arXiv:0705.3541 · Physical Review B

Compact quantum electrodynamics in 2+1 dimensions and spinon deconfinement: A renormalization group analysis

Flavio S. Nogueira, Hagen M. Kleinert

Published Jan 8, 2008• 25 citations

Abstract

We discuss compact $2+1$ dimensional Maxwell electrodynamics coupled to fermionic matter with $N$ replica. For large enough $N$, the latter corresponds to an effective theory for the nearest neighbor $\mathrm{SU}(N)$ Heisenberg antiferromagnet, in which the fermions represent solitonic excitations known as spinons. Here, we show that the spinons are deconfined for $N>{N}_{c}=36$, thus leading to an insulating state known as spin liquid. A previous analysis considerably underestimated the value of ${N}_{c}$. We show further that for $20<N\ensuremath{\leqslant}36$, there can be either a confined or a deconfined phase, depending on the instanton density. For $N\ensuremath{\leqslant}20$, only the confined phase exists. For the physically relevant value $N=2$, we argue that no paramagnetic phase can emerge since chiral symmetry breaking would disrupt it. In such a case, a spin liquid or any other nontrivial paramagnetic state (for instance, a valence-bond solid) is only possible if doping or frustrating interactions are included.

DOI: 10.1103/physrevb.77.045107

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Quantum Electrodynamics in 2+1 Dimensions, Confinement, and the Stability of U(1) Spin Liquids

Flavio S. Nogueira, Hagen M. Kleinert

arXiv:cond-mat/0501022Physical Review LettersOct 20, 200534 citations

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

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arXiv:1807.07574Physical Review XMay 2, 201998 citations

Dynamics of compact quantum electrodynamics at large fermion flavor

Wei Wang, Da-Chuan Lu, Xiao Yan Xu +2 more

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Stability of U(1) spin liquids in two dimensions

Michael Hermele, T. Senthil, Matthew P. A. Fisher +3 more

arXiv:cond-mat/0404751Physical Review BDec 30, 2004317 citations

Deconfined quantum criticality driven by Dirac fermions in SU(2) antiferromagnets

Flavio S. Nogueira

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Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet

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arXiv:2303.01585Physical Review XApr 15, 202438 citations

arXiv:0705.3541 · Physical Review B

Compact quantum electrodynamics in 2+1 dimensions and spinon deconfinement: A renormalization group analysis

Flavio S. Nogueira, Hagen M. Kleinert

Published Jan 8, 2008• 25 citations

arXiv HTML PDF

Abstract

We discuss compact

2+1

dimensional Maxwell electrodynamics coupled to fermionic matter with

N

replica. For large enough

N

, the latter corresponds to an effective theory for the nearest neighbor

\mathrm{SU}(N)

Heisenberg antiferromagnet, in which the fermions represent solitonic excitations known as spinons. Here, we show that the spinons are deconfined for

N>{N}_{c}=36

, thus leading to an insulating state known as spin liquid. A previous analysis considerably underestimated the value of

{N}_{c}

. We show further that for

20<N\ensuremath{\leqslant}36

, there can be either a confined or a deconfined phase, depending on the instanton density. For

N\ensuremath{\leqslant}20

, only the confined phase exists. For the physically relevant value

N=2

, we argue that no paramagnetic phase can emerge since chiral symmetry breaking would disrupt it. In such a case, a spin liquid or any other nontrivial paramagnetic state (for instance, a valence-bond solid) is only possible if doping or frustrating interactions are included.

DOI: 10.1103/physrevb.77.045107

Related papers

Quantum Electrodynamics in 2+1 Dimensions, Confinement, and the Stability of U(1) Spin Liquids

Flavio S. Nogueira, Hagen M. Kleinert

Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when N > Nc = 36/pi3 approximately to 1.161, where N is the number of fermion replica. For N < Nc, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as N --> Nc-. Our results imply the stability of a spin liquid at the physical value N = 2 for Mott insulators.Compact quantum electrodynamics in 2 + 1 dimensions often arises as an effective theory for a Mott insulator, with the Dirac fermions representing the low-energy spinons. An important and controversial issue in this context is whether a deconfinement transition takes place. We perform a renormalization group analysis to show that deconfinement occurs when N > Nc = 36/pi3 approximately to 1.161, where N is the number of fermion replica. For N < Nc, however, there are two stable fixed points separated by a line containing a unstable nontrivial fixed point: a fixed point corresponding to the scaling limit of the noncompact theory, and another one governing the scaling behavior of the compact theory. The string tension associated with the confining interspinon potential is shown to exhibit a universal jump as N --> Nc-. Our results imply the stability of a spin liquid at the physical value N = 2 for Mott insulators.

arXiv:cond-mat/0501022Physical Review LettersOct 20, 200534 citations

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

Xiao Yan Xu, Yang Qi, Long Zhang +3 more

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-symmetrybreaking phases such as the valence-bond solid (VBS) and Néel 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 N f ¼ 4, our data suggest a continuous quantum phase transition between the VBS and Néel order.We also provide preliminary theoretical analysis for these quantum phase transitions.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-symmetrybreaking phases such as the valence-bond solid (VBS) and Néel 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 N f ¼ 4, our data suggest a continuous quantum phase transition between the VBS and Néel order.We also provide preliminary theoretical analysis for these quantum phase transitions.

arXiv:1807.07574Physical Review XMay 2, 201998 citations

Dynamics of compact quantum electrodynamics at large fermion flavor

Wei Wang, Da-Chuan Lu, Xiao Yan Xu +2 more

Thanks to the development in quantum Monte Carlo technique, the compact U(1) lattice gauge theory coupled to fermionic matter at ( $2+1$ )D is now accessible with large-scale numerical simulations, and the ground state phase diagram as a function of fermion flavor ( ${N}_{f}$ ) and the strength of gauge fluctuations is mapped out. Here we focus on the large fermion flavor case ( ${N}_{f}=8$ ) to investigate the dynamic properties across the deconfinement-to-confinement phase transition. In the deconfined phase, fermions coupled to the fluctuating gauge field to form U(1) spin liquid with continua in both spin and dimer spectral functions, and in the confined phase fermions are gapped out into valence bond solid phase with translational symmetry breaking and gapped spectra. The dynamical behaviors provide supporting evidence for the existence of the U(1) deconfined phase and could shine light on the nature of the U(1)-to-VBS phase transition which is of the ${\mathrm{QED}}_{3}$ -Gross-Neveu chiral O(2) universality whose properties are still largely unknown.Thanks to the development in quantum Monte Carlo technique, the compact U(1) lattice gauge theory coupled to fermionic matter at ($2+1$)D is now accessible with large-scale numerical simulations, and the ground state phase diagram as a function of fermion flavor (${N}_{f}$) and the strength of gauge fluctuations is mapped out. Here we focus on the large fermion flavor case (${N}_{f}=8$) to investigate the dynamic properties across the deconfinement-to-confinement phase transition. In the deconfined phase, fermions coupled to the fluctuating gauge field to form U(1) spin liquid with continua in both spin and dimer spectral functions, and in the confined phase fermions are gapped out into valence bond solid phase with translational symmetry breaking and gapped spectra. The dynamical behaviors provide supporting evidence for the existence of the U(1) deconfined phase and could shine light on the nature of the U(1)-to-VBS phase transition which is of the ${\mathrm{QED}}_{3}$-Gross-Neveu chiral O(2) universality whose properties are still largely unknown.

arXiv:1906.06929Physical review. B./Physical review. BAug 13, 201928 citations

Stability of U(1) spin liquids in two dimensions

Michael Hermele, T. Senthil, Matthew P. A. Fisher +3 more

We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless $U(1)$ gauge fluctuations can be stable in two dimensions, at least when the physical $SU(2)$ spin symmetry is generalized to $SU(N)$ . Equivalently, we show that compact ${\mathrm{QED}}_{3}$ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological $U(1)$ symmetry corresponding to the conservation of gauge flux. Beginning with an $SU(N)$ generalization of the $S=1∕2$ square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large- $N$ spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact ${\mathrm{QED}}_{3}$ .We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless $U(1)$ gauge fluctuations can be stable in two dimensions, at least when the physical $SU(2)$ spin symmetry is generalized to $SU(N)$. Equivalently, we show that compact ${\mathrm{QED}}_{3}$ has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological $U(1)$ symmetry corresponding to the conservation of gauge flux. Beginning with an $SU(N)$ generalization of the $S=1∕2$ square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-$N$ spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact ${\mathrm{QED}}_{3}$.

arXiv:cond-mat/0404751Physical Review BDec 30, 2004317 citations

Deconfined quantum criticality driven by Dirac fermions in SU(2) antiferromagnets

Flavio S. Nogueira

Quantum electrodynamics in $2+1$ dimensions is an effective gauge theory for the so-called algebraic quantum liquids. A new type of such a liquid, the algebraic charge liquid, has been proposed recently in the context of deconfined quantum critical points [R. K. Kaul et al., Nat. Phys. 4, 28 (2008)]. In this context, by using the renormalization group in $d=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, we show that a deconfined quantum critical point occurs in a SU(2) system provided the number of Dirac fermion species ${N}_{f}\ensuremath{\ge}4$. The calculations are done in a representation where the Dirac fermions are given by four-component spinors. The critical exponents are calculated for several values of ${N}_{f}$ . In particular, for ${N}_{f}=4$ and $\ensuremath{\epsilon}=1$ $(d=2+1)$ , the anomalous dimension of the N\'eel field is given by ${\ensuremath{\eta}}_{N}=1/3$, with a correlation length exponent $\ensuremath{\nu}=1/2$. These values considerably change for ${N}_{f}>4$ . For instance, for ${N}_{f}=6$ , we find ${\ensuremath{\eta}}_{N}\ensuremath{\approx}0.75\text{ }191$ and $\ensuremath{\nu}\ensuremath{\approx}0.660\text{ }09$. We also investigate the effect of chiral symmetry breaking and analyze the scaling behavior of the chiral holon susceptibility, ${G}_{\ensuremath{\chi}}(x)\ensuremath{\equiv}⟨\overline{\ensuremath{\psi}}(x)\ensuremath{\psi}(x)\overline{\ensuremath{\psi}}(0)\ensuremath{\psi}(0)⟩$.Quantum electrodynamics in $2+1$ dimensions is an effective gauge theory for the so-called algebraic quantum liquids. A new type of such a liquid, the algebraic charge liquid, has been proposed recently in the context of deconfined quantum critical points [R. K. Kaul et al., Nat. Phys. 4, 28 (2008)]. In this context, by using the renormalization group in $d=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, we show that a deconfined quantum critical point occurs in a SU(2) system provided the number of Dirac fermion species ${N}_{f}\ensuremath{\ge}4$. The calculations are done in a representation where the Dirac fermions are given by four-component spinors. The critical exponents are calculated for several values of ${N}_{f}$. In particular, for ${N}_{f}=4$ and $\ensuremath{\epsilon}=1$ $(d=2+1)$, the anomalous dimension of the N\'eel field is given by ${\ensuremath{\eta}}_{N}=1/3$, with a correlation length exponent $\ensuremath{\nu}=1/2$. These values considerably change for ${N}_{f}>4$. For instance, for ${N}_{f}=6$, we find ${\ensuremath{\eta}}_{N}\ensuremath{\approx}0.75\text{ }191$ and $\ensuremath{\nu}\ensuremath{\approx}0.660\text{ }09$. We also investigate the effect of chiral symmetry breaking and analyze the scaling behavior of the chiral holon susceptibility, ${G}_{\ensuremath{\chi}}(x)\ensuremath{\equiv}⟨\overline{\ensuremath{\psi}}(x)\ensuremath{\psi}(x)\overline{\ensuremath{\psi}}(0)\ensuremath{\psi}(0)⟩$.

arXiv:0802.0500Physical Review BMay 5, 20088 citations

Quantum Electrodynamics in 2+1 Dimensions as the Organizing Principle of a Triangular Lattice Antiferromagnet

Alexander Wietek, Sylvain Capponi, Andreas Martin Läuchli

Quantum electrodynamics in 2+1 dimensions ( QED3 ) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated J1−J2 Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED3 , in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of QED3 , which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by QED3 . Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to N=48 sites. Published by the American Physical Society 2024Quantum electrodynamics in 2+1 dimensions ( QED3 ) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated J1−J2 Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED3 , in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of QED3 , which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by QED3 . Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to N=48 sites. Published by the American Physical Society 2024

arXiv:2303.01585Physical Review XApr 15, 202438 citations

We discuss compact

2+1

dimensional Maxwell electrodynamics coupled to fermionic matter with

N

replica. For large enough

N

, the latter corresponds to an effective theory for the nearest neighbor

\mathrm{SU}(N)

Heisenberg antiferromagnet, in which the fermions represent solitonic excitations known as spinons. Here, we show that the spinons are deconfined for

N>{N}_{c}=36

, thus leading to an insulating state known as spin liquid. A previous analysis considerably underestimated the value of

{N}_{c}

. We show further that for

20<N\ensuremath{\leqslant}36

, there can be either a confined or a deconfined phase, depending on the instanton density. For

N\ensuremath{\leqslant}20

, only the confined phase exists. For the physically relevant value

N=2

Thanks to the development in quantum Monte Carlo technique, the compact U(1) lattice gauge theory coupled to fermionic matter at (

2+1

)D is now accessible with large-scale numerical simulations, and the ground state phase diagram as a function of fermion flavor (

{N}_{f}

) and the strength of gauge fluctuations is mapped out. Here we focus on the large fermion flavor case (

{N}_{f}=8

) to investigate the dynamic properties across the deconfinement-to-confinement phase transition. In the deconfined phase, fermions coupled to the fluctuating gauge field to form U(1) spin liquid with continua in both spin and dimer spectral functions, and in the confined phase fermions are gapped out into valence bond solid phase with translational symmetry breaking and gapped spectra. The dynamical behaviors provide supporting evidence for the existence of the U(1) deconfined phase and could shine light on the nature of the U(1)-to-VBS phase transition which is of the

{\mathrm{QED}}_{3}

-Gross-Neveu chiral O(2) universality whose properties are still largely unknown.

We establish that spin liquids described in terms of gapless fermionic (Dirac) spinons and gapless

U(1)

gauge fluctuations can be stable in two dimensions, at least when the physical

SU(2)

spin symmetry is generalized to

SU(N)

. Equivalently, we show that compact

{\mathrm{QED}}_{3}

has a deconfined phase for a large number of fermion fields, in the sense that monopole fluctuations can be irrelevant at low energies. A precise characterization is provided by an emergent global topological

U(1)

symmetry corresponding to the conservation of gauge flux. Beginning with an

SU(N)

generalization of the

S=1∕2

square-lattice Heisenberg antiferromagnet, we consider the $\ensuremath{\pi}$-flux spin liquid and, via a systematic analysis of all operators, show that there are no relevant perturbations (in the renormalization-group sense) about the large-

N

spin-liquid fixed point, which is thus a stable phase. We provide a further illustration of this conclusion with an approximate renormalization-group calculation that treats the gapless fermions and the monopoles on an equal footing. This approach directly points out some of the flaws in the erroneous ``screening'' argument for the relevance of monopoles in compact

{\mathrm{QED}}_{3}

Quantum electrodynamics in

2+1

dimensions is an effective gauge theory for the so-called algebraic quantum liquids. A new type of such a liquid, the algebraic charge liquid, has been proposed recently in the context of deconfined quantum critical points [R. K. Kaul et al., Nat. Phys. 4, 28 (2008)]. In this context, by using the renormalization group in $d=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, we show that a deconfined quantum critical point occurs in a SU(2) system provided the number of Dirac fermion species ${N}_{f}\ensuremath{\ge}4$. The calculations are done in a representation where the Dirac fermions are given by four-component spinors. The critical exponents are calculated for several values of

{N}_{f}

. In particular, for

{N}_{f}=4

and $\ensuremath{\epsilon}=1$

(d=2+1)

, the anomalous dimension of the N\'eel field is given by ${\ensuremath{\eta}}_{N}=1/3$, with a correlation length exponent $\ensuremath{\nu}=1/2$. These values considerably change for

{N}_{f}>4

. For instance, for

{N}_{f}=6

, we find ${\ensuremath{\eta}}_{N}\ensuremath{\approx}0.75\text{ }191$ and $\ensuremath{\nu}\ensuremath{\approx}0.660\text{ }09$. We also investigate the effect of chiral symmetry breaking and analyze the scaling behavior of the chiral holon susceptibility, ${G}_{\ensuremath{\chi}}(x)\ensuremath{\equiv}⟨\overline{\ensuremath{\psi}}(x)\ensuremath{\psi}(x)\overline{\ensuremath{\psi}}(0)\ensuremath{\psi}(0)⟩$.