The Axial Charge in Hilbert Space and the Role in Chiral Gauge Theories

The Axi al Charge in Hilbert Space and the Rol e in Chiral Gauge Theories T atsuya Y amaoka ∗ Departmen t of Ph y sics, the U niv ersity of Osak a, T oyonaka , Os aka 560-004 3, Japan E-mail: t_yama oka@he t.phys.sci.osaka-u.ac.jp W e inv es tig ate the Hamiltonian f or mulation of 1+1- dimensional stagg ered f ermio ns and recon- struct the v ector and a x ial charge operators, o rig inally identified b y Ark ya Chatter jee et a l. , within the Wilson fermio n f or m alism. Th ese oper a to rs commute with the Hamiltonian an d red u ce, in the continuu m limit, t o the generators of the v ector and axial U ( 1 ) symmetr ies. A notable f eature of the axial charge op erator is th a t it acts locall y on operators and p ossesses quantized eigenv alues. Its eig enstates can theref ore be inter preted as f er mion states with well-defined in teger chirality , analogo u s to those in th e con tinuum th e o r y . This structu re enables the f or mulation of a gaug e theor y in which the axial U ( 1 ) 𝐴 symmetr y is promo ted to a gaug e sym metr y . W e c o nstruc t a Hamiltonian in te r m s o f the eigenstates o f the axial charg e op erator, thereby preser vin g e xact axial symmetr y on th e lattice while reco v erin g vector symmetr y in the con tin uum limit. As ap plica- tions, we study the implemen tation of the Symmetr ic Mass Gener ation (SMG) mechanism in the 3-4-5 -0 models. Our fram ew ork ad m its sym metr y-preser v ing inter a c tion ter ms with quantized chiral charg es, althou gh fur ther numer ical inv estig ation is required to confir m the realizatio n o f the SMG mechanism in interacting systems. OU-HET -1305 The 42nd Internatio nal Symposium on Lattice Field Theor y (L ATTICE2025) 2-8 N ov ember 20 25 T ata Institut e of F u ndamenta l Researc h, Mu mbai, Ind ia ∗ Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial- NoDerivativ es 4.0 I nternational License (CC B Y -NC-ND 4.0) A l l r ights for text and data mining, AI training, and similar technologies for commercial purposes, are reserved. ISSN 1824-8039 . Pu b lished by SISSA M edialab. https://pos.si ssa.it/ The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a 1. Introdu ction The lattice f ormulation of chi ral ga ug e theories remains a longs tandin g chall eng e in quantu m field theory . The primar y obstruction is the Nielsen –Ninomiy a no-g o theorem [ 1 , 2 ], which stat es t hat an y local, Hermitian, and transla tional ly in variant lattice f er mion action necessaril y yields f ermion doubl ers as a conse que nce of the per iodicity of the Brillouin zone . R emo ving these doubl ers inev itabl y breaks chiral symmetr y , making the non perturbativ e f ormulation of chiral gau g e theories highl y nontrivial. In the path -integral f orm alism, substa ntial progress has been ach ie v ed through the ov er lap f er mion construction [ 3 ], whic h is based on the Ginspar g–Wil son relat ion [ 4 ]. Ov erlap f er mions preserv e a modified chiral symmetry at finite lattice spacing and allo w f or a controlled tre atment of chiral properties on the latti ce. Ho we ver , the cor respon ding Hamiltonian f ormulation of lattice chi ral symmetry is sti ll under activ e in ve stig ation [ 5 – 13 ]. Rec entl y , Ref. [ 5 ] proposed a ne w f or mulation of 1+1-dimen sional stagg ered f ermions in which both v ector and axial cha rg e operato rs are local, commute with the Hamiltonian, and posses s qua ntized eig env alues. Thes e operat ors g enerat e independ ent U ( 1 ) symmetries that can in pr inciple be g aug ed separa tely . The noncommutativi ty betw een the vector and axial charg es e nc odes the chiral anomal y within the Hamiltonian frame work. In this w ork, w e revisi t this co nst ruction from the vie wpoint of Wilson f erm ions. The re- sult was alre ady published in Ref [ 14 ]. W e demonstra te tha t, in 1+1 dimens ions, the stagg ered f er mion H amilton ian can be smoothl y def ormed into the Wils on f ermion Hamiltonian. U sing this equ iv alence , w e reinterpret the quant ized axial cha rg e in terms of Wilson f er mion v ariables. The eig enstates of the axial char g e are constructed as linear combin ations of posit iv e-energ y creation and neg ativ e-ener gy annihilation operators , leading to a Hamiltonian that contains particle-number nonco nserving ter ms. Nev ertheless, in the con tinuum limit, the theory still admits conserve d v ec- tor and axia l U ( 1 ) charg es th at remai n noncommutativ e, there b y av oiding contra diction with the Nielsen– Ninomiy a theor em. The resultin g Hamiltonian f or mulation is particular l y useful f or constructing chir al gaug e theories via the symmetric mass g enera tion (SMG) mec hanism [ 15 , 16 ]. SMG pro vides a w a y to g ap f er mions w ithout f ermion bilinear mass terms while preserving symmetries, pro vided that the under lyin g theory is anomal y-fre e. Motiva ted b y this pers pectiv e, we reconsi der the 3-4- 5-0 chir al model and clarify the role of the cons erve d char g es in determining whether anomal y cance llation occurs nontriviall y or only in a trivial lattice sense. Based on this anal ysi s, w e recons truct the m odel within the pres ent frame wo rk. This paper is org anized as f ollo ws. In S ec. 2 , w e first demon stra te the equiv alence betw een stag g ered and Wilson f e rmion H amiltonians i n 1+1 dimensions. W e then rev ie w the quantized lattice cha rg e operato rs and ref or mulate the axial charg e in terms of Wilson f er mions in Sec. 3 . In S ec. 4 , w e redefine the axi al charg e operator 𝑄 𝐴 using Wil son f er mions and cons truct the Hamiltoni an in terms of the fields with definite char g e of 𝑄 𝐴 . W e v erify that this Hamilton ian still supports tw o conserv ed cha rg es correspondin g to v ector and axial U ( 1 ) symmetries in the continuum limit. Ne xt, in Sec. 5 w e anal yze the cons erv ed symmetries in the continuu m limit and discuss the applica tion to SM G cons tructions, the 3-4-5-0 models. Sec. 6 is dev oted to our conclusion. 2 The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a 2. Equ ivalence of Stagg ered a nd Wilson F ermions in 1 + 1 Dimensions In thi s section , w e de monstr ate that, in 1+1 di mensions , the Hamiltonian of massless free stag g ered f er mions is equiv alent to that of Wilson fe rm ions [ 17 ]. T his equiv alence pro vides the basis f or reinterpreting the qu antized axial c harg e in the Wil son f ermion frame wo rk. W e begin with the Hamiltonian of the 1+1-dimension al stagg ered fe rmion [ 18 – 21 ], 𝐻 KS = i 2 𝑁 Õ 𝑗 = 1  𝑐 † 𝑗 𝑐 𝑗 + 1 + 𝑐 𝑗 𝑐 † 𝑗 + 1  = i 2 2 𝑁 Õ 𝑗 = 1  𝑎 𝑗 𝑎 𝑗 + 1 + 𝑏 𝑗 𝑏 𝑗 + 1  , (1) where 𝑐 𝑗 is a single-compo nent comple x fermion satisfying { 𝑐 𝑗 , 𝑐 † 𝑗 ′ } = 𝛿 𝑗 , 𝑗 ′ . (2) Decomposi ng 𝑐 𝑗 into two Majoran a f er mions, 𝑐 𝑗 = 1 2  𝑎 𝑗 + 𝑏 𝑗  , (3) with 𝑎 𝑗 = 𝑎 † 𝑗 , 𝑏 𝑗 = 𝑏 † 𝑗 , { 𝑎 𝑗 , 𝑎 𝑗 ′ } = { 𝑏 𝑗 , 𝑏 𝑗 ′ } = 2 𝛿 𝑗 𝑗 ′ , (4) one sees that Eq. ( 1 ) describes a sing le massles s Dirac f er mion in the continuum limit. Introd ucing a two- compone nt field 𝜓 𝑗 = 𝜓 𝐿 , 𝑗 𝜓 𝑅 , 𝑗 ! = 𝑐 2 𝑗 i 𝑐 2 𝑗 + 1 ! , (5) and defining 𝛾 0 = 𝜎 1 , 𝛾 1 = − 𝜎 3 , 𝛾 5 = 𝜎 2 , (6) the Hamiltonian can be re wr itten as 𝐻 KS = 𝑁 Õ 𝑗 = 1 𝜓 † 𝑗 𝛾 0  𝛾 1 1 2 ( ∇ + ∇ † ) − 1 2 ∇ † ∇  𝜓 𝑗 = 𝑁 Õ 𝑗 = 1 𝜓 † 𝑗 𝛾 0 D 𝑊 ( 0 ) 𝜓 𝑗 ≡ 𝐻 W , (7) where the latti ce deriv ativ es are defined b y ∇ 𝜓 𝑗 = 𝜓 𝑗 + 1 − 𝜓 𝑗 , ∇ † 𝜓 𝑗 = 𝜓 𝑗 − 𝜓 𝑗 − 1 , ∇ † ∇ 𝜓 𝑗 = 𝜓 𝑗 + 1 + 𝜓 𝑗 − 1 − 2 𝜓 𝑗 , (8) and D 𝑊 ( 𝑚 ) den otes the Wilso n–Dirac opera tor D 𝑊 ( 𝑚 ) = 1 2 { 𝛾 1 ( ∇ + ∇ † ) − ∇ † ∇ } − 𝑚 . (9) Thus, in 1+1 dimensions, the st agg ered f er mion Hamilton ian is smoothl y conn ected to the Wilson f er mion Hamiltonia n in the massless free limit. 3 The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a 3. Conserv ed Charg es in the Hamiltonian o f Stagger ed F ermions A remar kable property of the 1+1-dimen sional stagg ered f ermion Hamiltonian with a finit e lattice H ilbert space is the e xis tence of two conserv ed charg e opera tors [ 5 ], denoted b y 𝑄 𝑉 and 𝑄 𝐴 . In the conti nuum limit, these cor respon d to the v ector and axial U ( 1 ) symmetries, respectiv el y . Both char ges are loca l operato rs and commute with the Hamiltonian, [ 𝐻 KS , 𝑄 𝑉 ] = [ 𝐻 KS , 𝑄 𝐴 ] = 0 . (10) The y are ex plicitl y giv en by 𝑄 𝑉 = 2 𝑁 Õ 𝑗 = 1  𝑐 † 𝑗 𝑐 𝑗 − 1 2  = i 2 2 𝑁 Õ 𝑗 = 1 𝑎 𝑗 𝑏 𝑗 ≡ 2 𝑁 Õ 𝑗 = 1 𝑞 𝑉 𝑗 , (11) 𝑄 𝐴 = 1 2 2 𝑁 Õ 𝑗 = 1  𝑐 𝑗 + 𝑐 † 𝑗   𝑐 𝑗 + 1 − 𝑐 † 𝑗 + 1  = i 2 2 𝑁 Õ 𝑗 = 1 𝑎 𝑗 𝑏 𝑗 + 1 ≡ 2 𝑁 Õ 𝑗 = 1 𝑞 𝐴 𝑗 + 1 2 . (12) Importantl y , both 𝑄 𝑉 and 𝑄 𝐴 ha v e quantize d eigen v alues and can be g aug ed independ entl y at fi nite lattice spacing. Ho we ver , on a fi nite lattice the y do not commute with eac h other , [ 𝑄 𝑉 , 𝑄 𝐴 ] = − 2 𝑁 Õ 𝑗 = 1  𝑐 𝑗 𝑐 𝑗 + 1 + 𝑐 † 𝑗 𝑐 † 𝑗 + 1  = i 𝐺 1 ≠ 0 , (13) where 𝐺 1 is one of the g enerators of th e O nsag er alg ebra . This non-A belian alg ebraic structure disapp ears in the cont inuum limit, where the commutator van ishes. The noncommutativi ty bet w een 𝑄 𝑉 and 𝑄 𝐴 encod es the mix ed anomal y betw een v ector and axial U ( 1 ) symmetr ies within the H amiltoni an f ormalism, in a manner consis tent w ith the Nielsen– Ninomiy a theorem. 4. Eige nstates of the Axial Charge and the Hamiltonian Using the eq uiv alence betw een stagg ered and Wilson f er mions estab lished in the pre vious sectio n, w e no w ref or mulate the conserv ed char g es 𝑄 𝑉 and 𝑄 𝐴 in the Wilson f er mion frame wor k. In par ticula r , the axial c harg e oper ator 𝑄 𝐴 is an on- site symmetr y with quan tized eig en value s in coordi nate space. Its eig enstates theref ore admit an interpretation as fermions with integ er chiralit y , analog ous to W e y l f ermions in the continuu m theory . This s tructure enables the cons truction of lattice gau g e theories with a gaug ed U ( 1 ) 𝐴 symmetry . T o mak e this e xplicit, w e construct f erm ionic operators that diagonal ize 𝑄 𝐴 . The f ollo wing operat ors are eigen stat es of 𝑄 𝐴 with eigen values ± 1: Ψ † 𝐿 , 𝑗 Ψ † 𝑅 , 𝑗 ! = 1 2 √ 2 − 2 𝑐 † 2 𝑗 + ( 𝑐 2 𝑗 + 1 − 𝑐 † 2 𝑗 + 1 ) − ( 𝑐 2 𝑗 − 1 + 𝑐 † 2 𝑗 − 1 ) 2 𝑐 † 2 𝑗 + ( 𝑐 2 𝑗 + 1 − 𝑐 † 2 𝑗 + 1 ) − ( 𝑐 2 𝑗 − 1 + 𝑐 † 2 𝑗 − 1 ) ! . (14) The y satisfy [ 𝑄 𝐴 , Ψ † 𝐿 , 𝑗 ] = Ψ † 𝐿 , 𝑗 , [ 𝑄 𝐴 , Ψ † 𝑅 , 𝑗 ] = − Ψ † 𝑅 , 𝑗 , (15) 4 The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a and obe y canonical anti commutatio n relations. In terms of these fields, the Wilson Hamiltonian becomes 𝐻 W = i 𝑁 Õ 𝑗 = 1  Ψ † 𝐿 , 𝑗 1 2 ( ∇ + ∇ † ) Ψ 𝐿 , 𝑗 − Ψ † 𝑅 , 𝑗 1 2 ( ∇ + ∇ † ) Ψ 𝑅 , 𝑗 − Ψ 𝑅 , 𝑗 1 2 ∇∇ † Ψ 𝐿 , 𝑗 − Ψ † 𝐿 , 𝑗 1 2 ∇∇ † Ψ † 𝑅 , 𝑗  . (16) The last line contai ns particle-numbe r noncons erving ter ms or iginati ng from the Wilson term. Thus, f ermion number is not manif estl y conserv ed in this basis. Nev ertheless, the Hamiltonian still admits conserve d char g es 𝑄 𝑉 and 𝑄 𝐴 , satisfyin g [ 𝐻 W , 𝑄 𝑉 ] = [ 𝐻 W , 𝑄 𝐴 ] = 0 . (17) In this basis, 𝑄 𝐴 = 𝑁 Õ 𝑗 = 1 ( Ψ † 𝐿 , 𝑗 Ψ 𝐿 , 𝑗 − Ψ † 𝑅 , 𝑗 Ψ 𝑅 , 𝑗 ) , (18) which mak es the intege r -va lued chi rality ex plicit. The ve ctor c harg e 𝑄 𝑉 tak es a more in vo l ved f orm , reflecti ng the mixing induced b y the Wilso n term. A clearer picture emer g es in momentum space. E xpand ing the stagg ered f er mion as 𝑐 𝑗 = 1 √ 2 𝑁 Õ 𝑘 𝛾 𝑘 𝑒 2 𝜋 𝑖 2 𝑁 𝑘 𝑗 , (19) one finds that the Hamiltonian beco mes 𝐻 W = 1 2 Õ − 𝑁 2 ≤ 𝑘 < 𝑁 2  ( ˜ Ψ † 𝐿 , 𝑘 ˜ Ψ 𝐿 , 𝑘 − ˜ Ψ † 𝑅 , 𝑘 ˜ Ψ 𝑅 , 𝑘 ) sin 2 𝜋 𝑘 𝑁 − i ( ˜ Ψ † 𝐿 , 𝑘 ˜ Ψ † 𝑅 , − 𝑘 − ˜ Ψ 𝑅 , − 𝑘 ˜ Ψ 𝐿 , 𝑘 )  1 − cos 2 𝜋 𝑘 𝑁   . (20) The second term correspon ds to the Wilso n term and va nishes linear l y in momentum near 𝑘 = 0. T aking the con tinuum limit 𝑁 → ∞ at fixe d phy sical momentum, the commutators redu ce to [ 𝑄 𝑉 , ˜ Ψ † 𝛼, 𝑘 ] = ˜ Ψ † 𝛼, 𝑘 , (21) [ 𝑄 𝑉 , ˜ Ψ 𝛼, 𝑘 ] = − ˜ Ψ 𝛼, 𝑘 , (22) sho wing that 𝑄 𝑉 flo ws to the g enerator of the v ector U ( 1 ) 𝑉 symmetry , while 𝑄 𝐴 g enerat es U ( 1 ) 𝐴 . Although the Hamiltonia n appears to violate particle number conserva tion at finite lattice spacin g, it preserves two noncommuting cons erv ed char g es. Because 𝑄 𝑉 and 𝑄 𝐴 do not commute on a finite la ttice and their eig en v alues are no t simult aneous l y quantized , the y cannot be ga ug ed simultan eousl y . This structure ensu res consist ency w ith th e Nielsen– Ninomiy a theorem, w hile allo wing w ell-de fined W ey l f ermions with integ er c hirality ov er the entire Brillouin zone. 5 The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a 5. Application to Symmetric Mass Generation A promisi ng strat egy f or constructing chiral g auge theo ries on the lattice is the m echanism of symmetric mass gener ation (SMG ) [ 22 – 24 ]. SMG allo ws f er mions to acq uire a mass g ap throug h multi-f ermion interactio ns without introducin g an y f ermion bili near mass terms, while prese rving the imp osed symmetries. Although suc h interact ions are typic all y irrele v ant in weak coupling, numerical studi es indicate that the y can become rele vant in s trong-c ouplin g regimes [ 25 ]. For a f er mionic sys tem with symmetry group 𝐺 , the realizati on of SMG req uires: (i) The symmetr y 𝐺 m ust be anomal y-fre e. (ii) 𝐺 must f orbid all f ermion bili near m ass ter m s. (iii) The instan ton satu ration condit ion must be satisfied. The third conditi on beco mes essential in the presence of bac kgrou nd or dynamic al 𝐺 g aug e fields with nontrivial topological cha rg e. Such bac kgrounds gene rate chi ral zero modes, which must be satura ted b y interacti on ter ms in order to maintain consi sten cy . In Sec. 4 , we const ructed W e y l f ermions with in teg er -v alued axial ch arg es. This s tructure enable s us to g auge the U ( 1 ) 𝐴 symmetry gen erated b y 𝑄 𝐴 and to inv estig ate SMG within th is frame wo rk. In the prese nt proce edings , we f ocus on the 3-4-5-0 m odel [ 15 , 16 , 25 – 27 ]. A detailed anal ysis of the 1 4 ( − 1 ) 4 model has bee n presented separatel y in Ref. [ 14 ], and will not be repea ted here. 5.1 3-4-5- 0 model W e begin with f our fla vor s of massless Dirac fermions, 𝐻 = i 4 Õ 𝑓 = 1 𝑁 Õ 𝑗 = 1  Ψ 𝐻 † 𝑓 , 𝐿 , 𝑗 1 2 ( ∇ + ∇ † ) Ψ 𝐻 𝑓 , 𝐿 , 𝑗 − Ψ 𝐻 † 𝑓 , 𝑅 , 𝑗 1 2 ( ∇ + ∇ † ) Ψ 𝐻 𝑓 , 𝑅 , 𝑗 − Ψ 𝐻 𝑓 , 𝑅 , 𝑗 1 2 ∇∇ † Ψ 𝐻 𝑓 , 𝐿 , 𝑗 − Ψ 𝐻 † 𝑓 , 𝐿 , 𝑗 1 2 ∇∇ † Ψ 𝐻 † 𝑓 , 𝑅 , 𝑗  . (23) The goal is to g ap out one chira lity of each fla v or , lea ving the desired chira l spectrum in the infrare d. The Hamiltonia n preserve s f our U ( 1 ) symmetries gene rated b y fla v or -resol ved vec tor and axial cha rg es, 𝑄 𝑉 𝑓 and 𝑄 𝐴 𝑓 . From these, w e define 1 𝑄 𝑎 1 = 3 𝑄 𝐴 1 + 4 𝑄 𝐴 2 − 5 𝑄 𝐴 3 , (24) 𝑄 𝑎 2 = 5 𝑄 𝐴 2 − 4 𝑄 𝐴 3 − 3 𝑄 𝐴 4 , (25) 𝑄 𝑣 1 = 3 𝑄 𝑉 1 + 4 𝑄 𝑉 2 − 5 𝑄 𝑉 3 , (26) 𝑄 𝑣 2 = 5 𝑄 𝑉 2 − 4 𝑄 𝑉 3 − 3 𝑄 𝑉 4 . (27 ) 1 In pr inciple, the char g e assignments of the U ( 1 ) gau g e symmetr y and the extra U ( 1 ) symmetry in this model can be uniqu ely determined, both in the latt ice and continuum theories, by the condition that anomal y matching is equiv alent to the boundary fully gap ping condition [ 15 , 16 ]. In our latti ce cons truction, how ev er , the anomal y matching conditions f or the U ( 1 ) gaug e symmetry and the e xtra U ( 1 ) symmetry become tri vial b y construction, i.e., [ 𝑄 𝑎 1 , 𝑄 𝑎 2 ] = 0. As a result, the ch arg e assignments are no l onger uniquel y fixed b y anomal y considerations alone. In this wo rk, we t heref ore choo se t he charg e assignments so that they consistentl y reproduce those of the 3-4-5-0 model in the continuum limit. A discussion of anomal y cancellation in this model can al so be fou nd in Ref. [ 28 ]. 6 The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a These reproduce the char ge assignments of the continuum 3-4-5-0 model. At this stag e the theory is vec tor -lik e, but fe rm ion bilinea r mass terms are f orbidden by U ( 1 ) 𝑎 1 × U ( 1 ) 𝑎 2 . In our construction , the anomal y cancel lation within U ( 1 ) 𝑎 1 × U ( 1 ) 𝑎 2 is realized triviall y at the latti ce le vel, while mix ed anomalies betw een axial and ve ctor symmetries remain : [ 𝑄 𝑎 , 𝑠 , 𝑄 𝑣 , 𝑠 ′ ] ≠ 0 . (28) Theref ore, to gap out selected ch iral modes w hile preserving U ( 1 ) 𝑎 1 × U ( 1 ) 𝑎 2 , one must e xplicitl y break the anomal ous U ( 1 ) 𝑣 1 × U ( 1 ) 𝑣 2 via multi-f er mion intera ctions. Such inter actions can be chosen as Δ 𝐻 1 = Õ 𝑗 ( Δ 1 + h . c . ) , (29) Δ 𝐻 2 = Õ 𝑗 ( Δ 2 + h . c . ) , (30) with Δ 1 ∝ Ψ 1 , 𝐿 , 𝑗 ( Ψ † 2 , 𝐿 , 𝑗 Ψ † 2 , 𝐿 , 𝑗 + 1 ) Ψ 3 , 𝑅 , 𝑗 ( Ψ 4 , 𝑅 , 𝑗 Ψ 4 , 𝑅 , 𝑗 + 1 ) , (31) Δ 2 ∝ ( Ψ 1 , 𝐿 , 𝑗 Ψ 1 , 𝐿 , 𝑗 + 1 ) Ψ 2 , 𝐿 , 𝑗 ( Ψ † 3 , 𝑅 , 𝑗 Ψ † 3 , 𝑅 , 𝑗 + 1 ) Ψ 4 , 𝑅 , 𝑗 . (32) These interaction terms commute with 𝑄 𝑎 1 and 𝑄 𝑎 2 throug hout the entire Brillouin zone, while e xplicitl y breaki ng the v ector symmetr ies. Moreo v er , appropriate combination s of Δ 1 and Δ 2 g enerat e the req uired ’ t Hooft v er tices, thereb y satisfy ing the instan ton saturat ion conditi on. Whether the intended chir al spectrum is dynamicall y real ized req uires further numerical in- v estig ation. Nev ertheless, the present cons tr uction demonstra tes th at the SM G frame w ork can be consis tently implemented with a gau g ed U ( 1 ) 𝐴 symmetry g enerated b y the qu antized axial charg e. 6. Conclusion In this w ork, we hav e recons tr ucted tw o lattice charg e op erators , 𝑄 𝑉 and 𝑄 𝐴 , that commute with the Hamiltonia n of the 1+1-d imension al stagg ered fermion, b y emplo ying the Wilson f er mion f orm alism. W e ha ve sho wn that these opera tors reduce, in the continuu m limit, to the g enerators of the v ector U ( 1 ) 𝑉 and axial U ( 1 ) 𝐴 symmetries, respecti v ely . Further more, w e cons tructed the Hamiltonia n e xplicitl y in ter ms of the eigen sta tes of 𝑄 𝐴 , there b y ensuring the e xac t preservati on of the axial U ( 1 ) 𝐴 symmetry on the latt ice. Although the resulting Hamiltonia n ma y appear to break the naiv e v ector U ( 1 ) 𝑉 symmetry at the latti ce lev el, it is, b y cons tr uction, compa tible with 𝑄 𝑉 and ther ef ore reproduces the correct U ( 1 ) 𝑉 symmetry in the continuum limit. While this observatio n ma y appe ar str aightf or war d, it is ne v er theles s nontrivial that the Hamiltonian remains consist ent with 𝑄 𝑉 e v en w hen it is e xplici tl y design ed to pres erve a xial symmetry on the lattice . A ke y adv antag e of maintai ning e xac t 𝑄 𝐴 symmetry is that the operator acts locall y , which in turn allo ws it to be g aug ed. This opens the possib ility of f or mulating chiral U ( 1 ) 𝐴 g aug e theories on the lattice, at least within cer tain controlled setting s. 7 The Axial Charg e in Hilber t Sp ace and the Role in Chiral Gaug e Theories T atsuy a Y amaok a As an appli cation, w e dis cussed ho w the S ymmetric Mass Generation (SMG) mechanis m can be implemen ted in free chiral U ( 1 ) 𝐴 g aug e the ories (without dynamic al gaug e fields), us ing the 3-4-5- 0 model as illustr ativ e e xa mples. Because the Hamiltonia n is f or mulated in terms of f ermions with well- defined, m omentum-indepen dent int eg er -v alued ch irality , the interacti on ter ms requir ed f or gap ping can be introduc ed consi sten tl y without violat ing the symmetries that must be preserv ed. In the case of the 3-4-5 -0 model, ho we ve r , further numerical in ve stig ation is necessary to determine whether the S MG mecha nism is genu inel y realize d in this frame wo rk. Sinc e the Hamiltonian contai ns Wilson-ty pe terms that g enericall y mix chirali ties, it is highl y nontrivial whether one chi rality can be full y gapp ed while the other remain s massl ess solely through interactio n effects. W e lea v e this important q ues tion f or future stud y . Finall y , one of the motiv ations f or adopting the Hamiltoni an f ormulation is its natural compat- ibility with quan tum simulation platf orms. 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