and a symmetry-broken confined phase with gapped fermionic excitations at strong coupling PhysRevX.6.041049 ; gazit2017emergent ; gazit2018confinement ; PhysRevB.96.205104 ; ChuangChen2019 ; Gonzalez2020 . Systems of gapless fermions coupled to gauge fields with continuous gauge groups are of significant current interest as well. An important example is (2+1)-dimensional quantum electrodynamics (QED₃) with U(1) gauge group, both in its compact and noncompact version Polyakov1975 ; Polyakov1977 ; hands2002 ; hands2004 ; armour2011 ; fiore2005 ; arakawa2005 ; arakawa2006 ; PhysRevD.94.065026 ; PhysRevLett.121.041602 ; PhysRevD.90.036002 ; dipietro2016 ; Janssen:2016nrm ; herbut2016 ; chester2016anomalous ; PhysRevD.94.056009 ; Kotikov:2016prf ; gusynin2016 . This theory is a natural candidate for the low-energy description of gapless U(1) spin liquids, which may be realizable in certain planar magnets PhysRevLett.86.3871 ; PhysRevB.66.144501 ; PhysRevB.70.214437 ; PhysRevB.72.104404 ; PhysRevB.71.075103 ; PhysRevX.7.031020 ; XYSong2019 .

Using Quantum Monte Carlo (QMC) simulations, the phase diagram of a lattice model of fermions coupled to a compact U(1) gauge field has recently been mapped out by some of the present authors xu2019 ; PhysRevB.100.085123 . The model has a free parameter that can be used to drive a transition from a U(1) deconfined (U1D) phase with nodal dispersion of the fermionic excitations to confined phases in which spin and/or lattice symmetries are spontaneously broken, see Fig. 1. The U1D phase represents a lattice realization of QED₃, while the confined phases describe different conventionally ordered phases.

Meanwhile, several analytical works, such as renormalization group (RG) and large-$N_{\mathrm{f}}$ calculations, have made quantitative predictions for the universal behavior of quantum critical fermion systems coupled to gauge fields Janssen:2017eeu ; Ihrig2018 ; PhysRevB.98.165125 ; PhysRevD.98.085012 ; PhysRevB.99.195135 ; Benvenuti:2018cwd ; Benvenuti:2019ujm ; Zerf2019 ; Dupuis:2019uhs ; Dupuis:2019xdo ; Boyack:2018urb . However, while there has been tremendous progress in the case of the ungauged (2+1)-dimensional Gross-Neveu-like transitions Janssen:2014gea ; PhysRevB.94.245102 ; PhysRevB.96.165133 ; PhysRevD.96.096010 ; PhysRevB.98.125109 ; Ray:2018gtp ; Gracey:2018qba ; Gracey2018review ; Iliesiu:2015qra ; Iliesiu:2017nrv ; Toldin:2014sxa ; Otsuka:2015iba ; TCLang2019 ; YZLiu2019 ; huffman2019fermion , the situation with fluctuating gauge fields remains significantly less clear. In particular, a quantitative comparison of universal properties between analytical predictions from the continuum quantum field theory and measurements in corresponding lattice simulations has so far not been possible. Such a comparison is the goal of this work.

The remainder of the paper is organized as follows: In Sec. II, we review the lattice model studied in Ref. xu2019 and present additional evidence for a continuous transition between deconfined and confined phases. The continuum field theory that we propose to describe the universal behavior of the confinement transition is discussed in Sec. III. Section IV contains the comparison of the field-theory predictions with the numerical results. Conclusions are drawn in Sec. V.

We simulate a square-lattice model described by the Euclidean action $S=\int_{0}^{\beta}d\tau(L_{F}+L_{\phi})$ with

where $\psi^{\dagger}_{i\alpha}$ and $\psi_{i\alpha}$ denote fermion creation and annihilation operators at lattice site $i$ and flavor index $\alpha=1,\dots,N_{\mathrm{f}}$. The compact U(1) gauge field lives on nearest-neighbor bonds $\langle ij\rangle$ and is denoted by $\phi_{ij}$. The nearest-neighbor fermion hopping amplitude $t$ is modulated with phase $\phi_{ij}$, thereby inserting magnetic flux in each plaquette. The expression $\operatorname{curl}\phi$ sums the gauge fields in each elemental plaquette $\square$ and the coupling $K>0$ stabilizes a $\pi$ flux in each plaquette, see Fig. 1(a).

In the $J\to 0$ limit, the gauge field has no imaginary time dynamics. The ground state in this limit is characterized by gapless Dirac excitations, as both $L_{F}$ and the flux term in $L_{\phi}$ with $K>0$ favor a $\pi$ flux in each plaquette. The Brillouin zone contains two Dirac cones per flavor and the number of (irreducible) Dirac fermions is $2N_{\mathrm{f}}$.

Turning on a small finite $J$ allows the U(1) gauge field to fluctuate. This model has been studied by sign-problem-free QMC simulations xu2019 . For $N_{\mathrm{f}}=2,4,6,8$, the low-temperature phase for small $J$ has been found to be characterized by a gapless spectrum and emergent scale invariance. This phase represents a lattice realization of QED₃ with deconfined excitations and as such has been dubbed U1D. In particular, for small $J$, the simulations suggested no evidence for confining monopole proliferation PhysRevB.100.085123 .

Upon increasing $J$ beyond a certain threshold, however, the fermions acquire a mass, monopoles of the compact gauge field start to proliferate, and the fermions exhibit a confining potential, see Fig. 1(b). For $N_{\mathrm{f}}=4,6,8$, the confined phase is described by a valence bond solid (VBS), characterized by an ordered array of spin singlets, thereby breaking lattice translation and rotation symmetries spontaneously.

To improve the understanding of the U1D-VBS transition, we have carried out additional QMC simulations near the transition point and on larger lattices. We also calculated larger $N_{\mathrm{f}}$ case with $N_{\mathrm{f}}=10$ and $12$. Technical details of the simulations, including the update scheme, the determination of the phase diagram, and the calculation of dynamical properties near the transition, are given in Refs. xu2019 ; PhysRevB.100.085123 . Fig. 2 shows the numerical data of VBS correlation ratios Pujari2016

where $\vec{M}=(\pi,0)$, $\delta\vec{q}=(\frac{2\pi}{L},0)$, for different flavor numbers $N_{\mathrm{f}}$ in the vicinity of the U1D-VBS phase transition. In the above equation, $\chi_{D}$ denotes the VBS structure factor, given as

where the dimer operator $D_{i}=\sum_{\alpha\beta}S_{\beta}^{\alpha}(i)S_{\alpha}^{\beta}(i+\hat{x})$ is defined along the $\hat{x}$ direction, and the $S_{\beta}^{\alpha}(i)$ is the spin operator defined as $S_{\beta}^{\alpha}(i)=\psi_{i\alpha}^{\dagger}\psi_{i\beta}-\frac{1}{N_{\mathrm{f}}}\delta_{\alpha\beta}\sum_{\gamma}\psi_{i\gamma}^{\dagger}\psi_{i\gamma}$. In the thermodynamic limit, $r_{\text{VBS}}$ is zero (one) in the U1D disordered (VBS ordered) phase. The curves on different lattices suggest a single crossing point in the thermodynamic limit, implying the existence of a continuous quantum phase transition.

More evidence for a continuous phase transition comes from the measurement of the spin and dimer correlation functions $C_{S}(r)$ and $C_{D}(r)$. Technical details concerning these measurements are deferred to Appendix A. Fig. 3 shows the real-space decay of the spin and dimer correlators at the crossing points of the correlation ratios for different flavor numbers $N_{\mathrm{f}}$. Instead of the usual exponential behavior, the correlators show a power-law decay with characteristic exponents $2\Delta_{\text{AFM}}$ and $2\Delta_{\text{VBS}}$.¹¹1$\Delta_{\text{AFM}}$ and $\Delta_{\text{VBS}}$ will be defined as scaling dimensions of AFM and VBS order parameters in Sec. III. This constitutes further strong evidence for quantum criticality beyond the findings previously presented in Ref. xu2019 .

The VBS order parameter has a discrete $\mathbbm{Z}_{4}$ rotation symmetry, associated with the four possible VBS ordering patterns on the square lattice. At criticality, such a $\mathbbm{Z}_{4}$ symmetry is usually expected to get enhanced to a continuous $\mathrm{U}(1)\simeq\mathrm{O}(2)$ rotation symmetry lou2007 ; NSMa2019 . This suggests that the low-energy description of the quantum critical point has to include three different types of excitations: $2N_{\mathrm{f}}$ flavors of (irreducible) gapless Dirac fermions $\psi$ and $\bar{\psi}$, coupled to a U(1) gauge field $a_{\mu}$, and a real two-component scalar field $\vec{\phi}=(\phi^{x},\phi^{y})$ serving as an order parameter for spontaneous O(2) symmetry breaking. The minimal theory that is consistent with these requirements is the QED₃-Gross-Neveu-XY (QED₃-GN-XY) model proposed in Ref. xu2019 . The purpose of the rest of this paper is to confirm that the numerical data is indeed consistent with this proposal.

In this section, we discuss the critical behavior of the QED₃-GN-XY model within two complementary analytical approaches: a $1/N_{\mathrm{f}}$ expansion in fixed dimension and a RG analysis in $D=4-\epsilon$ space-time dimensions. The Lagrangian of the QED₃-GN-XY model in $D=3$ reads

where $\psi$ and $\bar{\psi}=\psi^{\dagger}\gamma_{0}$ are two-component spinors, $(\phi^{a})=(\phi^{x},\phi^{y})$, $a=1,2$, is a real XY order-parameter field, and the $2\times 2$ Dirac matrices satisfy the Euclidean Clifford algebra $\{\gamma_{\mu},\gamma_{\nu}\}=2\delta_{\mu\nu}\mathbbm{1}_{2}$, $\mu,\nu=0,1,2$. In the Yukawa vertex $\mu^{a}\otimes\mathbbm{1}_{2}\otimes\mathbbm{1}_{N_{\mathrm{f}}/2}$, the $2\times 2$ Pauli matrices $(\mu^{a})=(\mu^{x},\mu^{y})$ connect the two Dirac nodes, $\mathbbm{1}_{2}$ acts on spin indices, and $\mathbbm{1}_{N_{\mathrm{f}}/2}$ acts on the additional flavor indices; see Ref. PhysRevB.72.104404 for details. Thus, the indices $i$ and $j$ run from $1$ to $2N_{\mathrm{f}}$, and we assume $N_{\mathrm{f}}$ even. $a_{\mu}$ denotes the U(1) gauge field and $\epsilon_{\mu\nu\rho}$ is the totally antisymmetric tensor. In addition to the gauge symmetry, the model features a global axial U(1) symmetry under which the two-component scalar field $(\phi^{a})$ transforms as an SO(2) vector. The parameters $e$, $g$, and $\lambda$ denote the charge, Yukawa, and quartic scalar couplings. The parameter $r$ can be used to tune through the U1D-VBS transition.

For $r>0$, the scalar field $\phi^{a}$ can be integrated out and the theory describes the U1D phase with enhanced $\mathrm{SU}(2N_{\mathrm{f}})$ symmetry. For $r<0$, the fermions are fully gapped, since $\gamma_{0}\mu^{a}$ anticommutes with the Hamiltonian $\mathcal{H}(p)=\gamma_{0}\gamma_{i}p_{i}$, and the gauge field confines due to the proliferation of monopoles Polyakov1975 ; Polyakov1977 ; PhysRevB.70.214437 ; XYSong2019 . This represents the VBS ordered phase. The critical point is given by $r=0$ at tree level.

Besides the universal exponents that describe the critical behavior in the vicinity of the quantum phase transition, we aim at computing the correlators of the staggered magnetization

where $\alpha=x,y,z$ denotes the spin components and $\vec{r}=r_{x}\hat{x}+r_{y}\hat{y}$ the lattice site, and the staggered dimer operator

with $a=x,y$ PhysRevB.72.104404 . In the above equations, two operators $\mathcal{O}$ and $\mathcal{O}^{\prime}$ are considered equivalent, $\mathcal{O}\sim\mathcal{O}^{\prime}$, if they transform in the same way under all symmetries. At criticality, the correlation functions are given by power laws

where $\Delta_{\mathrm{AFM}}$ and $\Delta_{\mathrm{VBS}}$ are the scaling dimensions of the operators $O_{\mathrm{AFM}}^{\alpha}$ and $O_{\mathrm{VBS}}^{a}$. The correlation function of a microscopic lattice operator will in general be a linear combination of all operators in the conformal field theory that are equivalent to the respective lattice operator. In the long-wavelength limit, however, the operator with the lowest scaling dimension will dominate.

We now discuss three possible universality classes that could in principle describe the continuous U1D-VBS transition observed in the lattice model. Among these three scenarios, we will see that the numerical data is consistent only with the QED₃-GN-XY universality class.

The transition between the U1D disordered phase and the VBS ordered phase is detected via a real two-component order parameter. Hence, if the U1D-VBS transition were a standard quantum transition within the Landau paradigm, quantum-to-classical mapping would apply, and one would expect the transition to be in the classical 3D XY universality class. Within Scenario 1 (Scen1), we thus assume the critical exponents Poland:2018epd ; Chester2019

However, it is well known that the presence of gapless fermions at the transition can change the universality class sachdevbook . Previous studies in systems with Dirac fermions (but without a fluctuating gauge field) coupled to a $\mathrm{O}(2)$ order parameter have put forward the Gross-Neveu-XY universality class TCLang2013 ; ZCZhou2016 ; Li:2017sbk ; Xu:2018kekule ; Otsuka:2018kcb ; Liao:2019vbs ; Li:2019hhu ; PhysRevD.96.096010 . This describes our Scenario 2 (Scen2), with exponents

cf. Sec. III. We note in particular that $\Delta_{\mathrm{VBS}}\Bigr{|}_{\text{GN-XY}}<1$. Below, we will show that this result is inconsistent with our numerical data, which we ascribe to the presence of gapless gauge-field excitations at the transition. We will demonstrate that the data is instead consistent, within error bars, with our theoretical predictions for the QED₃-GN-XY universality class. The universal exponents within this scenario, dubbed Scenario 3 (Scen3), have been computed in Sec. III,

Note that now $\Delta_{\mathrm{VBS}}\Bigr{|}_{\text{QED${}_{3}$-GN-XY}}>1$.

In the vicinity of the quantum critical point at coupling $J=J_{\mathrm{c}}$, the dimensionless correlation ratio $r_{\text{VBS}}$ should obey the finite-size scaling law

with scaling function $\mathcal{F}(x)$ and linear system size $L$. Fig. 8 shows the correlation ratios $r_{\text{VBS}}$ as function of the rescaled coordinate $x\coloneqq(J/J_{\mathrm{c}}-1)L^{1/\nu}$ for $N_{\mathrm{f}}=4,6,8$, using the predictions for $1/\nu$ within the three scenarios proposed above, as well as using fitted values. Due to the limited system sizes that are available within tenable simulation times, the data collapses are not perfect in all cases. However, for all flavor numbers checked, the data collapses using the standard XY prediction are significantly worse than for the other two scenarios. This rules out Scenario 1, revealing that the transition evades the usual quantum-to-classical mapping. The differences between the qualities of the data collapses in Scenario 2 and 3 are less significant, in particular for $N_{\mathrm{f}}=8$. The reason for this behavior is that the critical exponents in these two scenarios approach the same limiting value for large $N_{\mathrm{f}}$, $\lim_{N_{\mathrm{f}}\to\infty}(1/\nu)=1$. Nevertheless, for $N_{\mathrm{f}}=4$ and $6$, we note that the spread of values of $r_{\text{VBS}}$ within the ranges of $x$ shown is significantly larger within Scenario 3 than for the other two scenarios. We interpret this as first evidence for QED₃-GN-XY universality (Scenario 3).

To clearly rule out the pure Gross-Neveu-XY universality class (Scenario 2), we have performed fits to the dimer $C_{D}(r)$ and spin $C_{S}(r)$ correlation functions, assuming power-law behaviors

with exponents $\Delta_{\text{VBS}}$ and $\Delta_{\text{AFM}}$. The fits are shown as straight lines in the log-log plots of Fig. 3, and we plot the extracted exponents in Fig. 9 in comparison with the theoretical predictions for Gross-Neveu-XY universality (Scenario 2) and QED₃-GN-XY universality (Scenario 3). While the errors are large, the overall behavior is significantly closer to the QED₃-GN-XY predictions, in particular for larger flavor numbers $N_{\mathrm{f}}\geq 8$. We also note that we find $\Delta_{\text{VBS}}>1$ for all considered values of $N_{\mathrm{f}}$. This rules out Scenario 2, suggesting that the transition observed in the simulation is indeed described by a new universality class in which the fluctuating U(1) gauge field plays a significant role.

In conclusion, we have provided evidence for the existence of an unconventional quantum critical point between a deconfined phase, characterized by gapless fermionic degrees of freedom coupled to a U(1) gauge field, and a conventionally ordered phase with spontaneously broken XY symmetry, in which the fermions acquire a band gap and the gauge field confines. The critical point can hence be understood as an example of a continuous confinement transition in $2+1$ space-time dimensions.

Using a finite-size scaling analysis of our QMC data, we have demonstrated that this transition evades the usual quantum-to-classical mapping. We have also shown that the numerical data is inconsistent with the pure Gross-Neveu-XY universality class. Instead, the data is consistent with our predictions for a novel universality class, dubbed QED₃-GN-XY, which includes besides the fluctuating fermionic degrees of freedom also the effects of the coupling to the U(1) gauge field.

We have shown that the corresponding QED₃-GN-XY field theory can be studied within a large-$N_{\mathrm{f}}$ expansion in fixed dimension, or alternatively within an $\epsilon$ expansion in $D=4-\epsilon$ space-time dimensions. Using these expansion schemes, we have computed estimates for the characteristic exponents that describe the QED₃-GN-XY universality class, including the order-parameter anomalous dimension $\eta_{\phi}$ and the correlation-length exponent $\nu$, as well as the scaling dimensions of the spin and dimer correlation functions. The accuracy of these estimates increases for larger flavor number, and should be reasonable for our simulations of the cases $N_{\mathrm{f}}=6$ and $8$. (E.g., in the pure Gross-Neveu-XY universality class, the accuracy of the leading-order $1/N_{\mathrm{f}}$ results is expected to be of the order of $\lesssim 3\%$ for $N_{\mathrm{f}}=8$ Gracey:2018qba ; PhysRevD.96.096010 .) For $N_{\mathrm{f}}=4$, the corrections may be more significant. This could be one of the reasons for the reduced quality of the data collapse in this case. Improving on this would require to extend our analytical calculations to the next order in the large-$N_{\mathrm{f}}$ expansion (e.g., along the lines of the calculation in Ref. PhysRevD.98.085012 for the QED₃-Gross-Neveu-Ising case) and/or to a higher order in the $4-\epsilon$ expansion (e.g., along the lines of Refs. Ihrig2018 ; PhysRevB.98.165125 ). On the numerical side, refined quantitative predictions for the critical exponents and universal scaling dimensions require simulations on significantly larger lattices. Besides an improved characterization of the universality class, such simulations could also facilitate a search of the predicted emergent second length scale $\xi^{\prime}$, associated with the proliferation of monopoles on the VBS side of the transition.