The chiral phase transition of QED 3 around the critical number of fermion flavors
aa r X i v : . [ h e p - ph ] A ug The chiral phase transition of QED around the critical number of fermion flavors Pei-lin Yin , Zhu-fang Cui , , Hong-tao Feng , , and Hong-shi Zong , , ∗ Key Laboratory of Modern Acoustics, MOE, Institute of Acoustics,and Department of Physics, Nanjing University, Nanjing 210093, China Department of Physics, Nanjing University, Nanjing 210093, China Department of Physics, Southeast University, Nanjing 211189, China State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing, 100190, China and Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China
At zero temperature and density, the nature of the chiral phase transition in QED with N f massless fermion flavors is investigated. To this end, in Landau gauge, we numerically solve thecoupled Dyson-Schwinger equations for the fermion and boson propagator within the bare andsimplified Ball-Chiu vertices separately. It is found that, in the bare vertex approximation, thesystem undergoes a high-order continuous phase transition from the Nambu-Goldstone phase intothe Wigner phase when the number of fermion flavors N f reaches the critical number N f,c , while thesystem exhibits a typical characteristic of second-order phase transition for the simplified Ball-Chiuvertex.Keywords: QED , Chiral phase transition, Dyson-Schwinger equationsPACS Number(s): 11.10.Kk, 11.15.Tk, 11.30.Qc ∗ [email protected] I. INTRODUCTION
Dynamical chiral symmetry breaking (DCSB) and confinement are two fundamental features of Quantum Chromo-dynamics (QCD). Studying these two aspects, will provide profound insight into the origin of observable mass and thenature of the early Universe. However, due to the complicated non-Abelian character of QCD, it is difficult to havea thorough understanding of the mechanism of DCSB and confinement. In this case, to gain valuable comprehensionabout them, it is suggested to study some model which is similar to QCD and, at the same time, simpler. QuantumElectrodynamics in (2+1) dimensions (QED ) is such a model, and it has been studied quite intensively over thepast few years [1–18]. In addition, due to the coupling constant being dimensionful (its dimension is √ mass), QED is super-renormalizable, so it does not suffer from the ultraviolet divergences which are present in QED . Apartfrom these interesting features, QED with N f massless fermion flavors can also be regarded as a possible low energyeffective theory for strongly correlated electronic systems [19–24].In the last few decades, whether a critical number of fermion flavors N f,c , where the system undergoes a phasetransition from the Nambu-Goldstone phase (NG phase) into the Wigner phase (WN phase), exists or not has beendiscussed intensively. A breakthrough was achieved in 1988 by T.W. Appelquist et al. [3]. Using the bare vertexapproximation and the one-loop vacuum polarization, they solved the Dyson-Schwinger equations (DSEs) for thefermion self-energy and found that chiral phase transition occurs when the number of fermion flavors reaches a criticalnumber N f,c = 32 /π . Subsequently, P. Maris solved the coupled DSEs for the fermion self-energy and boson vacuumpolarization with a range of simplified fermion-boson vertices and obtained a critical number of fermion flavors N f,c =3.3 [8]. Recently, A. Bashir et al . analyzed the characters of fermion wave function renormalization and bosonvacuum polarization at the infrared momenta when the fermion mass function vanishes and arrived at N f,c ≈ N f,c are relatively scarce. The authors of Ref. [14] discussed this question bysolving DSEs for the fermion self-energy in the lowest-order approximation and drew the conclusion that the chiralphase transition in QED with N f massless fermion flavors is a continuous phase transition higher than second-order.In the present paper, we try to reanalyze the nature of this phase transition by numerically solving the coupled DSEsfor the fermion and boson propagator within the bare and simplified Ball-Chiu vertices separately at zero temperatureand density.This paper is organized as follows. In Section II, we firstly introduce the criteria determining the locations andcharacteristics of the chiral phase transition, and then derive the DSEs satisfied by the fermion and boson propagatorof QED . In Section III, we discuss the behaviors of the fermion and boson propagator in NG phase and WN phaseby numerically solving the coupled DSEs in the truncated scheme and then study how the chiral condensate, thechiral susceptibility, the differential pressure between these two phases, and the infrared values of self-energy functionchange with the variation of the number of fermion flavors. In Section IV, we will briefly summarize our results andgive the conclusions. II. CHIRAL PHASE TRANSITION AND DSES IN QED In quantum field theory, the dynamic properties of a system are fully characterized by the generating functionalcorresponding to the partition function in thermodynamics. It is commonly accepted that when the system is in acertain phase, the generating functional is usually analytic for some choice of parameters, such as the current massof the fermion, the temperature and the chemical potential; the generating functional often exhibits the non-analyticcharacter while the phase transitions occur. So the location and characteristic of the chiral phase transition in thesystem can be determined by the behaviors of this quantity with respect to the corresponding parameters (i.e., thecurrent mass, the temperature and the chemical potential). In this case, a phase transition, in which the first-orderderivative of the generating functional with some of the parameters is discontinuous, is referred to as first-order ordiscontinuous phase transition. Second-order or continuous phase transition exhibits the continuity in first-orderderivative and the discontinuity or infinity in second-order derivative.
A. Criteria for chiral phase transition
The chiral condensate is the vacuum expectation value for the scalar operator ¯ ψψ . The character, the nonzero valueof it indicates that chiral symmetry reflected on the Lagrangian level is spontaneously broken on the vacuum level andthe chiral symmetry gets restored when the chiral condensate vanishes for the chiral limit, makes it possible to definethe chiral condensate as the order parameter for the chiral phase transition. The chiral condensate is commonly givenby the first-order derivative of the generating functional with respect to the current mass of the fermion h ¯ ψψ i = − ∂LnZ∂m = − Z d p (2 π ) Tr[ S ( p )] , (1)where S is the dressed fermion propagator and the notation Tr denotes trace operation over Dirac indices of thefermion propagator.In addition to the above chiral condensate, the susceptibility representing a response of the system to an externalperturbation is often investigated. The feature, the susceptibility usually exhibits some of singular behaviors, such asdiscontinuity or infinity, when the chiral phase transition occurs, enables it to be used for studying the chiral phasetransition [25, 26]. Herein we consider the chiral susceptibility that can be written as the first-order derivative ofchiral condensate with respect to the current mass of the fermion χ c = ∂ ( −h ¯ ψψ i m ) ∂m (cid:12)(cid:12)(cid:12)(cid:12) m → = ∂ LnZ∂m (cid:12)(cid:12)(cid:12)(cid:12) m → = − Z d p (2 π ) Tr[ S ( m, p ) ∂ S − ( m, p ) ∂m S ( m, p )] (cid:12)(cid:12)(cid:12)(cid:12) m → , (2)This equation indicates that the chiral susceptibility measures the response of the chiral condensate to an infinitesimalchange of the fermion mass.Furthermore, in order to analyze the effect of the number of fermion flavors on the chiral phase transition moredirectly, we need to calculate the derivative of the generating functional with respect to the number of fermion flavors.Because of the character of non-perturbation, we cannot obtain an exact expression for the generating functional.However, in some situations, an expression for the effective potential can be given in terms of the fermion and bosonpropagator, which permits us to carry out the calculation of the derivative. In the present paper, we adopt the CJTeffective potential [27], which corresponds to the bare vertex approximation for solving the DSEs for the fermion andthe boson propagator, to study the nature of the chiral phase transition around the critical number of fermion flavors.The pressure is the negative of the CJT effective potential density. At zero temperature and density the effectivepressure is given as P ( N f ) = − N f × Tr[Ln( S − S ) + 12 (1 − S − S )] + 12 × Tr[Ln( D − D ) + (1 − D − D )] , (3)where the trace, the logarithm, and the product of propagator are taken in the functional sense. The first itemrepresents, the contribution of N f massless fermion flavors, and the last item denotes the contribution of photon,which is rarely discussed in the existing literature. Because of the divergent integral, the differential pressure betweenthe NG phase and WN phase is often calculated∆ P ( N f ) = P NG ( N f ) − P WN ( N f ) , (4)To determine the order of the chiral phase transition that we concern, herein we regard ∆ P ( N f ) as a continuousfunction of N f . More specifically, we can expand ∆ P ( N f ) by the Taylor series near the critical number of fermionflavors N f,c ∆ P ( N f ) = ∆ P ( N f,c ) + ( N f − N f,c ) ∂ (∆ P ( N f )) ∂ N f (cid:12)(cid:12)(cid:12)(cid:12) N f = N f , c + ( N f − N f,c ) ∂ (∆ P ( N f )) ∂ N f (cid:12)(cid:12)(cid:12)(cid:12) N f = N f , c + · · · , (5)with ∆ P ′ ( N f ) = ∂ (∆ P ( N f )) ∂ N f , (6)∆ P ′′ ( N f ) = ∂ (∆ P ( N f )) ∂ N f , (7)The phase transition is a first-order one when ∆ P ′ ( N f ) is discontinuous at N f = N f,c and its continuity might implya second-order phase transition. B. Dyson-Schwinger equations in QED According to the fundamental theory of group representations, the dimension of a spinorial representation for theLorentz group must be even. In (2+1) dimension only three γ matrices that satisfy the corresponding Clifford algebraare needed, meanwhile, in quantum mechanics there have been three anticommutative matrices that are just 2 × × γ matrices. There is, therefore, nothing to generate a chiralsymmetry and so we cannot discuss the chiral symmetry. Besides, the possible mass term has the undesirable propertythat it is odd under the parity transformation. Given this, we employ the fourdimensional matrix representation andfour-component spinors as in four spaceCtime dimensions in this paper.In Euclidean space, the Lagrangian density of QED with N f massless fermion flavors reads L = N X i =1 ¯ ψ i ( ∂ + ie A ) ψ i + 14 F µν + 12 ξ ( ∂ µ A µ ) , (8)where the spinor ψ i is the fermion field with the indices i =1,..., N f representing different fermion flavors, A µ is theelectromagnetic vector potential, F µν is the electromagnetic field strength tensor, and ξ represents the gauge parameter(we will adopt the Landau gauge ξ = 0 throughout this paper). Using this Lagrangian density one can derive in thestandard way, for instance through functional analysis, the DSEs for the propagators.For the fermion propagator the DSEs can be written as S − ( p ) = S − ( p ) + Σ( p ) , (9)Σ( p ) = Z d k (2 π ) γ µ S ( k )Γ ν ( p, k ) D µν ( q ) , (10)where S ( p ) and S ( p ) = 1 /iγ · p are the dressed fermion propagator and the free fermion propagator in the chirallimit, respectively, Σ( p ) is the fermion self-energy, Γ ν ( p, k ) is the full fermionCboson vertex, and D µν ( q ) is the dressedphoton propagator. Meanwhile, based on Lorentz structure analysis, the fermion propagator can be written as S − ( p ) = i pA ( p ) + B ( p ) , (11)where both A ( p ) and B ( p ) are scalar functions of p . Substituting Eq. (10) and Eq. (11) into Eq.(9), one canimmediately obtain A ( p ) = 1 − i p Z d k (2 π ) Tr[ γ · pγ µ S ( k )Γ ν ( p, k ) D µν ( q )] , (12) B ( p ) = 14 Z d k (2 π ) Tr[ γ µ S ( k )Γ ν ( p, k ) D µν ( q )] , (13)The DSEs for the photon propagator have the form D − µν ( q ) = D , − µν ( q ) + Π µν ( q ) , (14)Π µν ( q ) = − N f Z d k (2 π ) Tr[ γ µ S ( k )Γ ν ( p, k ) S ( p )] , (15)where D µν ( q ) = ( δ µν − q µ q ν /q ) /q is the free photon propagator, Π µν ( q ) is the vacuum polarization tensor. At thesame time, in order to ensure the Ward-Takahashi identity, Π µν ( q ) has the formΠ µν ( q ) = ( q δ µν − q µ q ν )Π( q ) , (16)where Π( q ) is the photon self-energy, i.e., the vacuum polarization. Substituting Eq. (16) into Eq. (14), one has D µν ( q ) = ( δ µν − q µ q ν /q ) q (1 + Π( q )) , (17)However, we also note that the vacuum polarization tensor Π µν ( q ) has an ultraviolet divergence that is present onlyin the longitudinal part. By applying the following projection operator [16]: P µν = δ µν − q µ q ν q , (18)one can remove this divergence and project out a finite vacuum polarization Π( q ),Π( q ) = δ µν − q µ q ν /q q Π µν ( q ) , (19)From Eq. (12), Eq. (13), Eq. (17), Eq. (19), and Eq. (15), one can see that if the full fermion-boson vertexΓ ν ( p, k ) is known, the fermion and boson gap equations form a set of equations that can be solved numerically bythe iteration method. In the past few years, there are several attempts to determine the form of Γ ν ( p, k ) in theliterature [5–7]. It should be mentioned here that the authors of Refs. [28, 29] analyzed the general Lorentz structureof dressed fermionCphoton vertex in the context of considering the constraints of the gauge symmetry and presenteda workable model for the dressed fermionCphoton vertex. In the present paper, following the Ref. [8], we choose thefollowing Ans¨atze for the fermionCphoton vertexΓ ν ( p, k ) = f ( A ( p ) , A ( k )) γ ν , (20)and the form of f ( A ( p ) , A ( k )) is: (1) 1; (2) A ( p )+ A ( k )2 . The first one is the bare vertex and plays the most dominantrole in the large momentum limit. The second form is inspired by the Ball-Chiu (BC) vertex. Previous works showthat the numerical results of DSEs obtained employing this choice are in good agreement with the results obtainedemploying the BC and Curtis-Pennington (CP) vertices, so we choose this one to be used in following calculation.Substituting Eq. (17) into Eq. (12) and Eq. (13), and similarly substituting Eq. (11) and Eq. (15) into Eq. (19),we can write down the coupled fermion and boson gap equations in the following form: A ( p ) = 1 + 2 p Z d k (2 π ) A ( k ) A ( k ) k + B ( k ) ( p · q )( k · q ) /q f ( A ( p ) , A ( k )) q (1 + Π( q )) , (21) B ( p ) = 2 Z d k (2 π ) B ( k ) A ( k ) k + B ( k ) f ( A ( p ) , A ( k )) q (1 + Π( q )) , (22)Π( q ) = 4 N f q Z d k (2 π ) A ( k ) A ( k ) k + B ( k ) A ( p ) A ( p ) p + B ( p ) ( k − k · q ) − k · q ) /q ) f ( A ( p ) , A ( k )) , (23) III. NUMERICAL RESULTSA. The behavior of the propagator in the two phases
In order to study the phase transition between the NG phase and WN phase, one should study the different behaviorsof the fermion and photon propagator in these two phases. We can numerically solve the coupled equations by theiteration method. For the NG solution, the DSEs have a non-trivial solution B ( p ) >
0. We start from A ( p ) = 1, B ( p ) = 1, Π( p ) = 1 and iterate the three coupled equations until all three functions converge to a stable solution.Similarly, we can set A ( p ) = 1, B ( p ) = 0, Π( p ) = 1 and obtain the results for A ( p ), Π( p ) in the WN phase byiteration of the two coupled equations. The typical behaviors of the three functions A ( p ), B ( p ), Π( p ) in the NGphase and WN phase are plotted in Figs. 1, Fig. 2 and Fig. 3 respectively.From Fig. 1, Fig. 2 and Fig. 3 it can be seen that A ( p ), B ( p ), Π( p ) in these two phases show completelydifferent behaviors. For the NG phase, the three scalar functions are almost constant in the infrared region, and theirbehaviors in the ultraviolet region are: A ( p ) → B ( p ) ∝ /p , and Π( q ) ∝ / p q , respectively. For the WNphase, in the ultraviolet region the two scalar functions coincide with their corresponding NG solutions, while in theinfrared region, A ( p ) approaches zero while Π( p ) tends to divergency when p is close to zero, which confirms thepower laws governing the infrared behavior of QED in the symmetric phase (i.e., the WN phase). -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 A ( p ) p Nambu Solution with Ansatz 1 Wigner Solution with Ansatz 1 Nambu Solution with Ansatz 2 Wigner Solution with Ansatz 2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 A ( p ) p Nambu Solution with Ansatz 1 Wigner Solution with Ansatz 1 Nambu Solution with Ansatz 2 Wigner Solution with Ansatz 2
FIG. 1. The behavior of A ( p ) with the variation of p for N f = 2 (left) and N f = 3 (right). -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 -9 -8 -7 -6 -5 -4 -3 -2 -1 B ( p ) p Nambu Solution with Ansatz 1 Nambu Solution with Ansatz 2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 -10 -9 -8 -7 -6 -5 -4 B ( p ) p Nambu Solution with Ansatz 1 Nambu Solution with Ansatz 2
FIG. 2. The behavior of B ( p ) with the variation of p for N f = 2 (left) and N f = 3 (right). B. The nature of the chiral phase transition around the critical number of fermion flavors
Because the DSEs for the fermion and boson propagators have been reduced to three coupled equations for A ( p ), B ( p ), Π( p ), we can rewrite Eq. (1) in terms of A ( p ) and B ( p ) h ¯ ψψ i = − Z d p (2 π ) B N ( p ) A N ( p ) p + B N ( p ) , (24)For the Ans¨atze 1 of the fermion-photon vertex, substituting the NG solutions of A ( p ), B ( p ) into Eq. (24) andEq. (2), we can know how the chiral condensate and chiral susceptibility change with the variation of the number offermion flavors. The behaviors of them are plotted in Fig. 4. From Fig. 4, it can be seen that with the increase of thenumber of fermion flavors N f , the chiral condensate decreases rapidly and the chiral susceptibility increases slowly.When the number of fermion flavors reaches 3.1, the chiral condensate is zero and the chiral susceptibility reaches itsmaximum, which shows that chiral symmetry gets restored.In addition, we can also rewrite the Eq. (4) in terms of these three functions∆ P ( N f ) = − N f Z d p (2 π ) [ln A W ( p ) p A N ( p ) p + B N ( p ) + A N ( p )( A N ( p ) − p + B N ( p ) A N ( p ) p + B N ( p ) − A W ( p ) − A W ( p ) ]+ Z d p (2 π ) [ln 1 + Π W ( p )1 + Π N ( p ) + Π N ( p ) − Π W ( p )(1 + Π N ( p ))(1 + Π W ( p )) ] , (25)If we substitute different solutions of A ( p ), B ( p ), Π( p ) into Eq. (25) and Eq. (5), we can also know how thedifferential pressure and its derivatives change with the variation of N f . Their behaviors are plotted in Fig. 5. FromFig. 5, it is found that ∆ P ( N f ), ∆ P ′ ( N f ) and ∆ P ′′ ( N f ) all fall monotonically to zero as N f increases and the curvesshow no singularity around the critical number of fermion flavors. This means that the transformation from theNG phase to the WN phase is neither of first-order nor of second-order, but may be a high-order continuous phasetransition, when the number of fermion flavors reaches the critical value. -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 -2 ( p ) p Nambu Solution with Ansatz 1 Wigner Solution with Ansatz 1 Nambu Solution with Ansatz 2 Wigner Solution with Ansatz 2 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 -2 ( p ) p Nambu Solution with Ansatz 1 Wigner Solution with Ansatz 1 Nambu Solution with Ansatz 2 Wigner Solution with Ansatz 2
FIG. 3. The behavior of Π( p ) with the variation of p for N f = 2 (left) and N f = 3 (right). N f c - FIG. 4. The dependence of - h ¯ ψψ i and χ c on fermion flavors, where parameters are normalized by their value at N f = 2 . In order to consider the impact of a non-trivial extension of the bare vertex approximation on the nature ofchiral phase transitions, here we adopt Ans¨atze 2 of the fermionCphoton vertex (the BC simplified vertex) to studythis problem. It is well known that the CJT effective potential adopted in our manuscript corresponds to thebare vertex approximation for solving the fermion and the boson propagator. If one tries to go beyond the barevertex approximation (for example, the BC-simplified vertex), up to now people do not know how to construct thecorresponding effective potential like CJT. Therefore, we cannot use the differential pressure between NG phase andWN phase to study the phase transition. In this circumstance, a method based on the chiral susceptibility can work, -7 -6 -5 -4 -3 -2 -1 N f p (Nf) p ’ (Nf) p ’’ (Nf) FIG. 5. The dependence of ∆ P ( N f ), ∆ P ′ ( N f ), ∆ P ′′ ( N f ) on fermion flavors, where parameters are normalized by their valueat N f = 2 . -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 B ( p = ) N f Numerical result Fitting line
FIG. 6. The infrared values of the self-energy function B ( p ) as a function of the number of fermion flavors N f . as explained in Ref. [30]. In this work we will employ a different order parameter instead, namely, the infrared valuesof self-energy function B ( p = 0) [31, 32], of which a nonzero value implies a nonzero condensate. The analyticalexpression for this quantity is investigated and given by T.W. Appelquist et al. [33] B (0) = aN f exp − π/ √ N f,c /N f − , (26)where the notations a and N f,c are the fitting parameters. For our numerical result, it can be found that the bestfitting parameters are ( a, N f,c ) = (2 . , . B ( p ) decreases with the fermion flavors increasing. When the numberof fermion flavors approaches the critical value 3.43, the infrared value of B ( p ) drops rapidly, which signals a typicalcharacteristic of second-order phase transition. IV. SUMMARY AND CONCLUSIONS
In this paper, we discuss the nature of the chiral phase transition around the critical number of fermion flavors inQED with N f massless fermion flavors at zero temperature and density. For this, we firstly introduce the criteriadetermining the locations and characteristics of the chiral phase transition, and then in the Landau gauge, we nu-merically solve the coupled DysonCSchwinger equations within the bare and simplified BC vertices separately. Thenumerical results show that, in the bare vertex approximation, the system undergoes a high-order continuous phasetransition from the NG phase into the WN phase when the number of fermion flavors N f reaches the critical number N f,c . Finally, in order to analyze the impacts of different choices of vertex Ans¨atze on the results of our paper, wehave also adopted the simplified BC vertex to study this problem. As is shown in Fig. 6, for the simplified BC vertexthe chiral phase transition is a typical second-order phase transition. This shows that the impacts of the differentchoices of vertex Ans¨atze are important for the study of chiral phase transition of QED . Undoubtedly this problemdeserves further investigation. ACKNOWLEDGMENTS
This work is supported in part by the National Natural Science Foundation of China (under Grants 11275097,10935001, 11274166 and 11075075), the National Basic Research Program of China (under Grant 2012CB921504) andthe Research Fund for the Doctoral Program of Higher Education (under Grant No 2012009111002). [1] R. D. Pisarski, Phys. Rev. D , 2423 (1984).[2] T. Appelquist, M. J. Bowick, E. Cohler, and L. C. R. Wijewardhana, Phys. Rev. Lett. , 1715 (1985).[3] T. Appelquist, D. Nash, and L. C. R. Wijewardhana, Phys. Rev. Lett. , 2575 (1988).[4] D. Nash, Phys. Rev. Lett. , 3024 (1989).[5] C. J. Burden and C. D. Roberts, Phys. Rev. D , 540 (1991). [6] D. Curtis, M. Pennington, and D. Walsh, Phys. Lett. B , 313 (1992).[7] K. I. Kondo and P. Maris, Phys. Rev. Lett. , 18 (1995).[8] P. Maris, Phys. Rev. D , 4049 (1996).[9] A. Bashir, Phys. Lett. B , 280 (2000).[10] V. P. Gusynin and M. Reenders, Phys. Rev. D , 025017 (2003).[11] H.-t. Feng, F.-y. Hou, X. He, W.-m. Sun, and H.-s. Zong, Phys. Rev. D , 016004 (2006).[12] A. Bashir, A. Raya, I. C. Clo¨et, and C. D. Roberts, Phys. Rev. C , 055201 (2008).[13] A. Bashir, A. Raya, and S. S´anchez-Madrigal, Phys. Rev. D , 036013 (2011).[14] H.-t. Feng, B. Wang, W.-m. Sun, and H.-s. Zong, Phys. Rev. D , 105042 (2012).[15] M. G¨opfert and G. Mack, Commun. Math. Phys. , 545 (1982).[16] C. J. Burden, J. Praschifka, and C. D. Roberts, Phys. Rev. D , 2695 (1992).[17] P. Maris, Phys. Rev. D , 6087 (1995).[18] I. F. Herbut and B. H. Seradjeh, Phys. Rev. Lett. , 171601 (2003).[19] W. Rantner and X.-g. Wen, Phys. Rev. Lett. , 3871 (2001).[20] M. Franz, Z. Teˇsanovi´c, and O. Vafek, Phys. Rev. B , 054535 (2002).[21] G.-z. Liu, Phys. Rev. B , 172501 (2005).[22] P. A. Lee, N. Nagaosa, and X.-g. Wen, Rev. Mod. Phys. , 17 (2006).[23] J. E. Drut and T. A. L¨ahde, Phys. Rev. Lett. , 026802 (2009).[24] C.-x. Zhang, G.-z. Liu, and M.-q. Huang, Phys. Rev. B , 115438 (2011).[25] Y. Aoki, G. Endr˝odi, Z. Fodor, S. Katz, and K. Szabo, Nature , 675 (2006).[26] G. A. Contrera, M. Orsaria, and N. N. Scoccola, Phys. Rev. D , 054026 (2010).[27] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D , 2428 (1974).[28] A. Bashir, R. Bermudez, L. Chang, and C. D. Roberts, Phys. Rev. C , 045205 (2012).[29] S.-x. Qin, L. Chang, Y.-x. Liu, C. D. Roberts, and S. M. Schmidt, Phys. Lett. B , 384 (2013).[30] S.-x. Qin, L. Chang, H. Chen, Y.-x. Liu, and C. D. Roberts, Phys. Rev. Lett. , 172301 (2011).[31] C. Roberts and S. Schmidt, Prog. Part. Nucl. Phys. , S1 (2000).[32] M. Blank and A. Krassnigg, Phys. Rev. D , 034006 (2010).[33] T. W. Appelquist, M. Bowick, D. Karabali, and L. C. R. Wijewardhana, Phys. Rev. D33