High-temperature domain walls of QCD with imaginary chemical potentials
PPrepared for submission to JHEP
High-temperature domain walls of QCD withimaginary chemical potentials
Hiromichi Nishimura, Yuya Tanizaki , RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973 USA Department of Physics, North Carolina State University, Raleigh, NC 27607, USA
E-mail: [email protected] , [email protected] Abstract:
We study QCD with massless quarks on R × S under symmetry-twistedboundary conditions with small compactification radius, i.e. at high temperatures. Undersuitable boundary conditions, the theory acquires a part of the center symmetry and it isspontaneously broken at high temperatures. We show that these vacua at high temperaturescan be regarded as different symmetry-protected topological orders, and the domain wallsbetween them support nontrivial massless gauge theories as a consequence of anomaly-inflowmechanism. At sufficiently high temperatures, we can perform the semiclassical analysisto obtain the domain-wall theory, and d U ( N c − gauge theories with massless fermionsmatch the ’t Hooft anomaly. We perform these analysis for the high-temperature domainwall of Z N c -QCD and also of Roberge-Weiss phase transitions. a r X i v : . [ h e p - t h ] J un ontents SU ( N c ) gauge theory in high-temperature phase 33 Domain wall at the Roberge-Weiss phase transitions 4 T (cid:29) Λ d bulk 8 Z N -QCD at high temperatures 8 Z N -QCD and center symmetry 94.2 Semiclassical analysis of the domain wall theory at T (cid:29) Λ Confinement is one of the most important properties of non-Abelian gauge theory, and itstill acquires a lot of interest to uncover its property. Although we cannot characterizeconfinement or deconfinement by using local order parameter, they are characterized byinfrared behaviors of the loop operator, called Wilson loops, for certain SU ( N c ) gaugetheories [1–3]. In this sense, SU ( N c ) Yang-Mills theory has the center symmetry Z N c ,which is recently called Z N c one-form symmetry [4]. Spontaneous breakdown of higher-formsymmetries imply the appearance of topological gauge theories in the infrared behaviors,and thus confinement and Higgs phases are separated as topological orders [5–7]. In ourreal world, the strong-interaction sector of the Standard Model is described by quantumchromodynamics (QCD), which is the SU ( N c ) gauge theory coupled to the Dirac fermionsin the fundamental representation. In this case, since the color flux between two test quarkscan break up by pair productions of dynamical quarks, we do not have a clear separationbetween confinement and Higgs phases as quantum phases of matters. In other words,the center symmetry is explicitly broken by the existence of dynamical quarks. Instead,QCD acquires chiral symmetry when quark masses are quite small, and chiral symmetry isspontaneously broken so as to generate mass scale [8, 9].– 1 –hese interesting behaviors, confinement and chiral symmetry breaking, are the conse-quence of strong infrared dynamics, and it is usually very difficult to extract such informa-tion starting from QCD. An important direction is to perform the first-principle numericalcomputations of these systems, and the most established one is the Monte Carlo simulationof lattice gauge theories [1, 10]. Another important direction is to discuss mathematicallyrigorous nature of QFTs. For this purpose, we have to find a quantity that is easily com-putable but is not affected by renormalization. Historically, this turned out to be true for’t Hooft anomaly of QCD with massless quarks [11–13], and we can conclude the existence ofmassless bound states when color degrees of freedom cannot be seen in the infrared. Thanksto the development of symmetry-protected topological phases [14–19], people understandthat the applicability of ’t Hooft anomaly matching condition is much broader, and anomalymatching conditions are providing new insights on strongly-coupled QFTs [20–54].Using this development of knowledge, we study properties of confinement/deconfinementfor QCD with massless quarks. Although the center symmetry does not exist for QCD asfour-dimensional quantum field theories, we can find its interesting remnant by consideringthe symmetry-twisted boundary condition on the compactified spacetime M = M × S (cid:51) ( x , τ ) . This is first discussed by Roberge and Weiss [55]: They consider QCD and introducethe non-thermal boundary condition for the quark field ψ as ψ ( x , τ + β ) = e i φ ψ ( x , τ ) . (1.1)Because of the gauge invariance, the partition function has the periodicity π/N c as afunction of φ , Z ( φ + 2 π/N c ) = Z ( φ ) , instead of the naive periodicity π . This periodicity π/N c can be easily understood when quarks are confined inside hadrons. However, thisperiodicity is nontrivial when quarks are deconfined, and we have to introduce N c branchstructure of the free energy to make consistency. As a consequence, there exist the first-order phase transitions between N c quasi-vacua at high temperatures, which are calledRoberge-Weiss (RW) phase transitions. There are many studies on the property of RWphase transitions [56–65].Since the RW point has the first-order phase transition, we can consider the domainwall connecting those two pure states, and we will call it the high-temperature domain wall.High-temperature domain walls are recently studied for QCD with adjoint fermions [49, 66],and the d gauge theories coupled to chiral fermions appear on the domain wall. In adjointQCD, the existence of chiral fermions on the domain wall is protected by mixed ’t Hooftanomaly between the center symmetry and the discrete chiral symmetry. In this paper,we will also find that the high-temperature domain wall at RW point supports d gaugetheories with massless fermions despite the fact that there is no one-form symmetry. Weconstruct those gauge theories explicitly at sufficiently high temperatures, and compute the’t Hooft anomaly for those theories on the domain wall.Our computation of the ’t Hooft anomaly of high-temperature domain walls suggestthat the RW phase transition can be regarded as the phase transition between differentsymmetry-protected topological (SPT) orders. This new insight is consistent with therecent result in Refs. [31, 54]: the RW point has the mixed anomaly between “RW parity”symmetry and the chiral symmetry. One can also regard that our computation gives an– 2 –xplicit explanation about how the subtle parity anomaly found in Refs. [31, 54] is realizedin the high-temperature QCD with imaginary chemical potential.In this paper, we also discuss the massless Z N -QCD [36, 37, 67–76], which is SU ( N ) gauge theory with N -flavor massless fundamental quarks with symmetry twisted boundarycondition, ψ f ( x , τ + β ) = e π i f/N +i φ ψ f ( x , τ ) . (1.2)This boundary condition is a special one since we have Z N center-related symmetry as athree-dimensional QFT [36, 37, 76], and we can discuss the domain wall connecting thosevacua. Those domain walls are again given by d U ( N − gauge theory coupled to masslessfermions, but they produce the different chiral anomaly from that of RW domain wall. Wewill again see that it satisfies the anomaly-inflow mechanism from the bulk SPT phases bythe help of the result in Refs. [36, 37].The paper is organized as follows. In Sec. 2, we give a brief description about thehigh-temperature domain wall in pure SU ( N c ) Yang-Mills theory. In Sec. 3, we discuss themassless gauge theories on high-temperature domain walls at the RW phase transition, andcompute its ’t Hooft anomaly. In Sec. 4, we study the high-temperature domain walls of Z N -QCD. We summarize the result in Sec. 5. In Appendix A, we set our convention aboutthe chirality of massless fermions on the domain wall. In Appendix B, we give justificationof our ansatz about high-temperature domain wall used in this paper. SU ( N c ) gauge theory in high-temperature phase Here, we consider the pure SU ( N c ) Yang-Mills theory, for simplicity, and assume sufficientlyhigh temperatures compared with the strong scale Λ , T (cid:29) Λ . At high-temperature, Z N c center symmetry is spontaneously broken and there are N c discrete vacua, described by theexpectations values of the Polyakov loop, Φ( x ) = P exp (cid:90) β a ( x , τ )d τ. (2.1)Let us take the Polyakov gauge [77], and then the effective action becomes S − loop = βg (cid:90) R d x (cid:18)
12 tr[ F IJ ] + tr[( D I a ) ] + g V ( a ) (cid:19) , (2.2)with the one-loop effective action [78, 79] V ( a ) = − π β (cid:88) n ≥ n (cid:18)(cid:12)(cid:12)(cid:12) tr c (Φ n ) (cid:12)(cid:12)(cid:12) − N (cid:19) . (2.3)We here give an offset to V ( a ) so that V ( a ) ≥ and V (0) = 0 . This -dimensional theoryis again a strongly-coupled non-Abelian gauge theory, and we cannot solve it analytically.Throughout this paper, we assume the standard lore saying that -dimensional gluons getmass gap and the spatial Wilson loop shows area law, and the Z N c zero-form symmetry Φ (cid:55)→ e π i /N c Φ is spontaneously broken as suggested by the one-loop effective action.– 3 –here are N c vacua characterized by the Polyakov-loop expectation values, N c (cid:104) tr(Φ) (cid:105) = e π i k/N c , (2.4)with k = 0 , , . . . , N c − . We can consider a domain wall connecting these pure states, andcall it a high-temperature domain wall. We put the following ansatz of the high-temperaturedomain wall [80], Φ( x ) = exp (i ρ ( x ) T N c − ) , (2.5)where T N c − = πN c diag(1 , . . . , , − N c ) is the last Cartan element of su ( N c ) , and ρ ( x ) should be determined by the classical equation of motion with the boundary condition ρ ( x = −∞ ) = 0 and ρ ( x = ∞ ) = 1 . This domain wall connects the vacua k = 0 and k = 1 . We can discuss the BPS bound [81, 82] in terms of ρ ( x ) within this ansatz.Justification of this ansatz will be discussed in Appendix B.With this ansatz, the adjoint Higgsing SU ( N c ) → U ( N c −
1) = [ SU ( N c − × U (1)] / Z N c − (2.6)occurs near the domain wall, ρ (cid:39) / . The division by Z N c − can be understood as follows:The embedding SU ( N c − × U (1) (cid:44) → SU ( N c ) is given by ( U N c − , e i φ ) (cid:55)→ (cid:32) e i φ U N c −
00 e − i( N c − φ (cid:33) . (2.7)The kernel of this embedding is given by Z N c − , whose generator is given by the element (e π i / ( N c − N c − , e − π i / ( N c − ) , and thus the image of the map inside SU ( N c ) is isomorphicto [ SU ( N c − × U (1)] / Z N c − . The defining representation N c breaks up into ( N c − ) +1 ⊕ ( ) − ( N c − , where ( R ) Q denotes the representation R of SU ( N c − with the U (1) charge Q . The possible value of Q is constrained by the ( N c − -ality of R .Although we obtain a nontrivial gauge theory on the domain wall, we do not expect thatit causes any interesting low-energy physics like spontaneous symmetry breaking after takinginto account the quantum fluctuation because even d U (1) pure gauge theory acquires thetrivial mass gap. Only exception would be the case with topological term with θ = π ,as discussed in Refs. [25, 66]. This is a good lesson for us; even if we find the masslessLagrangian on the domain wall, it is important to discuss whether that massless naturesurvives under quantum and thermal fluctuations (see also Refs. [83, 84]). This motivatesus to compute the ’t Hooft anomaly of the domain wall theory [29]. In this section, we consider QCD with quark imaginary chemical potential φ/β , which isrelated to the U (1) V -twisted boundary condition on the quark field, ψ ( τ + β ) = e i φ ψ ( τ ) .Roberge and Weiss showed that the QCD partition function has a fractional periodicity φ ∼ φ + 2 π/N c by the gauge invariance [55]. This Roberge-Weiss periodicity concludes thatthese is a Z center symmetry for quantized values of φ [31, 54], and we call it the RW– 4 –arity. At high temperatures, this RW parity is spontaneously broken, and there are twodegenerate vacua with the mass gap as the three-dimensional quantum field theory. Weshow that these distinct vacua can be regarded as the different SPT phases by studyingthe domain wall between them. The RW domain wall supports massless field theories byanomaly-inflow arguments. Let us start with a review of RW phase transition [55]. We consider the QCD partitionfunction with U (1) imaginary chemical potential, Z ( T, φ ) = tr (cid:104) exp (cid:16) − β ( (cid:98) H + i µ I (cid:98) Q ) (cid:17)(cid:105) , (3.1)where (cid:98) H is the QCD Hamiltonian with massless quarks, (cid:98) Q is the quark number operator, T = 1 /β is the temperature, µ I is the imaginary chemical potential, and we set φ = βµ I + π .In the path-integral expression, this is the SU ( N c ) Yang-Mills theory coupled to N f -flavormassless Dirac fermions, S = 1 g (cid:90) tr[ F c ∧ (cid:63)F c ] + (cid:90) d x N f (cid:88) f =1 ψ f γ µ ( ∂ µ + a µ ) ψ f , (3.2)where a is the SU ( N c ) gauge field, F c = d a + a ∧ a is the SU ( N c ) field strength, and ψ f , ψ f are the four-dimensional Dirac fermions with the flavor label f = 1 , . . . , N f . Our spacetimeis R × S , and the boundary condition of the quark fields is twisted by U (1) phase, ψ ( x , τ + β ) = e i φ ψ ( x , τ ) . (3.3)Naively, we expect the periodicity φ ∼ φ +2 π from this expression. Since the gauge-invariantoperators have the charge in N c Z , however, the partition function has a shorter periodicity φ ∼ φ +2 π/N c . In other words, we can consider the transformation, Φ( x ) (cid:55)→ e π i /N c Φ( x ) , onthe Polyakov loop associated with simultaneously change the boundary condition of quarkfields as ψ ( x , τ + β ) = e i( φ +2 π/N c ) ψ ( x , τ ) , (3.4)and then the value of the partition function does not change. This is called the Roberge-Weiss (RW) periodicity [55]. Here, it is important to notice that the above transformation is not symmetry because the boundary condition of matter fields are different. RW periodicityjust says that the theory with two different boundary conditions have the same free energy.The continuous symmetry of massless QCD is given by [31, 36, 37, 50, 54] G = SU ( N f ) L × SU ( N f ) R × U (1) V Z N c × Z N f . (3.5)For generic values of φ , the charge conjugation C : ψ (cid:55)→ C ψ t with C = i γ γ is explicitlybroken by the boundary condition, because (3.3) is changed as ψ ( x , τ + β ) = e − i φ ψ ( x , τ ) . (3.6)– 5 –f φ is quantized to π/N c , we can construct the charge-conjugation symmetry by simultane-ously inserting the appropriate color ’t Hooft magnetic flux [31, 54] : When φ = − ( π/N c ) k ,we define the following Z transformation, Φ( x ) (cid:55)→ e π i k/N c Φ( x ) † , ψ ( x , τ ) (cid:55)→ e (2 π i /N c ) k C ψ ( x , τ ) t . (3.7)We call this Z transformation as the RW parity. After this transformation, the boundarycondition is changed as ψ ( x , τ + β ) = e − π i k/N c − i φ ψ ( x , τ ) , (3.8)which is the same with (3.3) when φ = − πk/N c . Therefore, the RW parity is the symmetryof the theory, and thus the symmetry group is enhanced at φ = − πk/N c as G (cid:111) ( Z ) RW . (3.9)Now, let us discuss the high-temperature phase of massless QCD with imaginary chem-ical potential. High- T behavior is controlled by the one-loop potential, and they are givenby V gluon = − π β (cid:88) n ≥ n (cid:18)(cid:12)(cid:12)(cid:12) tr c (Φ n ) (cid:12)(cid:12)(cid:12) − (cid:19) , (3.10) V quark = 2 N f π β (cid:88) n ≥ n (cid:16) e i nφ tr c (Φ n ) + e − i nφ tr c ((Φ † ) n ) (cid:17) . (3.11)We immediately see that, at φ = − π − πN c k , the classical vacuum Φ k = e π i k/N c is chosen.At the intermediate point, φ = − π − πN c , the vacua connected to k = 0 and k = 1 has thesame lowest energy, and we have the first-order phase transition between those two states:this is called the RW phase transition. Since these pure states k = 0 and k = 1 are relatedby ( Z ) RW , the RW phase transition is the consequence of spontaneous breakdown of RWparity symmetry. T (cid:29) Λ At really high temperatures, T (cid:29) Λ , we can do a certain semi-classical calculations dueto asymptotic freedom, and obtain the three-dimensional effective theory as we have ex-plained. Strictly speaking, we cannot still solve the problem even in that regime, since sucha dimensionally-reduced theory is again typically strongly coupled and cannot be solved.Therefore, let us adopt a standard lore that the three-dimensional Yang-Mills theory (with-out Chern-Simons terms) is trivially gapped with the mass scale ∼ g T , and we will derivethe nontrivial domain-wall theory under this assumption.Since the RW parity is spontaneously broken at the RW phase transition, we canconsider the high-temperature domain wall connecting two pure states. In the following,we take the RW point, φ = − π − πN c , (3.12) In Ref. [54], the author considered the PT transformation R : ( x , τ ) (cid:55)→ ( − x , − τ ) instead of C . Thesetwo are equivalent because of CPT theorem. – 6 –nd consider the domain wall connecting k = 0 and k = 1 : Φ( x ) = exp (i ρ ( x ) T N c − ) , (3.13)with ρ ( −∞ ) = 0 and ρ ( ∞ ) = 1 .The d kinetic term of the f -th flavor fermion is given by ψ f γ I D I ψ f + ψ f γ (cid:18) ∂ + i β ρ ( x ) T N c − (cid:19) ψ f , (3.14)and the imaginary-time direction gives the real mass term for d fermions with the mass m n ( x ) = 2 πn + ρ ( x ) T N c − − π − πN . (3.15)Under the Higgsing SU ( N c ) → [ SU ( N c − × U (1)] / Z N c − , the fermion is decomposed into ( N c − ) and ( ) − ( N c − . The mass function for ( N c − ) is m ( N c − ) n ( x ) = 2 πn + 2 πN ρ ( x ) − π − πN (cid:54) = 0 . (3.16)Thus, ( N c − ) completely decouples in the low-energy limit. The mass function for ( ) − ( N c − is m ( ) n ( x ) = 2 πn − πN ( N − ρ ( x ) − π − πN . (3.17)For n = 1 , this takes zero at x = 0 , and others cannot be zero. As a consequence, we get N f -flavor d Dirac fermions with the gauge charge ( ) − ( N c − living on the domain wall(see Appendix A for details).The quark kinetic term on the domain wall is given by N f (cid:88) f =1 ψ ( ) f (cid:0) σ I [ ∂ I − tr( a (cid:48) I )] (cid:1) ψ ( ) f , (3.18)where a (cid:48) is the U ( N c − gauge field and ψ ( ) f is the d massless Dirac fermion in therepresentation ( ) − ( N c − , which comes out of the normalizable zero mode of d Diracfermion ψ f . This low-energy theory has the chiral symmetry, SU ( N f ) L × SU ( N f ) R Z N f ⊂ G. (3.19)The U (1) baryon number symmetry in G cannot be seen within this low-energy Lagrangiansince the fermions in the ( N c − ) representation is completely neglected because of theirnonzero thermal mass. This two-dimensional field theory has the chiral anomaly character-ized by three-dimensional level- Chern-Simons action, CS [ L ] − CS [ R ] = 14 π tr (cid:18) L d L + 23 L (cid:19) − π tr (cid:18) R d R + 23 R (cid:19) , (3.20)where L and R are background gauge fields for SU ( N f ) L and SU ( N f ) R , respectively.– 7 –ere, we demonstrate that the high-temperature domain wall at the RW phase transi-tion supports U ( N c − gauge theory coupled to N f massless d Dirac fermions by semiclas-sical analysis at sufficiently high temperatures. We expect that the validity of that effectivetheory is limited to sufficiently high temperatures because the theory is expected to be-come strongly coupled near the chiral restoration temperatures. The anomaly, however, isa topological obstruction of gauging the global symmetry and it cannot be changed underlocal deformation of the Lagrangian so long as the symmetry is respected. This suggeststhat, so long as the RW parity is spontaneously broken, the high- T domain wall shouldsupports the d massless field theory with an ’t Hooft anomaly characterized by (3.20).Since the same anomaly is carried by the level- SU ( N f ) Wess-Zumino-Witten model, itseems to be natural to expect that the long-range behavior on the high- T domain wall isgiven by that conformal field theory. d bulk In this section, we confirm more explicitly that the domain-wall theory should have ananomaly (3.20) by using the anomaly-inflow mechanism [85]. Recent understanding ofthe ’t Hooft anomaly says that the system with ’t Hooft anomaly should be realized as aboundary of symmetry-protected topological (SPT) orders if anomalous symmetry is weaklygauged [14–17]. Then, the anomaly of the boundary theory is canceled by anomaly inflowfrom the bulk SPT order, and the combined system has no anomaly. In the case of the RWphase transition, since -dimensional theory has no chiral anomaly, the bulk gapped statesseparated by the high-temperature domain wall can be regarded as different SPT phasesprotected by the chiral symmetry.The above anomaly-inflow discussion has a nice consistency with the recent discussionin Refs. [31, 54]. These papers show that the RW point has a mixed ’t Hooft anomaly be-tween the RW parity and the chiral symmetry: Let us consider the QCD partition function Z RW [ L, R ] on R × S with the background -dimensional SU ( N f ) L , R gauge fields L, R .With these backgrounds, the RW parity at φ = − π − π/N c is anomalously broken, ( Z ) RW : Z RW [ L, R ] (cid:55)→ Z RW [ L, R ] exp (CS [ L ] − CS [ R ]) . (3.21)This relation shows that, when RW parity is broken, the partition functions of those twopure states are different by the Chern-Simons action, exp (CS [ L ] − CS [ R ]) . Since we areassuming that the d bulk is gapped, this justifies that these two pure states are differentas SPT orders protected by the chiral symmetry. Z N -QCD at high temperatures In this section, we consider massless QCD with N c = N f = N , and we take the flavor-dependent boundary condition. This theory has the color-flavor locked center symmetry Z N , and it is called Z N -QCD [36, 37, 67–76]. At high temperatures, this center symmetryis spontaneously broken, and there are N distinct vacua with the mass gap as the three-dimensional quantum field theory. As in the case of the RW phase transition, we show that– 8 –hese distinct vacua can be regarded as the different SPT phases. As a consequence, thedomain wall connecting them support massless field theories by anomaly-inflow arguments. Z N -QCD and center symmetry Z N -QCD is SU ( N ) gauge theory with degenerate N -flavor fundamental quarks with boththe flavor-twisted and U (1) V -twisted boundary conditions: ψ f ( x , τ + β ) = exp (cid:18) i 2 πN f + i φ (cid:19) ψ f ( x , τ ) (4.1)where f = 1 , . . . , N . As we show below, this theory has the color-flavor locked Z N centersymmetry, so it is called Z N -QCD [36, 37, 67–76]. We take the fermion mass to be zero inthis paper.Let us give a detailed comment on the symmetry of massless Z N -QCD. We start withthe internal symmetry of massless QCD for generic numbers of color N c and flavor N f : G = SU ( N f ) L × SU ( N f ) R × U (1) V Z N c × Z N f . (4.2)Representing the Dirac fields in the chiral basis, ψ = ( ψ R , ψ L ) , SU ( N f ) L × SU ( N f ) R × U (1) V acts on the quark field ψ as ( g L , g R , e i α ) : ψ (cid:55)→ e i α ( g R ψ R , g L ψ L ) . (4.3)Since ( g L , g R , e i α ) and ( g L e π i /N f , g R e π i /N f , e i α − π i /N f ) give the same mapping, the symme-try group must be divided by Z N f to remove this redundancy. Furthermore, any gauge-invariant local operator has the quantized charge N c under U (1) V , and thus we also haveto introduce the identification, α ∼ α + 2 π/N c , and obtain the above symmetry group. Weare interested in the subgroup of G : G sub = SU ( N f ) V × U (1) V Z N c × Z N f × ( Z N f ) L . (4.4)Let us now set N c = N f = N and consider the symmetry of Z N -QCD. The vector-like flavorsymmetry SU ( N ) is broken down to U (1) N − because of the flavor-dependent boundaryconditions, so naively the symmetry group seems to become U (1) N ( Z N ) c × ( Z N ) L , (4.5)but there is an extra Z N symmetry, called the color-flavor center symmetry [36, 37, 76].The center symmetry of the pure Yang-Mills theory is the Z N one-form symmetryacting on the Wilson lines. On R × S , it induces the Z N zero-form symmetry acting onthe Polyakov loop Φ( x ) as Φ( x ) (cid:55)→ e π i /N Φ( x ) . (4.6)This symmetry is explicitly broken in QCD, because it changes the boundary condition ofthe quark field as ψ f ( x , τ + β ) = e π i /N exp (cid:18) i 2 πN f + i φ (cid:19) ψ f ( x , τ ) . (4.7)– 9 –or Z N -QCD, however, one can compensate this violation by performing the shift of theflavor label, ψ f (cid:55)→ ψ f +1 , which is a part of the vector-like SU ( N f ) × U (1) symmetry.Therefore, Z N -QCD has a symmetry generated by [36, 37, 76] Φ( x ) (cid:55)→ e π i /N Φ( x ) , ψ f (cid:55)→ ψ f +1 , (4.8)and we call it the center symmetry ( Z N ) center . The symmetry group of massless Z N -QCDis thus obtained: ( Z N ) center (cid:110) U (1) N Z N × ( Z N ) L . (4.9)The noncommutativity between ( Z N ) center and U (1) N originates from the fact that U (1) N is the maximal Abelian subgroup of U ( N ) and the shift of the flavor label is given by thenon-diagonal matrix of U ( N ) . T (cid:29) Λ Let us make the quark field periodic by mapping ψ f ( τ ) (cid:55)→ exp (cid:16) i β (cid:0) πN f + φ (cid:1) τ (cid:17) ψ f ( τ ) .Then, the Dirac operator becomes (cid:88) I =1 ψ f γ I D I ψ f + ψ f γ (cid:18) ∂ + a + i 2 πN β f + i φβ (cid:19) ψ f . (4.10)We take the Polyakov gauge, so that a is diagonal and τ -independent. We take a domain-wall solution, which connects two perturbative vacuum of the gluon one-loop potential, Φ = and Φ = e π i /N , as (cid:104) Φ( x ) (cid:105) = (cid:104) e i βa ( x ) (cid:105) = exp (i ρ ( x ) T N − ) , (4.11)where T N − = πN diag(1 , . . . , , − N ) . Since the fluctuation of a should be small at high temperatures, we can take (cid:104) a ( x ) (cid:105) = i ρ ( x ) T N − β . (4.12)Therefore, the quark kinetic term becomes (cid:88) i =1 ψ f γ i D i ψ f + ψ f (cid:20) γ ∂ + 2 π i β γ (cid:18) n + ρ ( x )2 π T N − + fN + φ π (cid:19)(cid:21) ψ f . (4.13)As we mentioned, we want to take a special φ so that the real mass is non-zero for any n and f = 0 , . . . , N − when ρ ( x ) = 0 . As such an example, let us take φ = − π/N, (4.14) Exact expression or exact location of the classical vacuum does not matter in the following argument.But we point out that the gluon potential is O ( N ) while the quark potential is O ( N − ) under this twistedboundary condition [76], and there are the factor difference already for N = 3 . So the use of theclassical vacuum of gluon potential should be a good approximation. – 10 –hen the real mass on the bulk becomes πβ (cid:18) n + 2 f − N (cid:19) (cid:54) = 0 . (4.15)This ensures that the quarks are classically massive as it acquires the real mass, m (cid:38) N πT .The gauge group SU ( N ) is Higgsed to [ SU ( N − × U (1)] / Z N − near the domainwall. The fundamental quark in the representation N is thus breaks into ( N − ) and ( ) − ( N − . First ( N − color of fermions (i.e. ( N − ) ) have the mass m ( N − ) n,f ( x ) = 2 πN β (cid:18) N n + ρ ( x ) + f − (cid:19) , (4.16)with f = 0 , , . . . , N − and ρ ( −∞ ) ≤ ρ ( x ) ≤ ρ ( ∞ ) = 1 . The mass function flipits sign only for n = 0 and f = 0 , and others have the definite sign. Therefore, only themode with n = 0 and f = 0 can be a candidate of the domain-wall fermions with the gaugerepresentation ( N − ) . Second the last color component of the fermion (i.e. ( ) − ( N − )has the mass function m ( ) − ( N − n,f = 2 πN β (cid:18) N n − ρ ( x )( N −
1) + f − (cid:19) . (4.17)For n (cid:54) = 0 , this always has the definite mass, and so does for f = 0 . Thus the candidate ofthe domain wall fermions are n = 0 and f = 1 , . . . , N − .Since the direction of the sign of those domain wall masses are flipped between ( N − ) and ( ) − ( N − , the chirality between these two representations of D fermions are opposite.For the convention of the chirality, one can see Appendix A. Thus, the gauged quark kineticterm now becomes ψ N − σ i ( ∂ i + a (cid:48) i + A ,i − A χ,i P L ) ψ N − + N − (cid:88) f (cid:48) =1 ψ f (cid:48) σ i ( ∂ i − tr[ a (cid:48) i ] + A f (cid:48) ,i + A χ,i P L ) ψ f (cid:48) , (4.18)where a (cid:48) is the U ( N − dynamical gauge field, A f are U (1) background gauge fieldsfor f -th flavor rotation, and A χ is the Z N background gauge field for the discrete chiralsymmetry. We can find that the mixed anomaly between the gauge symmetry U ( N − and the discrete chiral symmetry ( Z N ) L cancels among N domain-wall fermions, and thenwe find the following SPT action for the ’t Hooft anomaly of the domain-wall theory, S SPT , DW = N − (cid:88) f =0 π (cid:90) A χ ∧ d A f . (4.19)In order to find this result, it is convenient to use the Stora-Zumino chain. The starting– 11 –oint is the -dimensional Abelian anomaly A ; A = 2 π π ) (cid:90) tr ( N − (cid:2) ( F (cid:48) + d A − d A χ ) − ( F (cid:48) + d A ) (cid:3) + N − (cid:88) f (cid:48) =1 π π ) (cid:90) (cid:2) ( − tr F (cid:48) + d A f (cid:48) + d A χ ) − ( − tr F (cid:48) + d A f (cid:48) ) (cid:3) = N − (cid:88) f =0 π (cid:90) d A χ ∧ d A f − π (cid:90) N d A χ ∧ (cid:0) tr F (cid:48) + d A (cid:1) . (4.20)Here, F (cid:48) = d a (cid:48) + a (cid:48) ∧ a (cid:48) is the U ( N − field strength, and the second term of the lastline vanishes modulo π . We obtain the -dimensional topological action S SPT , DW as aboundary theory of A .Here, we elucidated that the domain wall connecting different vacua related by ( Z N ) center supports the (1 + 1) -dimensional gauge theory with massless Dirac fermions. The computa-tion is done in the semiclassical regime, T (cid:29) Λ , but we can argue its persistence because ofthe topological nature of anomaly. In other words, the gapped vacua related by ( Z N ) center are different as symmetry-protected topological order, and the difference is characterizedby the Z N topological action (4.19). Let us emphasize that this facts survive even at T (cid:38) Λ so long as the system is in the deconfined phase.Moreover, we can also prove this statement from the anomaly-inflow mechanism as wehave done for the Roberge-Weiss high-temperature domain wall in Sec. 3.3. In Refs. [36, 37],it is found that Z N -QCD has the mixed ’t Hooft anomaly among ( Z N ) center , U (1) N / Z N ,and ( Z N ) L symmetries. In our context, it is useful to summarize this result as the partitionfunction Z Z N of Z N -QCD breaks ( Z N ) center symmetry anomalously under the existence ofbackground gauge fields: ( Z N ) center : Z Z N [ A f , A χ ] (cid:55)→ Z Z N [ A f , A χ ] exp (i S SPT , DW [ A f , A χ ]) . (4.21)This says that the d high-temperature states related by the broken center symmetry canbe regarded as the different symmetry protected topological states, and the domain wallbetween them should cancel the anomaly inflow from the bulk. This argument does notuse any concrete information of the construction of domain-wall theories, and thus it showsthe robustness of the existence of nontrivial ground states under the effect of quantum andthermal fluctuations. We study the domain-wall localized theories at the high-temperature phase of QCD withmassless fundamental quarks under symmetry-twisted boundary conditions, especially forthe Roberge-Weiss phase transition and the Z N -QCD. These theories has the center-relateddiscrete symmetry, and it is spontaneously broken at high-temperature phases. We findthat the domain wall connecting distinct states related by the broken center symmetrysupports U ( N − gauge theory with d massless Dirac fermions by explicit weak-couplingcomputation at sufficiently high temperatures.– 12 –hese domain-wall localized theories has the chiral flavor symmetry with an ’t Hooftanomaly. Since ’t Hooft anomaly is a topological object, we argue the persistence of gapplessexcitations on the domain wall even in the strongly-coupled region of the QCD phasediagram as long as the center symmetry is spontaneously broken. We prove this statementusing the recent developments about the relation between ’t Hooft anomaly and SPT orderswith anomaly-inflow mechanism. In other words, we give an interpretation of the purestates related by the broken center symmetry as different SPT orders protected by chiralsymmetry. Acknowledgments
The work of H. N. and Y. T. were supported by Special Postdoctoral Researchers Programof RIKEN. After April, the work of Y. T. was supported by JSPS Overseas Fellowships.
A Domain-wall Dirac fermions and chirality
First, we clarify our convention of the gamma matrices in four dimensions. We only considerthe flat Euclidean spacetime.In four dimensions, the gamma matrices are the × matrices γ µ ( µ = 1 , . . . , ),satisfying { γ µ , γ ν } = 2 δ µν . (A.1)The γ matrix is introduced by γ = γ γ γ γ . We realize this algebra by the chiralrepresentation, γ I = (cid:32) σ I σ I (cid:33) , γ = (cid:32) − i (cid:33) , (A.2)where σ I ( I = 1 , , ) are the × Pauli matrices. The γ matrix is expressed by thediagonal matrix in this representation, γ = γ γ γ γ = (cid:32) − (cid:33) . (A.3)The left- and right-handed spinors are defined by the projectors P L = (1 − γ ) / and P R = (1+ γ ) / , respectively. Therefore, the four-component Dirac fermion ψ is representedas ψ = ( ψ R , ψ L ) = ( ψ R+ , ψ R − , ψ L+ , ψ L − ) . That is, the first two components are right-handed and the last ones are left-handed.We now consider the domain wall fermion. Our set up is that the fourth direction iscompactified τ ∼ τ + β , and β is sufficiently small. The domain wall is set at x = 0 along the x - x directions. Using the real mass function m ( x ) , the domain wall fermion isobtained as the zero-mode solution of the Dirac equation, [ γ ∂ + γ i m ( x )] ψ = 0 . (A.4)We can easily solve this equation of motion as ψ ( x ) = exp (cid:20) − i γ γ (cid:90) x m ( s )d s (cid:21) ψ (0) . (A.5)– 13 –ere, i γ γ = diag( − σ , σ ) = diag( − , , , − .When m (+ ∞ ) > and m ( −∞ ) < , the normalizability requires that the first andfourth components must vanish, so that the normalizable zero-mode is given by ψ =(0 , ψ R − , ψ L+ , . This is the two-dimensional Dirac fermion on the domain wall, and thechirality of two-dimensions and that of four-dimensions are flipped.When m (+ ∞ ) < and m ( −∞ ) > , the normalizability requires that the secondand third components must vanish, so that the normalizable zero-mode is given by ψ =( ψ R+ , , , ψ L − ) . This is also the two-dimensional Dirac fermion on the domain wall, butthe chirality of two-dimensions and that of four-dimensions are the same. B Justification of the ansatz of the domain wall
In this appendix, we will show that the ansatz for the domain wall is correct. This is partlydiscussed in Ref. [80] for N = 3 and N = ∞ , and we here provide the discussion for generalvalues of N . For simplicity, we restrict our attention to the pure gluon potential in thisAppendix.We take the following basis of the Cartan matrices of the su ( N ) Lie algebra: H = diag(1 , − , , . . . , , H = diag(0 , , − , . . . , , . . . , H N − = diag(0 , . . . , , − , ,T N − = 2 πN diag(1 , . . . , , − ( N − . (B.1)This satisfies tr( H i H j ) = 2 δ i j − δ i +1 j − δ i j +1 , tr( H i T N − ) = 0 . (B.2)The matrix tr( H i H j ) is the tridiagonal Toeplitz matrix. It is a positive matrix, and itseigenvalues are given as (cid:16) πkN − (cid:17) > with k = 1 , . . . , N − . Another importantproperty in this Cartan basis is that H i T N − = 2 πN H i . (B.3)In this basis, we can write the Polyakov loop as Φ = exp (cid:16) i (cid:126)θ · (cid:126)H + i ρT N − (cid:17) = diag (cid:16) e i( θ +(2 π/N ) ρ ) , e i( θ − θ +(2 π/N ) ρ ) , . . . , e i( − θ N − +(2 π/N ) ρ ) , e − π i( N − ρ/N (cid:17) , (B.4)where (cid:126)θ = ( θ , . . . , θ N − ) and (cid:126)θ · (cid:126)H = θ H + · · · + θ N − H N − .We note that the gradient of tr(Φ n ) vanishes at (cid:126)θ = 0 : ∂∂θ i tr(Φ n ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ =0 = i n tr (cid:16) H i exp(i nρT N − ) (cid:17) = 0 . (B.5)Let us also compute the Hesse matrix, then we get ∂ ∂θ i ∂θ j tr(Φ n ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ =0 = − n tr (cid:16) H i H j exp(i nρT N − ) (cid:17) = − n e π i nρ/N tr( H i H j ) . (B.6)– 14 –sing these properties, we show that at each fixed ρ a local minimum of the -loop Polyakov-loop potential (2.3) locates at (cid:126)θ = 0 . First we take a derivative of V with respect to θ i , ∂∂θ i V = − iπ β (cid:88) n (cid:54) =0 tr c ( H i Φ n ) n tr c Φ − n , (B.7)and find that it is zero at (cid:126)θ = 0 . Second we compute the Hesse matrix, ∂ ∂θ i ∂θ j V = 2 π β (cid:88) n (cid:54) =0 tr c ( H i H j Φ n ) n tr c Φ − n − π β (cid:88) n (cid:54) =0 tr c ( H i Φ n ) tr c ( H j Φ − n ) n . (B.8)At (cid:126)θ = 0 , it becomes ∂ ∂θ i ∂θ j V (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ =0 = (cid:34) π β ∞ (cid:88) n =1 N − πnρ ) n (cid:35) tr ( H i H j ) . (B.9)This is a Toeplitz matrix, whose overall factor in the bracket is nonzero and positive forany value of ρ . Therefore (cid:126)θ = 0 is a local minimum of the 1-loop Polyakov loop potentialfor any N . References [1] K. G. Wilson, “Confinement of Quarks,”
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