PHYSICAL REVIEW X 9, 021022 (2019)
Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: The Parent State of Quantum Phases
– † ‡ Xiao Yan Xu,1,* Yang Qi,2 4, Long Zhang,5 Fakher F. Assaad,6 Cenke Xu,7 and Zi Yang Meng8,9,10,11, 1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China 2Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, China 3State Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China 4Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China 5Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 6Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074, Würzburg, Germany 7Department of Physics, University of California, Santa Barbara, California 93106, USA 8Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 9Department of Physics, The University of Hong Kong, China 10CAS Center of Excellence in Topological Quantum Computation and School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 11Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
(Received 1 August 2018; revised manuscript received 19 March 2019; published 2 May 2019)
The interplay between lattice gauge theories and fermionic matter accounts for fundamental physical phenomena ranging from the deconfinement of quarks in particle physics to quantum spin liquid with fractionalized anyons and emergent gauge structures in condensed matter physics. However, except for certain limits (for instance, a large number of flavors of matter fields), analytical methods can provide few concrete results. Here we show that the problem of compact Uð1Þ lattice gauge theory coupled to fermionic matter in ð2 þ 1ÞD is possible to access via sign-problem-free quantum Monte Carlo simulations. One can hence map out the phase diagram as a function of fermion flavors and the strength of gauge fluctuations. By increasing the coupling constant of the gauge field, gauge confinement in the form of various spontaneous-symmetry- breaking phases such as the valence-bond solid (VBS) and N´eel antiferromagnet emerge. Deconfined phases with algebraic spin and VBS correlation functions are also observed. Such deconfined phases are incarnations of exotic states of matter, i.e., the algebraic spin liquid, which is generally viewed as the parent state of various quantum phases. The phase transitions between the deconfined and confined phases, as well as that between the different confined phases provide various manifestations of deconfined quantum criticality. In particular, for four flavors Nf ¼ 4, our data suggest a continuous quantum phase transition between the VBS and N´eel order. We also provide preliminary theoretical analysis for these quantum phase transitions.
DOI: 10.1103/PhysRevX.9.021022 Subject Areas: Condensed Matter Physics, Particles and Fields
I. INTRODUCTION several decades [1–15]. This is because gauge fields coupled to matter fields is a fundamental concept in many The interplay between lattice gauge theories and fer- areas of physics. For example, in condensed matter, mionic matter has allured the imagination of physicists for ð2 þ 1ÞD field theories with a compact Uð1Þ gauge field coupled to gapless relativistic fermions often serve as the *[email protected] low-energy effective field theories in 2D strongly correlated † [email protected] electron systems including cuprate superconductors [1,2, ‡ [email protected] 5–7] and quantum spin liquids [11–14,16]. In high-energy physics, the mechanism of quark confinement in gauge Published by the American Physical Society under the terms of theories with dynamical fermions such as quantum the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to chromodynamics (QCD) is among the most elusive sub- the author(s) and the published article’s title, journal citation, jects, and the absence or presence of a deconfined phase in and DOI. 3D compact quantum electrodynamics (cQED3) coupled to
2160-3308=19=9(2)=021022(17) 021022-1 Published by the American Physical Society XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019)
(not necessarily large) Nf massless fermions has attracted a flavors of fermionic matter is particularly important to lot of attention [4,10–15] and remains unsolved to this day. condensed matter physics because these cases actually In recent years, collective efforts from both condensed correspond to the low-energy field theory description of matter and high-energy physics have started to generate many interesting strongly correlated electron systems, and promising outcomes [4,12,17–26]. There exist concrete therefore host the potential promise of establishing the new examples, by now, of discrete Z2 gauge field theories paradigms in condensed matter physics. Furthermore, coupled to fermionic matter at ð2 þ 1ÞD, deconfined phase perturbative renormalization-group calculations to higher with fractionalized fermionic excitations at weak gauge orders have recently been carried out in attempt to acquire fluctuation, as well as symmetry-breaking phase with the critical properties of the deconfinement-to-confinement gapped fermionic excitations at strong gauge fluctuation transition in the form of QED3-Gross-Neveu universality have been observed [18–20]. The apparently continuous classes [38–41]. transition between the deconfined and confined phases is Based on these considerations, in this work, we succeed highly nontrivial [20], as it is driven by the condensation of in performing large-scale quantum Monte Carlo (QMC) emergent fractionalized excitations and is hence beyond the simulations on the cQED3 coupled to Nf flavor of fermions scope of the Landau-Ginzburg-Wilson paradigm of critical and eventually map out the phase diagram (Fig. 1) in the phenomena in which symmetry breaking is described by a fermion flavor and gauge field fluctuations strength plane. local order parameter. Deconfined phases—Uð1Þ deconfined phase (Uð1ÞD here- The cQED3 is the simplest theory to discuss confinement after) to be more precise—are indeed found in the phase – and chiral symmetry breaking [27 29]. The pure cQED3 diagram for Nf ¼ 6 and 8, and even at Nf ¼ 2 and 4 there without a matter field is known to be always confining are very positive signatures of their existence. Various [10,27,30,31]. However, when there is fermionic matter, confined phases in the form of different symmetry break- the gapless fermionic fluctuations may drive the system ings, such as antiferromagnetic order (AFM) and valence- towards deconfinement. The large-Nf limit of cQED3 bond solid (VBS), are also discovered. Interesting quantum with fermionic matter is believed to belong to this case phase transitions between deconfined and confined phases [11,13,15,32], but the existence of the deconfined phase for and between different confined phases [18,42–46] are small Nf is still under debate [12,17,33–36]. Analytically, revealed as well. the perturbative calculation at small Nf is uncontrolled. For the sake of a smoother narrative, the rest of the paper Numerically, recent hybrid Monte Carlo (HMC) calcula- is organized in the following order. In Secs. II A and II B, tions in Ref. [17] faced difficulties caused by fermion zero we first start with a quantum rotor model coupled to modes. Though these difficulties may be cured by turning fermions, which can be formulated as cQED3 coupled to on a four-fermion interaction term, the scaling dimension of fermionic matter. Then in Sec. II C, we discuss the sign the four-fermion interaction will receive corrections from structure of this model, where we find that a pseudounitary gauge fluctuation at the order of 1=Nf, which may lead to a group can be used to avoid the phase problem at odd Nf relevant runaway flow in renormalization group (RG) and the sign problem at even Nf. In Secs. II D and II E,we calculation [37]. Thus, a combined RG flow of monopole explain the challenges in the QMC simulation even without and four-fermion interactions may be complicated, and a the sign problem and provide our solution with a fast deconfined phase could still exist in the phase diagram but update algorithm for simulating gauge fields with continu- evades the previous study of HMC. cQED3 with finite Nf ous symmetries. In Sec. III, we discuss the whole phase
(a)J (b) (c) c or c † 2 i i U(1)D AFM i c † te b i 4 U(1)D VBS AFM 6 U(1)D VBS cj Nf 8 U(1)D VBS
FIG. 1. (a) Phase diagram spanned by the Fermi flavors Nf and the strength of gauge field fluctuations J of the model shown in (b). Uð1ÞD stands for the Uð1Þ deconfined phase where the fermions dynamically form a Dirac system. This phase corresponds to the algebraic spin liquid where all correlation functions show slow power-law decay. VBS stands for valence-bond-solid phase and AFM stands for the antiferromagnetic long-range ordered phase (N´eel phase). (b) Sketch of the model of Eq. (1). The yellow circles represent the gauge field attached to each fermion hopping, and the yellow dashed lines stand for the flux term per plaquette. (c) The gauge- invariant propagator for fermions with a string of gauge fields attached.
021022-2 MONTE CARLO STUDY OF LATTICE COMPACT QUANTUM … PHYS. REV. X 9, 021022 (2019) X † ϕ diagram and then focus on the physical properties and ψ ∂ − μ δ − i ij ψ LF ¼ iα½ð τ Þ ij te jα þ H:c:; understanding of the Uð1ÞD phase, in particular, the reason hijiα for it being the parent state of various quantum phases, the 4 X 1 − ϕ τ 1 − ϕ τ deconfinement-confinement phase transitions, and VBS-to- Lϕ ¼ 2 f cos½ ijð þ Þ ijð Þ g JNfΔτ AFM phase transition at Nf ¼ 4. Preliminary theoretical hiji analysis of these transitions is also given in Sec. III. Finally, 1 X ϕ the discussion and conclusions are given in Sec. IV. þ 2 KNf cos ðcurl Þ; ð3Þ □
II. MODEL AND METHOD respectively, where μ is set to zero for the half-filled case. β ¼ð1=TÞ is the inverse temperature. The model in Eq. (1) A. Rotor model with fermions is now explicitly formulated as (unconstrained) cQED3 The system we are interested in can be most conveniently coupled to fermionic matter [11,13,17]. We now consider formulated as a 2D quantum rotor model coupled to the symmetries of the model and show that the Gauss law is fermions with Hamiltonian dynamically imposed in the low-temperature limit.
X X 1 1 2 † θˆ B. Symmetries and limiting cases ˆ − ˆ i ij ˆ H ¼ 2 JNf 4 Lij t ðciαe cjα þ H:c:Þ hi;ji hi;jiα Our model [see Eq. (1)] has global and local symmetries. 1 X It enjoys a manifest global SUðNfÞ spin symmetry as well θˆ þ 2 KNf cosðcurl Þ; ð1Þ as a particle-hole symmetry: □ ˆ −1 ˆ† ˆ ¯ −1 i ˆ P zciαP ¼ zð Þ ciα: ð4Þ ˆ ˆ where Lij and θij are canonical angular momentum, ˆ θˆ θˆ In the above, z is a complex number that makes it clear that ½L ;e i ij ¼ e i ij , and its coordinate operator of rotors ij the particle-hole symmetry is antiunitary, and ð−1Þi takes on each bond b ¼hiji of a 2D square lattice, as depicted in the value 1 (−1) on sublattice A (B). Fig. 1(b). The fermion flavor α runs from 1 to N , and the f The local Uð1Þ gauge transformation fermions are minimally coupled to the rotor via nearest- neighbor hopping on the square lattice. The flux term with φ ˆ → ˆ i i θˆ → θˆ φ − φ K>0 favors π flux in each elementary plaquette □, where ciα ciαe ; ij ij þ i j ð5Þ □ the magneticP flux of each plaquette is defined as θˆ θˆ θˆ is an invariant. The generator of this local symmetry curl ¼ b∈□ b, and the summation over b is taken in either clockwise or anticlockwise orientation around an corresponds to a local conserved charge (Gauss law) elementary plaquette. X X ˆ − ˆ ˆ† ˆ − 1 2 For the Monte Carlo simulations, it is convenient to Qi ¼ Lij þ ðciαciα = Þð6Þ ˆ j α work in a representation where θij is diagonal. That is, omitting the bond index θˆjϕi¼ϕjϕi with ϕ ∈ ½0; 2πÞ.In with ½Qˆ ; Hˆ ¼0. In our simulations, we sample over all Qˆ this representation, Lˆ ¼ −i½∂=ð∂ϕÞ with eigenvectors i i ˆ ˆ ϕ ϕ sectors, such that our Hamiltonian corresponds to an Lhϕjli ≡ Lei l ¼ lei l and l ∈ Z. With theseR defini- 2π ϕ ϕ unconstrained gauge theory. As a consequence, the corre- tions, theP resolution of unity reads 0 j ih j¼ lation functions of gauge-dependent quantities such as the 1 2π 1ˆ ½ =ð Þ l jlihlj¼ . To formulate the path integral, we single-particle operator are local in space but not in time: −Δτ ˆ 2 8 † † have to estimate the matrix element: ϕ0 e JNfL = ϕ .To ˆ τ ˆ δ ˆ τ ˆ h j j i hciαð Þcjαi¼ i;jhciαð Þciαi. Below, we argue that the this end, we insert resolution of the unit operator and use Gauss law constraint is dynamically imposed in the the Poisson summation formula to obtain zero-temperature limit. ˆ At J ¼ ∞, Lij vanishes and charges are completely −ΔτJN Lˆ 2=8 − 4= ΔτJN 1−cos ϕ−ϕ0 localized since hopping on a given bond involves excita- hϕ0je f jϕi ∼ e ½ ð fÞ ½ ð Þ ; ð2Þ tions of the rotor mode. In this limit, charge configurations ˆ corresponding to specific values of Qi are degenerate, and where the Villain approximation is used. With the at any finite value of J the degeneracy will be lifted. The above, the Hamiltonian in Eq. (1) can be formulated in dynamical generation of the term Ra coherent-state path integral with action S ¼ SF þ Sϕ ¼ X β ˆ ˆ ˆ 0 dτðLF þ LϕÞ, and the Lagrangian for the fermion and HQ ¼ Ki;jQiQj þ ð7Þ gauge field parts is i;j
021022-3 XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019) Z Z accounts for the lifting of this degeneracy. Note that since − − − ¯ ðSϕþSFÞ Sϕ SF ˆ ˆ Z ¼ Dðϕ; ψ; ψÞe ¼ Dϕe Trψ ½e : ð9Þ fP; Qig¼0, the terms containing products of odd numbers ˆ ’ of Qi s are forbidden. The above equation defines a R β classical model with a finite-temperature Kosterlitz- As the action of the gauge field part Sϕ ¼ 0 dτ Lϕ with Lϕ ˆ shown in Eq. (3) is always real (thus, its exponential is Thouless transition. At zero temperature, the Qi are frozen in a given pattern, and the Gauss law is imposed. always positive), the sign structure of the Monte Carlo For J → ∞, our model maps onto an SUðNfÞ quantum weight will come only from the trace over fermions. To antiferromagnetic. We again start from the J ¼ ∞ degen- trace out fermions, we first discretize the imaginary time τ Δτ 1 2 … erate case and consider t in second-order degenerate ¼ z (z ¼ ; ; ;Lτ) where Lτ is the total number of Δτ β perturbation theory. As we mention above, hopping of a time slices (Lτ ¼ ). Then, performing the fermion fermion with flavor index α from site i to nearest-neighbor trace, we have site j leaves the rotor in an excited state associated with Lτ Y Nf energy cost J. The only way to remove this excitation is for −S Trψ ½e F ¼ det I þ B ; ð10Þ a fermion with flavor index α0 to hop back from site j to z z¼1 site i. These processes are encoded in the SUðN Þ f R Heisenberg model β with SF ¼ 0 dτ LF and LF shown in Eq. (3). After the −Δτ β Vz 2 X discretization of , Bz ¼ e with Vz the coupling ˆ t ˆ † ˆ ˆ ˆ † H →∞ ∝− ðD D þ D D Þð8Þ matrix for each fermion flavor [we have Nf in total; J J ij ij ij ij hiji therefore, there is power Nf in Eq. (10)], which has only elements connecting the sites between different sublattices P ϕ N † − i ij ˆ f ˆ ˆ of the square lattice ðVzÞij ¼ te . We recognize that with Dij ¼ α¼1 ciαcjα. In our simulations, we have on such kinds of B matrices form a pseudounitary group average Nf=2 fermions per site such that the representation z SUðn; nÞ where 2n is a dimension of Bz (total number of of the SUðNfÞ group corresponds to the antisymmetric self- sites). As proved in the Appendix B, ∀ D ∈ SUðn; nÞ, adjoint representation (i.e., the Young tableau correspond- 1 ∈ ℜ ing to a column of N =2 boxes) [47]. On the square lattice detð þ DÞ holds. Therefore, the fermion weight will f always be real for all integers N and, most importantly, be and for even values of N where the negative sign problem f f semi-positive-definite for all even values of N such that is absent, this model has been considered in former f QMC simulations can be performed. Although the current auxiliary field QMC simulations [12].AtN ¼ 2, one f paper focuses only on the compact Uð1Þ gauge fields, the finds an antiferromagnetic state, and at and beyond N ¼ 6, f SUðn; nÞ group actually allows one to add extra non- a VBS state. In the large-Nf limit, we recover the Marston- Hermitian terms to the model, such as a staggered imagi- Affleck [2] saddle point accounting for dimerization. At nary chemical potential, that are also sign-problem-free for Nf ¼ 4 and in the absence of charge fluctuations, Ref. [12] even values of Nf. It will be very interesting to study the finds no compelling evidence of VBS and AFM orders non-Hermitian models and their properties in the presence 24 24 when considering lattices up to × . On the other hand, of gauge field fluctuations and electronic interactions in simulations of the corresponding SU(4) Hubbard model future investigations. [48] are consistent with an AFM state in the large-U limit albeit with decreasing value of the order parameter as a function of U. In our simulations, charge fluctuations are D. Difficulties of the QMC simulation present, and the phase diagram is consistent with AFM Although there is no sign problem for even Nf, the order in the large-J limit. simulation of Eq. (1) is by no means simple. Earlier ˆ At J ¼ 0, θij becomes a classical variable in the sense attempts in the high-energy community have been devoted that it has no imaginary time dynamics. Even in the absence to simulate similar models by means of the hybrid of the flux term (K term) the coupling to the fermions Monte Carlo [4,17,35,50] with (dynamical) mass term. favors, according to Lieb’s theorem [49],aπ flux per Mass terms are essential to avoid divergences when plaquette, with associated dynamically generated Dirac calculating forces in the realm of Hamiltonian [51] and dispersion relation of the cˆ fermions. The fate of this state Langevin [52] dynamics. Mass terms, however, introduce biases the severity of which have to be a posteriori at low values of Nf and when gauge fluctuations are accounted for is one of the central aims of our research. clarified. On the other hand, in the condensed matter community, the determinantal QMC (DQMC) is more popular, and it C. Absence of the sign problem for even Nf usually uses local updates and the mass terms are not To simulate the above model with the quantum essential here [53–56]. However, as far as we are aware, Monte Carlo method, we start with the partition function there exists no DQMC simulation on cQED3 coupled to
021022-4 MONTE CARLO STUDY OF LATTICE COMPACT QUANTUM … PHYS. REV. X 9, 021022 (2019) fermionic matter, and our work hence serves as a first YNb attempt. As we explain below, to be able to simulate the Bz ¼ Bz;b; ð12Þ 1 model in Eq. (1), there are several obstacles one needs to b¼ hz;b overcome. where Bz;b ¼ e and hz;b is an N × N matrix with only The most obvious obstacle is the computational com- nonzero elements Bz;bði; jÞ¼Bz;bðj; iÞ ¼ fðϕijÞ where plexity. For general models, the computation complexity θ ϕ fðϕ Þ¼ρðϕ Þei ð ijÞ is the complex function of auxiliary O β 4 ij ij of DQMC for one sweep is ð N Þ (here, N is the total field ϕ in polar form. It is obvious that only when number of sites) for models where a fast update is ij θðϕ Þ¼0 or π, B ði; jÞ will be diagonalized by an applicable, the complexity can be reduced to OðβN3Þ. ij z;b auxiliary-field-independent unitary transformation, and The most common example is the Hubbard model with then the aforementioned fast update scheme applies. on-site interaction; when we flip a single auxiliary field at There are many known models that belong to this case, time slice z and site i, it changes only one diagonal value in such as models with Heisenberg-type interaction [57–59], the Bz matrix, then the new equal-time Green’s function models with the Z2 (bosonic) gauge field coupled to G0ðτ; τÞ can be calculated as fermionic matter [18,19,60,61], etc. Novel physics has 0 … Δ … −1 been found in these models; e.g., in Ref. [58] a continuous G ¼½I þ B1B2; ;Bz−1ðI þ ÞBz; ;BLτ−1BLτ phase transition with fermion mass generation without Δ − −1 ¼ G½I þ ðI GÞ ; ð11Þ spontaneously breaking any symmetry was identified where G is the Green’s function at the previous step. (similar physics was also found in the lattice QCD community [62–65]). Unfortunately, our model in As Δii is the only nonzero element in the N × N matrix Δ, Δ 1 − Eq. (1) involves Uð1Þ gauge fields, and the auxiliary field ð GÞ can be written as the outer product of two ϕ θ ϕ ϕ vectors, and then the Sherman-Morrison formula can be ij is therefore a continuous variable and ð ijÞ¼ ij; used to reduce the complexity of calculating Eq. (11) from thus, our model does not belong to the cases discussed OðN3Þ to OðN2Þ [53,54]. Such an update scheme is above, and naively the fast update cannot be applied. referred as the fast update. For one sweep over all auxiliary U fields [scales as OðβNÞ], the scaling will be OðβN3) instead E. Fast update algorithm designed for ð1Þ gauge fields of OðβN4Þ. For models with off-site interaction, e.g., in our There is indeed an alternative way to design a fast update case due to the coupling between the fermion and gauge algorithm for the model in Eq. (1). It is based on the field, usually we need to make a further Suzuki-Trotter Woodbury matrix identity that effectively generalizes the decomposition over all bonds (assume bond b connects site Sherman-Morrison formula to higher-rank matrices. We 0 i and j, and the total number of bonds is Nb), and the Bz recognize that the new Bz;b after a single update can directly matrix is written as be factorized into ð1 þ ΔÞBz;b with
0 0 −iðϕij−ϕ Þ 2 iϕ iϕ Δii Δij ð1 − e ij Þsinh Δτ ð−e ij þ e ij Þ sinh Δτ cosh Δτ ¼ ; ð13Þ Δ Δ − ϕ −iϕ0 iðϕ −ϕ0 Þ 2 ji jj ð−e i ij þ e ij Þ sinh Δτ cosh Δτ ð1 − e ij ij Þsinh Δτ and other elements of the N × N matrix Δ are zero. With the discussion of QMC results, we first introduce the QMC such kind of structure and note that Eq. (11) still holds, observables that are used to characterize the symmetric and Δð1 − GÞ can be written as product of two matrices with symmetry-breaking phases and their phase transitions. dimension N × 2 and 2 × N. Therefore, we can use the Since physical observables are Hermitian, we construct generalized version of the Sherman-Morrison formula (the and measure various gauge-invariant structure factors, Woodbury matrix identity) to calculate Eq. (11), which also including spin χSðkÞ and dimer χDðkÞ structure factors. 2 has complexity OðN Þ. They are defined as With such a specially designed fast update, we are now X X 1 α β k r −r ready to simulate cQED3 coupled to fermionic matter χ k i ·ð i jÞ Sð Þ¼ 4 hSβðiÞSαðjÞie ; ð14Þ without an artificial mass term and still enjoy the L ij αβ OðβN3Þ computational complexity. X 1 k r −r χ ðkÞ¼ ðhD D i − hD ihD iÞei ·ð i jÞ; ð15Þ III. RESULTS D 4 i j i j L ij A. Physical observables α † where the spin operator S ðiÞ¼c c β −½1=ðN Þ δαβ Our model and major results are schematically summa- P β Piα i f † α β ˆ rized in Figs. 1(a) and 1(b), respectively, but before starting γ ciγciγ, and the dimer operator Di ¼ αβ SβðiÞSαðiþxÞ
021022-5 XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019) is defined as the dimer along the nearest-neighbor bond in into various symmetry-breaking phases. Therefore, the the xˆ direction. algebraic spin liquid phase, i.e., Uð1ÞD discovered in this From these structure factors, one can further construct work, actually serves as the original state of many interest- dimensionless quantities—the correlation ratio [66]—to ing quantum phases, hence, dubbed the “parent state of determine the precise position of the phase transitions. If quantum phases.” The discovery of such a deconfined one would like to detect the transition towards antiferro- phase is the most important result of this work. magnetic long-range order, the antiferromagnetic correla- As J increases, the system goes through deconfined- tion ratio is confined phase transitions to various symmetry-breaking phases. In the case of Nf ¼ 2, the symmetry-breaking χ X δq 1 − Sð þ Þ phase is the AFM (N´eel) phase, whereas in the case of rAFM ¼ χ X ; ð16Þ Sð Þ Nf ¼ 4, the symmetry-breaking phases are the VBS and AFM phases. In addition, for further increases of N , the X π π f where ¼ð ; Þ is the order wave vector for the AFM on symmetry-breaking phases are solely VBS. According to δq 2π 0 the square lattice, and ¼ f½ð Þ=L ; g is the smallest Ref. [20], the deconfined-confined phase transition could X momentum away from . In the same vein, we define the be a version of the deconfined quantum critical point with correlation ratio for the VBS order from the dimer structure emergent continuous symmetry, and in addition to the factor QMC data, we also provide a preliminary field theoretical χ M δq description of this transition. The transition from the VBS 1 − Dð þ Þ to AFM phases inside the confined regime at N ¼ 4,ifitis rVBS ¼ χ M ; ð17Þ f Dð Þ indeed continuous as our data suggest, is also a deconfined quantum critical point [18,42–46,68,69] whose theory is where M π; 0 is the order wave vector for the VBS. ¼ð Þ explained later. Other quantities, such as the energy density and various Overall, the presence of the Uð1ÞD deconfined phase and correlation functions (spin, dimer) in real space, are also the phase transitions between the deconfined to confined measured in the QMC simulation. phases and within the confined phases, are all intriguing and show the rich physics behind the simple model B. Phase diagram of Eq. (1). Now we can discuss the results from the QMC simu- Below, we discuss some exotic aspects of the phases and lation of Eq. (1). Starting with the final phase diagram phase transitions we obtain. that schematically summarizes all the data, as shown in Fig. 1(a), the phase diagram is spanned along the axes of C. Uð1ÞD phase and confinement transition N and J. We set K ¼ t ¼ 1 as the energy unit and choose f 1. N =2 the gauge fluctuation strength J as the tuning parameter to f 2 study different Nf cases. For each Nf, there are different First we focus on the case Nf ¼ . In the static limit phases and phase transitions, but there are similarities for (J → 0), the gauge fields are frozen into a π-flux pattern per all Nf investigated; that is, at small J,Uð1Þ deconfined plaquette [49,70] that results in a Dirac gapless dispersion phases [Uð1ÞD in Fig. 1(a)] are universally present in the relation. At finite J, the gauge fields fluctuate. and pro- phase diagram. This finding is highly nontrivial, as we liferation of monopoles of the gauge fields may drive explain in the Introduction (Sec. I), from both the high- spinons (the fermions) to confine. On the other hand, at energy physics and condensed matter physics communities, large values of J,anSUðNfÞ antiferromagnetic effective the existence of such a deconfined phase in cQED3 is still low-energy model in the self-adjoint antisymmetric repre- under debate due to the lack of controlled calculation at sentation emerges (see Sec. II B). At Nf ¼ 2, the ground finite and small Nf [4–15]. Our finding presented here state of this model is known to host an AFM. provides the first set of concrete evidence to support the Figure 2(a) shows the AFM correlation ratio rAFM as a existence of this phase. function of J for different system sizes. Figure 2(b) shows Moreover, as we further elucidate in later sections, such the crossing points of pairs of adjacent system sizes, and a a deconfined phase is expected to be the algebraic spin power-law extrapolation in 1=L gives rise to an estimate of liquid [6–9,13,14,67], in which critical correlations of the confinement transition in the thermodynamic limit: many competing order parameters, such as antiferromag- Jc ¼ 1.6ð2Þ. Since the correlation ratio shows no abrupt netic order, valence-bond-solid order, charge-density wave, features, the transition from the deconfined phase to the and superconductivity, coexist and share the same power- AFM is more likely to be continuous. This is consistent law decay due to the Uð1Þ gauge deconfinement and the with the flux energy per plaquette data presented in Fig. 13 subsequential conformally invariant, interacting fixed point in Appendix D. For the same J values as considered in [13,14]. Starting from the algebraic spin liquid phase, one Fig. 2, the flux energy per plaquette does not develop a can easily apply various perturbations and drive the system sharp change of slope.
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(a) (b)
FIG. 3. Photon mass m measured by the flux correlationP CðτÞ¼ hθðτ0Þθðτ0 þ τÞi ∼ expð−mτÞ with θðτÞ¼ □ sin ½curlϕðτÞ . FIG. 2. (a) The antiferromagnetic correlation ratio through the Zero photon mass is observed in the Uð1ÞD phase; finite photon Uð1ÞD-to-AFM transition at Nf ¼ 2. Here, β ¼ 4L, Δτ ¼ 0.2. mass is observed in the confined phase. Here we plot the The crossing points are the transition points separating the Nf ¼ 2 case. deconfined phase and N´eel phase. (b) The 1=L extrapolation of the crossings estimates the Uð1ÞD-to-AFM transition point algebraic spin liquid. It has the unique property that as a Jc ¼ 1.6ð2Þ for Nf ¼ 2. deconfined state emerging from competing orders, the correlation functions of these competing orders, such as The simplest way to detect the deconfinement- spin-spin, dimer-dimer, and bond-bond, have the same confinement transition may be the Wilson loops, but it power-law decay. If the data in Fig. 4 were deep inside the is known that in the presence of matter field, it cannot be confined phase, the decay of spin-spin and dimer-dimer used to detect the topological order of the deconfined phase (the Uð1ÞD phase); such effects have been dis- cussed in the literature [71,72]. One suitable way to (a) demonstrate the deconfinement-confinement transition here is to show how the photon mass changes over J. As soon as the matter fields bind to form the particle-hole condensate, we expect monopoles to proliferate and to generate a photon mass. The photon mass can be measured by the correlationP of flux quantity θðτÞ [73], which is defined as θðτÞ¼ □ sin ½curlϕðτÞ . The photon mass m isrelatedtothecorrelationofθðτÞ by CðτÞ¼hθðτ0Þθðτ0 þ τÞi ∼ expð−mτÞ. Figure 3 plots the estimated photon mass for different system sizes. We find (b) a signature of an absence of photon mass in the Uð1ÞD phase and a growth of the photon mass in the AFM phase as J increases. However, we want to point out that due to finite-size effects, i.e., uncertainties in extracting the exponential decay in θðτÞ close to the transition, the estimation of photon mass near the phase transition is more qualitative than quantitative. To further understand the properties of the deconfined phase, we measure the real-space correlation functions in the Uð1ÞD phase (at J ¼ 1.25 021022-7 XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019) correlations will be very different. For example, in the N´eel (a) phase, spin-spin will decay to a constant value and dimer- dimer correlation will decay exponentially. Therefore, our data in Fig. 4 provide supporting evidence of the algebraic spin liquid behavior of the Uð1ÞD in Fig. 1 at Nf ¼ 2. 2. Nf =4 Next we turn to the Nf ¼ 4 case where we also observe a Uð1ÞD phase at small J. As we show in Fig. 5(a), we can follow the crossing points of the correlation ratio of the (b) VBS order parameter for different system sizes so as to extract [see Fig. 5(b)] Jc1 ¼ 1.2ð3Þ. The data are consistent with a continuous phase transition from the deconfined phase to the VBS phase. Furthermore, the flux energy per plaquette also supports a continuous transition. Figure 6 depicts the real-space decay of the spin-spin and dimer-dimer correlation functions in the Uð1ÞD phase for different system sizes. Again, they show similar power-law decay, and the power is estimated to be 2ΔS ¼ 3.6ð3Þ for the spin-spin correlation and 2ΔD ¼ 3.5ð3Þ for the dimer- dimer correlation. This power law is faster than at N ¼ 2 f FIG. 6. The log-log plot of real-space decay of (a) spin and is hence consistent with the large-Nf prediction correlation functions and (b) dimer correlation functions for [7,13,14,67]. Nf ¼ 4 in the Uð1ÞD phase (at J ¼ 1.00 4. Theory for confinement transition We find a Uð1ÞD-to-AFM phase transition for the Nf ¼ 2 case and a Uð1ÞD-to-VBS phase transition for (a) (b) the Nf ¼ 4, 6, and 8 cases. These phase transitions should belong to the QED3-Gross-Neveu Oð3Þ or XY transitions [38] depending on the order parameters in the confined FIG. 5. The VBS correlation ratio through the Uð1ÞD-to-VBS phases. For example, at least with large enough (but still finite) N when the higher-order fermion interactions are transition at Nf ¼ 4. Here, β ¼ 3L, Δτ ¼ 0.2. (b) The 1=L f extrapolation of the crossings estimates the Uð1ÞD-to-VBS clearly irrelevant, the transition between the Uð1ÞDtoVBS transition point Jc1 ¼ 1.2ð3Þ for Nf ¼ 4. phase can be described by the following action: 021022-8 MONTE CARLO STUDY OF LATTICE COMPACT QUANTUM … PHYS. REV. X 9, 021022 (2019) (a) 0.64 L=8 10 12 0.62 14 16 0.6 AFM r 40 30 0.58 2 c J 20 10 0 0.56 0 0.1 0.2 1/L 6 8 10 12 14 16 18 20 22 24 J (b) 1 40 30 0.8 2 c FIG. 7. Dimension of spin (gren circles) and dimer (blue J 20 10 squares) in the Uð1ÞD phase as a function of Nf. The dashed 0 red line corresponds to the 1=Nf perturbative calculation, 0.6 0 0.1 0.2 2 1/L 1 þ η ¼ 4 − ½64=ð3π NfÞ , taken from Ref. [7]. Note here Nf VBS r corresponds to the number of four-component Dirac fermions. 0.4 L=6 8 Z 2 10 XNf 0.2 2 12 S ¼ d xdτ ψ¯ jγ · ð∂ − iaÞψ j þ uϕ · ψμψ¯ 14 16 j¼1 0 6 8 10 12 14 16 18 20 22 24 2 2 4 þj∂ϕj þ rjϕj þ gjϕj ; ð19Þ J (c) 4 L=6 T 8 where ψ ψ 1; …; ψ 2 has 2N components, and ϕ is 3.5 ¼ð Nf Þ f 10 12 an Oð2Þ vector in which the VBS order parameter is 3 x y 14 embedded. μ ¼ðμ ; μ Þ are two 2Nf × 2Nf matrices in 16 2.5 the fermion flavor space. ψμ¯ xψ, ψμ¯ yψ are two fermion mass operators that correspond to the VBS in the x and y 2 AFM-VBS 40 0 ϕ r directions, respectively. When r> , is gapped out, and 1.5 30 2 1 c the system is in the Uð ÞD phase due to the screening of J 20 1 massless fermions to the gauge field. When r<0, ϕ 10 0 condenses, the fermions are gapped out, then the compact 0.5 0 0.1 0.2 1/L gauge field is in the confined phase. In the Uð1ÞD phase, 0 6 8 10 12 14 16 18 20 22 24 the VBS and antiferromagnetic order parameters should J have the same scaling dimension due to the enlarged 2 SUð NfÞ symmetry in the low-energy field theory, con- FIG. 8. (a) Antiferromagnetic correlation ratio rAFM for 4 β 3 Δτ 0 2 1 sistent with our data in Figs. 4, 6, 10, and 11, while at the Nf ¼ . Here, ¼ L, ¼ . . Insets show the =L extrapo- 18 3 critical point r ¼ 0, these two order parameters still have lation of the crossing point in rAFM and Jc2 ¼ ð Þ. (b) VBS 4 1 power-law correlation functions but with different scaling correlation ratio rVBS for Nf ¼ . The inset is the =L extrapo- 19 5 lation of the crossing point in rVBS and Jc2 ¼ ð Þ. This is dimensions due to the loss of the SU(2Nf) symmetry in the consistent with Jc2 obtained from rAFM in (a). (c) AFM-VBS infrared. 4 1 correlation ratio rAFM-VBS for Nf ¼ . The inset is the =L 17 4 extrapolation of the crossing point in rAFM-VBS and Jc2 ¼ ð Þ. 5. AFM-VBS transition at Nf =4 4 The situation at Nf ¼ is even more interesting. As we Figure 8(b) is the rVBS correlation ratio, and the crossing further increase J, we observe another quantum phase point in it signifies the vanishing of the VBS order. The transition from the VBS to the AFM phase. This transition inset of Fig. 8(b) gives rise to Jc2 ¼ 19ð5Þ, consistent with is consistently revealed in three steps in Fig. 8. the onset of the AFM order in Fig. 8(a). Figure 8(a) shows the rAFM correlation ratio, and clearly The transition from VBS to AFM deserves more there is a crossing point signifying the establishment of the attention; apparently the data in Fig. 8 suggest a continuous AFM long-range order. The inset shows the 1=L extrapo- transition, and if it were the case, there is then the lation of the crossing point and gives rise to Jc2 ¼ 18ð3Þ. possibility that the critical point will acquire a larger 021022-9 XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019) Z X 2 symmetry group than that in the model in Eq. (1). In the 2 2 S ¼ d xdτ ∂ −ia−i alτl z þrjzj þ ; ð22Þ case of the Z2 gauge field coupled to the fermion, as shown 1 3 in Refs. [18,20], two similar situations with emergent l¼ continuous symmetry are also investigated. In the first l where aμ and aμ are gauge fields corresponding to the case, it is at N ¼ 3 that a continuous VBS-to-AFM phase f Uð1Þ and SU(2) subgroups of Uð2Þ. Note that these gauge transition occurs [18], and in the second case, it is the fields are “emergent” gauge fields, which are different deconfinement-confinement phase transition itself at from the explicit gauge field in our original simulation. Nf ¼ 2 [20]. Our phase transition at Jc2 is closer to the When r<0, zα;a condenses and leads to the antiferro- former. To further understand the nature of the transition magnetic state with ground-state manifold f½Uð4Þ =½Uð2Þ× from VBS to AFM, we plot the ratio of the AFM structure Uð2Þ g. When r>0, zα is gapped out, and the gauge factor and VBS structure factor, ;a fields will be confined. Here we assume that the Uð1Þ compact gauge field aμ still has the quadrumonopole χ X Sð Þ proliferation, which leads to the VBS phase like the original rAFM-VBS ¼ : ð20Þ χDðMÞ deconfined QCP theory for the SU(2) spins [42]. One of the crucial properties of the deconfined QCP is “ ” The results are shown in Fig. 8(c). Indeed, there is a the intertwinement between order parameters on two crossing point in the r , and the 1=L extrapolation sides of the transition, which can be captured by a AFM-VBS topological term which treats the N´eel and VBS order in the inset of Fig. 8(c) gives rise to J 2 ¼ 17ð4Þ, very c parameters on equal footing [77]. In the current case with consistent with the J 2 obtained from the crossings of r c AFM SU(4) spin symmetry, one can also introduce a topological in Fig. 8(a) and rVBS in Fig. 8(b). This transition is similar to the AFM-to-VBS transition in the SU(4) J-Q model [74], term that captures the intertwinement between the SU(4) where a continuous transition is observed. The transition N´eel and VBS orders with a topological term. To do this, in that case can be described by a noncompact CPN−1 we need to embed both the N´eel and VBS order parameters into a larger manifold. One way to parametrize the N´eel description with N ¼ 4 [75], and it is shown that the N †Ω Ω 4 4 monopoles are irrelevant at this fixed point [76]. Therefore, order manifold is ¼ U U, where is a × diagonal Ω 1 −1 a deconfined quantum critical point [42] is realized. matrix ¼ diagð 2×2; 2×2Þ, and U is an SU(4) matrix. N 4 4 N 2 1 However, in Ref. [75], on sublattice A of the lattice, there is a × Hermitian matrix with constraint ¼ . 8 8 P is a fundamental representation of SU(4), while on sub- Now we introduce a larger × matrix which lattice B, there is an antifundamental representation, which includes both the N´eel and VBS order parameters: is different from our case. z x y P ¼ cosðθÞN ⊗ τ þ sinðθÞ14 4 ⊗ ðV τ þ V τ Þ; ð23Þ In our case, with Nf ¼ 4 there is effectively a self- × x y conjugate representation of the SU(4) group on every site; thus, the field theory for the AFM-to-VBS transition is where ðVx;VyÞ is a two-component order parameter for the 2 2 different from Ref. [75]. According to Ref. [47], the AFM VBS phase, and Vx þ Vy ¼ 1. The order parameter P N´eel order in this case has the following Grassmannian unifies the SU(4) N´eel and VBS order parameters, just like ground-state manifold M: the Oð5Þ vector order parameter introduced in Ref. [77]. The topological term that captures the intertwinement Uð4Þ between N´eel and VBS order parameters is a Wess- M ¼ : ð21Þ Zumino-Witten (WZW) term: Uð2Þ ×Uð2Þ Z Z 1 S ∼ 2 τ ϵ P∂ P∂ P∂ P∂ P WZW d xd du μνρσtr½ μ ν ρ σ : ð24Þ To describe this antiferromagnetic state, one can either 0 introduce Nf ¼ 4 flavors of fermionic spinons with half- filling, or introduce two color species of bosonic spinons Using the same technique in Ref. [78], one can show that at zα;a (α ¼ 1; …; 4, a ¼ 1, 2), and couple them to a Uð2Þ the vortex core of the VBS order parameter, there is a gauge field [to describe the simplest N´eel order of SU(2) spinon with self-conjugate representation, which is con- spins, we need only one two component of bosonic sistent with intuition. In fact, the Oð5Þ WZW term spinon coupled with a Uð2Þ gauge field, as in introduced in Ref. [77] can be written in the same form Ref. [42] ]. The Uð2Þ gauge constraint will guarantee as Eq. (24), as long as we replace P in Eq. (24) by a 4 × 4 that on every site there are fixed numbers of spinons, and Hermitian matrix order parameter P ¼ n · Γ, where Γ are the color space is fully antisymmetric; thus, on every site, five Gamma matrices, and n is the Oð5Þ vector introduced the SU(4) spin is automatically in an antisymmetric self- in Ref. [77]. conjugate representation. Then, the field theory for the This topological term can be viewed as the low-energy N´eel-VBS transition is effective field theory of the π-flux state of the SU(4) 021022-10 MONTE CARLO STUDY OF LATTICE COMPACT QUANTUM … PHYS. REV. X 9, 021022 (2019) antiferromagnet, which again is described by a QED3 with calculations. In particular, the QED3-Gross-Neveu Ising eight flavors of Dirac fermions [13], but again the gauge transition has been investigated recently with perturbative field of this QED3 is an emergent gauge field which is RG calculations [38,39]. different from the gauge field introduced in the original Aside from the QED3-Gross-Neveu transitions, we find model that we simulate. The WZW term Eq. (24) can be evidence for a direct and continuous transition between derived by coupling the 8 × 8 matrix order parameter P to the AFM and VBS states in the confined region of the the eight flavors of Dirac fermions of the π-flux state and phase diagram at Nf ¼ 4. Since we have on average two integrate out the fermions following the standard procedure fermions per site, we should consider the antisymmetric of Ref. [79]. self-conjugate representation of the SU(4) group. We Our data, the crossing of rAFM-VBS in Fig. 8(c), suggest present various theoretical descriptions of this putative that the AFM and VBS order parameters have the same deconfined quantum phase transition in terms of multi- scaling dimension at this transition, which is consistent flavored spinons coupled to emergent Uð1Þ and SU(2) with the emergent SU(8) symmetry of the π-flux state of the gauge fields. We also discuss the effective low-energy field SU(4) antiferromagnet. The large SU(8) symmetry, if it theory with a topological term that captures the intertwine- indeed exists at the AFM-VBS transition, will ensure that ment between the magnetic and VBS orders and its many other order parameters have the same scaling connection to the π-flux state of the SU(4) spin system dimension as the AFM and VBS order parameters [13]. discussed in Ref. [13]. Numerical support for emergent These order parameters will also have similar fractionali- symmetries is provided. In the future, measurements of the zation dynamical signatures in their spectral functions as conserved current operators related with such emergent the AFM and VBS order parameters. continuous symmetries [80] can be performed. Finally, the confining phase transitions in our model, as IV. CONCLUSIONS well as the possible deconfined quantum critical point at Nf ¼ 4, will have distinct and very interesting dynamical Using large-scale DQMC, we investigate the compact properties in their spectral functions that can be further Uð1Þ gauge field theory coupled to Dirac fermion matter explored in QMC simulations. Such calculations provide 2 1 fields in ð þ ÞD and variable flavor number Nf, i.e., experimentally accessible signatures of exotic states of cQED3. With our simulations, we map out the entire matter where emergent gauge fields, fractionalized exci- ground-state phase diagram in the flavor Nf and gauge tations, can be traced. Similar attempts have recently been field fluctuation J strength plane. Our results are summa- applied to the deconfined quantum critical point in the rized in Fig. 1(a). pure spin model [69], emergent Z2 spin liquid at ð2 þ 1ÞD Most importantly, signatures supporting stable Uð1ÞD [81],andUð1Þ spin liquid at ð3 þ 1ÞD [82] and the Z2 phases are discovered at Nf ¼ 8 and 6, and evidence of the counterpart of our model [18]. In the present cQED3 Uð1ÞD phase at Nf ¼ 4 and 2 is also found. The properties model, dynamical measurements in the QMC simulation of the deconfined phase are consistent with the proposal of plus state-of-the-art analytical continuation [81,83,84] can algebraic spin liquid, in which various competing orders help reveal more fundamental physical understanding of (AFM order and VBS order, for example) all have algebraic these exotic quantum phase transitions. correlation with identical power laws in real space. The decay power is found to quantitatively converge to the ACKNOWLEDGMENTS large-Nf predictions [7,13,14,67]. The authors thank Lukas Janssen, Ribhu Kaul, Steven The transition between the deconfined and confined Kivelson, Thomas Lang, Srinivas Raghu, Subir Sachdev, phases at various Nf are determined using the RG- Michael Scherer, Yi-Zhuang You, Ashvin Vishwanath, and 2 invariant correlation ratios. At Nf ¼ , the transition Chong Wang for helpful discussions. F. F. A. would like to occurs between the Uð1ÞD and AFM phases. Since the thank T. Grover and Zhenjiu Wang for discussions on related AFM corresponds to Oð3Þ symmetry breaking, the critical projects. X. Y.X., Y.Q., L. Z., and C. X. acknowledge the theory should be described by the QED3-Gross-Neveu hospitality of the International Collaboration Center at 3 Oð Þ universality class. In contrast, at larger values of Nf Institute of Physics, Chinese Academy of Sciences. This the ordered phases correspond to VBS. The dynamical work is finalized during the program “International generation of the two VBS mass terms is described by the Workshop on New Paradigms in Quantum Matter” at the QED3-Gross-Neveu Oð2Þ universality class. As far as we center. X. Y.X. is thankful for the support of Hong Kong know, the QED3-Gross-Neveu Oð2Þ or Oð3Þ transition Research Grants Council (HKRGC) through C6026-16W, have not been investigated numerically before in an 16324216 and 16307117. Y. Q. acknowledges support from unbiased simulation. It is certainly worthwhile to carefully the Ministry of Science and Technology of China under study the critical properties of these transitions via QMC Grant No. 2015CB921700 and from the National Science simulations further and compare with future analytical Foundation of China under Grant No. 11874115. Z. Y.M. 021022-11 XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019) acknowledges support from the National Key R&D Program APPENDIX B: PSEUDOUNITARY GROUP (Grant No. 2016YFA0300502), the Strategic Priority SUðn;mÞ AND THE ABSENCE Research Program of CAS (Grant No. XDB28000000), OF THE SIGN PROBLEM and the National Science Foundation of China (Grants No. As we discuss in the main text, the fermion determinant 11574359 and No. 11674370). F. F. A. thanks the DFG Q I Lτ B n m research unit FOR1807 for financial support. L. Z. is for one flavor is detð nþm þ z¼1 zÞ, where and are supported by the National Key R&D Program of China the numbers of sites in the two sublattices, and Inþm hz (Grant No. 2018YFA0305802) and the Key Research denotes the (n þ m)-dimensional identity matrix. Bz ¼ e , Program of Chinese Academy of Sciences (Grants where hz has the following structure, No. XDPB08-4 and No. XDB28040200). We thank the Center for Quantum Simulation Sciences in the Institute of 0n Tz Physics, Chinese Academy of Sciences, the Tianhe-1A h ¼ ; ðB1Þ z † 0 platform at the National Supercomputer Center in Tianjin, Tz m and the Tianhe-2 platform at the National Supercomputer Center in Guangzhou for their technical support and gen- and Tz is the hopping matrix between different sublattices. erous allocation of CPU time. † Bz matrices satisfy (1) Bz ηBz ¼ η, where η ¼ diagðIn; −ImÞ, and (2) det Bz ¼ 1; thus, their products generate the pseudounitary group SUðn; mÞ. APPENDIX A: CONNECTION TO HIGH-ENERGY Theorem 1.—For any D ∈ SU n; m , cQED ð Þ LATTICE 3 ACTION ∈ R detðInþm þ DÞ . As we discuss in the Sec. II A, after the path integral of Proof.—First, suppose that λ is an eigenvalue of D, † −1 −1 the rotor degrees of freedom in a rotor model with a fermion Dv ¼ λv, then D ηv ¼ λ ηv, and DTηv ¼ðλ Þ ηv ; −1 T in Eq. (1), the action of cQED3 coupled to fermionic matter hence, ðλ Þ is an eigenvalue of D and is thus also an is obtained explicitly. In the high-energy lattice cQED3 eigenvalue of D. λ 1 ≤ ≤ action, the Lagrangian for the pure gauge field part takes Denote the eigenvaluesQ of D by i, i n þ m, then 1 λ the form detðInþm þ DÞ¼ ið þ iÞ. We then treat the eigenval- ues on the unit circle and those not on the unit circle X X separately. For those not on the unit circle, Lϕ ¼ Kτ fcos½ϕijðτ þ1Þ−ϕijðτÞ gþKr cosðcurlϕÞ; □ hiji Y Y Y 2 j1þλ j ð1þλ Þ¼ ð1þλ Þ½1þðλ Þ−1 ¼ i : ðA1Þ i i i λ i i;jλij≠1 i;jλij<1 i;jλij<1 where Kτ < 0 and Kr < 0 with jKτj¼jKrj. Compared ðB2Þ with the Lagrangian defined in Eq. (3), we study the case θ Kτ < 0 and K > 0 with Kτ ≠ K . As we can always i i r j j j rj For those on the unit circle, denoting λi ¼ e , rescale space and time to restore the Lorentz symmetry, the −π < θi ≤ π,wehave difference between jKτj¼jKrj and jKτ ≠ jKrj is trivial. Actually, our model can be exactly mapped to the case of Y Y θ 2 1 λ 2 θ 2 i i= Kr < 0 and the fermion hopping with a staggered phase ð þ iÞ¼ cosð i= Þe : ðB3Þ λ 1 λ 1 factor as follows: i;j ij¼ i;j ij¼ ϕ → ϕ π i;iþxˆ i;iþxˆ þ myðiÞ ; ðA2Þ Therefore, we find Y λ where myðiÞ is 1 (0) if the y coordinate of i is odd (even), 2 2 i ½detðInþm þ DÞ ¼jdetðInþm þ DÞj λ respectively. Therefore, our convention is equivalent to the i j ij high-energy lattice cQED3 action. 2 0 As we mention in the main text, the Dirac fermion in our ¼jdetðInþm þ DÞj > ; ðB4Þ model is realized because the Kr term prefers π flux π ∈ R through each plaquette, and the flux doubles the unit hence, detðInþm þ DÞ . cell. Following the standard literature such as Ref. [13],if This theorem implies that for an even number of flavor we start with Nf flavors of one-component fermions on the fermions, the model is free of the sign problem for any 2 lattice [like Eq. (1)], at low energy there will be Nf flavors hopping matrices Tz. For instance, this is true for both of two-component Dirac fermions. Abelian and non-Abelian gauge fields. 021022-12 MONTE CARLO STUDY OF LATTICE COMPACT QUANTUM … PHYS. REV. X 9, 021022 (2019) (a) It is easy to find an example pffiffiffi − 2 1 D ¼ pffiffiffi ∈ SUðn; mÞðB5Þ 1 − 2 such that detðI þ DÞ < 0. Therefore, the absence of the sign problem does not hold for an odd number of fermions in general. However, it does hold for models with fermions coupled to Z2 gauge fields [18–20]. (b) APPENDIX C: CONFINEMENT TRANSITION FOR Nf = 6 AND Nf =8 In this Appendix we discuss the results for Nf ¼ 6 and Nf ¼ 8. As shown in Figs. 9 and 12 and the corresponding insets, we estimate Jc ¼ 1.9ð3Þ for Nf ¼ 6 and Jc ¼ 2.5ð1Þ for Nf ¼ 8. Again, the data are consistent with a continuous transition between the deconfined UID and confined VBS phases. Correspondingly, the flux energy per pla- quette behaves as a smooth function across the criti- FIG. 10. The log-log plot of real-space decay of (a) spin cal point. correlation functions and (b) dimer correlation functions for 6 1 1 40 More interestingly, we plot the spin-spin and dimer- Nf ¼ in the Uð ÞD phase (at J ¼ . (b) (a) (b) FIG. 9. The VBS correlation ratio through the Uð1ÞD-to-VBS FIG. 11. The log-log plot of real-space decay of (a) spin transition at Nf ¼ 6. Here, β ¼ 2L, Δτ ¼ 0.1. (b) The 1=L correlation functions and (b) dimer correlation functions for extrapolation of the crossings estimates the Uð1ÞD-to-VBS Nf ¼ 8 in the Uð1ÞD phase (at J ¼ 2.00 021022-13 XU, QI, ZHANG, ASSAAD, XU, and MENG PHYS. REV. X 9, 021022 (2019) 0 -0.5 2 L / -1 Q C -1.5 J = 2.50 1.25 -2 0 0.05 0.1 0.15 0.2 1/L ˆ FIG. 14. Uniform structure factor of Qi [defined in Eq. (E1)] for FIG. 12. The VBS correlation ratio through the Uð1ÞD-to-VBS N ¼ 2. Note that β ¼ 4L here. For all J’s, this uniform structure 8 β 2 Δτ 0 1 1 f transition at Nf ¼ . Here, ¼ L, ¼ . . (b) The =L factor extrapolated to zero in thermodynamic limit. Thus, the 1 ˆ extrapolation of the crossings estimates the Uð ÞD-to-VBS constraint of Qi ¼ 0 is dynamically enforced. transition point at Jc ¼ 2.5ð1Þ for Nf ¼ 8. ˆ limit. We study the uniform structure factor of Qi by -0.2 calculating L=8 10 1 X -0.3 12 ˆ ˆ 14 CQ ¼ 2 hQiQji: ðE1Þ 16 L ij -0.4 ˆ We find that the uniform structure factor of Qi defined -0.5 above extrapolates to zero in the thermodynamic limit as shown in Fig. 14 for Nf ¼ 2. Other Nf cases show similar -0.6 behavior. Flux energy per plaquette -0.7 APPENDIX F: PERFORMANCE OF DQMC ON cQED 0.5 1 1.5 2 2.5 3 3.5 4 3 COUPLED TO FERMIONIC MATTER J As we discuss in the main text, in the DQMC simu- lation, we use local updates which flip the gauge variables FIG. 13. Flux energy per plaquette along the same J path. There ϕbðτÞ [∈ ½0; 2πÞ] on the space-time lattice one by one, and is no singularity around Jc, suggesting it is a continuous phase transition. we call one scan of the whole space-time lattice as one sweep, which is usually called one Monte Carlo step in DQMC. For the cQED3 problem, we design a specific fast APPENDIX D: FLUX ENERGY PER PLAQUETTE update method, which greatly improves the computation To characterize the continuous nature of confined and efficiency more accurately by making the fast update still work here, thus reducing the huge time cost for deconfined phase transition,P we also measure the flux 2 ˆ each sweep. energy per plaquette hð1=L Þ □ cos ðcurlθÞi. Figure 13 As a first attempt to study this challenging problem of depicts our result at Nf ¼ 2. For other Nf’s, the flux energy cQED3 coupled to fermionic matter in the condensed per plaquette has a similar continuous behavior. matter field by the DQMC method with the local update strategy, we need to demonstrate how well it works here. The following is a detailed discussion of the performance of APPENDIX E: DYNAMICALLY GENERATED the method. CONSTRAINT The first important quantity associated with the effi- As we mention in the main text, our model corresponds ciency of the method is the acceptance ratio. Figure 15 to an unconstrained gauge theory. As such, the Gauss law illustrates the acceptance ratio for different J at Nf ¼ 2. ˆ will be dynamically imposed, and Qi defined in Eq. (6) The acceptance ratio reduces as J becomes smaller. should converge to constant value in the zero-temperature Fortunately, the acceptance ratio deep in the Uð1ÞD phase 021022-14 MONTE CARLO STUDY OF LATTICE COMPACT QUANTUM … PHYS. REV. X 9, 021022 (2019) defined as the sumP of m□ of each plaquette in the time-slice plane τ, MðτÞ¼ □ m□ðτÞ. Figure 16 shows such net flux sweep series both inside the Uð1ÞD phase [Fig. 16(a)] and inside the AFM phase [Fig. 16(b)]atNf ¼ 2 with L ¼ 12 at different time slices τ and τ0. In the Uð1ÞD phase, it favors the πð−πÞ flux in each plaquette, and the net flux in each time-slice plane seldom changes and is weakly fluctuating between different time slices, while in the AFM phase, the net flux changes almost randomly with more extended values and large fluctuations between different time-slice planes, as a consequence of a proliferate of monopoles. FIG. 15. Local update acceptance ratio at Nf ¼ 2 with L ¼ 6. [1] I. Affleck and J. B. Marston, Large-n Limit of the Note the acceptance ratio for other sizes is almost the same, not Heisenberg-Hubbard Model: Implications for High-Tc shown here. Superconductors, Phys. Rev. B 37, 3774 (1988). [2] J. B. Marston and I. 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