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Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: the Parent State of Quantum Phases

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Monte Carlo Study of Lattice Compact Quantum Electrodynamics with Fermionic Matter: the Parent State of Quantum Phases 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 Þg JNfΔτ AFM phase transition at Nf ¼ 4.
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