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The interplay between lattice gauge theories and fermionic matter accountsfor fundamental physical phenomena ranging from the deconfinement of quarks inparticle physics to quantum spin liquid with fractionalized anyons and emergentgauge structures in condensed matter physics. However, except for certainlimits (for instance large number of flavors of matter fields), analyticalmethods can provide few concrete results. Here we show that the problem ofcompact $U(1)$ lattice gauge theory coupled to fermionic matter in $(2+1)$D ispossible to access via sign-problem-free quantum Monte Carlo simulations. Onecan hence map out the phase diagram as a function of fermion flavors and thestrength of gauge fluctuations. By increasing the coupling constant of thegauge field, gauge confinement in the form of various spontaneous symmetrybreaking phases such as valence bond solid (VBS) and N\'eel antiferromagnetemerge. Deconfined phases with algebraic spin and VBS correlation functions arealso observed. Such deconfined phases are an incarnation of exotic states ofmatter, $i.e.$ the algebraic spin liquid, which is generally viewed as theparent state of various quantum phases. The phase transitions betweendeconfined and confined phases, as well as that between the different confinedphases provide various manifestations of deconfined quantum criticality. Inparticular, for four flavors, $N_f = 4$, our data suggests a continuous quantumphase transition between the VBS and N\'{e}el order. We also providepreliminary theoretical analysis for these quantum phase transitions.

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