# Structure of the Entanglement Entropy of (3+1)D Gapped Phases of Matter

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12 12 Usig Eq. (39, Tr H ρ N = C 2N C E0 N I /2 a I,ã I =1 α I =e,o = C 2N N 0 /2 a 0=1 α 0=e,o CE 0 a 0 α0 N N (C EI CE I a I αi CE I ã I αi C E0,a 0,α 0 I=1 I=1 N ( N (C EI C EI,a I,ã I,α I ( C EI CE 0 a 0 α0 δ CE0 C E1 δ CE1 C E2 δ CEN C E0 δ a0a 1 δã1a 2 δã2a 3 δãn 1 a N δãn a 0 δ α0α 1 δ α1α 2 δ αn α 0 = C 2N C E0 = C 2N C E0 N (C E0 N α 0=o,e N (C E0 /2 a 1=1 2N (C E0 N ( N (C E0 2 N (C E0 /2 a N =1 1 N = 2 Σ (N 1. (43 I the first equatio, we expad the trae over the Hilbert spae i regio. I the seod equatio, we use the orthogoal oditio CE a α C E a α = δ CEC E δ aa δ αα. I the third equatio, we simplify the formula usig the delta futios C E0 = C E1 = = C EN, α 0 = α 1 = = α N, ad elimiate {a 0, ã I } by {a I }. I the last equatio, we used Eq. (41. Moreover, otie that Tr H ρ = 1, we obtai the etaglemet etropy S( = d dn 2 Σ (N 1 N=1 = Σ log 2. (44 Sie Σ is the umber of 2-simplies o Σ, whih is proportioal to the area of Σ, hee it is the area law term. Sie there is o ostat term, the topologial etaglemet etropy is trivial, refletig the absee of topologial order i this model. start by writig dow the groud state wave futio, ψ =C C E N (C E/ a=1 N (C E/ b=1 e i2πpαβ la,αl CE CE b,β sce a,αs CE b,β ÃCE a 1 α,β=0 α ÃCE b β, (45 where la,α, CE l CE b,β, sce a,α, s CE b,β are straightforward geeralizatios of Eq. (37 to the ases with arbitrary oeffiiets p ad,.f. Eq. (33. The redued desity matrix is ρ = C 2 C E N (C E i2πp(α αγ e N (C E/ a,ã=1 1 α, α,γ=0 CE a α CE ã α, (46 where we agai performed the uitary trasformatios to absorb the self-likig ad mutual-likig fators, ad deote the resultig ew basis as CE a α ad CE b β. For the same reaso as i Eq. (41, C 2 N (C E N (C E = 1, (47 Σ 1 where Σ is the umber of 2-simplies o the etaglemet surfae. I order to ompute the etaglemet etropy S = Tr H ρ log ρ, (48 we first alulate the etaglemet spetrum, i.e., we diagoalize ρ. s a first step, we arry out the sum over γ i Eq. (46. We ote that the sum is ovaishig oly if p(α α/ is a iteger, i whih ase the sum takes the value. Thus, 2. EE for the Torus: geeral (, p We arry out the aalogous alulatios for a geeral GWW theory with arbitrary oeffiiets ad p. We 1 γ=0 We fid i2πp(α αγ e ( = δ α α = 0 mod. gd(, p (49 ρ = C 2 C E N (C E = C E,a,α,ã, α [ ρ CE ] N (C E/ a,α;ã α a,ã=1 1 α, α ( δ α α = 0 mod CE a α CE ã gd(, p α (50a CE a α CE ã α, (50b

14 14 2π 2π BF BF + p 4π S 2 T 2 [(0, 0,, (g, g ] TQFT log log b 0 log S topo log log b 0 log log log gd(, p ( b 0 + χ log gd(, p χ log 2 2 S topo log gd(, p log gd(, p b 0 log gd(, p BB STQFT Table I: Costat part ad topologial part of the etaglemet etropy for geeralized Walker-Wag models. TQFT is the ostat part of the EE for the TQFT, while S topo is the TEE for a geeral theory whih belogs to the same phase of the TQFT. b 0 is the zeroth Betti umber of etaglemet surfae b 0 = g g=0 g. χ = g g=0 (2 2gg is the Euler harateristi of the etaglemet surfae. I partiular, we have S topo(s 2 = S topo(t 2. with geus g. (The results are summarized i Table I. For eah hole i (i = 1,, g of the etaglemet surfae, we itrodue a pair of additioal idies α i ad β i that out the umber of loops (modulo widig aroud the o-otratible yles aroud the hole i regio ad regio, respetively. The the wavefutio is ψ =C C E g i=1 N (C E g a=1 e i2πpα i β i N (C E g b=1 1 1 α 1 α g=0 β 1 β g=0 CE a α CE b β. (59 We ollet the set of idies α 1,, α g ito a idex vetor α. We first osider the ofiguratios i regio. Sie eah hole is assoiated with a idex α i, whih a take differet values, the omplete set of idies α a take g differet values. Hee, the N (C E ofiguratios are partitioed ito g lasses, where eah lass otais N (C E / g ofiguratios. For this reaso the summatio i Eq. (59 reahes oly up to N (C E / g. For regio, similar argumets hold. The the redued desity matrix o a geus g surfae takes the form ρ = C 2 C E N (C E g = C 2 C E N (C E = C E α, α N (C E/ g a,ã= α 1,,α g=0 α 1,, α g=0 γ 1,,γ g=0 1 1 α 1,,α g=0 α 1,, α g=0 [ ρ CE ] CE a aα,ã α N (C E/ g a,ã=1 α CE ã α, N (C E/ g a,ã=1 g i=1 g ( δ α i α i = 0 i=1 e i2πp(α i α i γ i mod CE a α CE ã α gd(, p CE a α CE ã α (60 where [ ] ρ CE = C 2 N (C E aα,ã α g i=1 [ ] 11 J gd(,p gd(,p = C 2 N (C E gd(, p g N (C E = g α i α i [ ] gd(, pg 11 Σ +g 1 g 0 gd(,p g N (C E. g gd(,p g aα,ã α [ ] J N (C E g aã [ ] 11 g 0 gd(,p g N (C E g gd(,p g aα,ã α (61 I the seod lie of Eq. (60, we summed over γ 1,, γ g usig Eq. (49. I the last lie of Eq. (60 ad the first lie of Eq. (61, we reorgaized the oeffiiets CE a α CE ã α ito a matrix form, where 11 is the gd(,p idetity matrix due to the delta futio, ad J gd(,p is beause all elemets of α = gd(,p k, α = k gd(,p with k, k = 0, 1,, gd(, p 1 are eumerated, ad similar for J N (C E. I the seod lie of Eq. (61, we expad the tesor g produt. I the last lie, we use the ormalizatio oditio C 2 N (C E N (C E = 1. We see that all of the o-zero eigevalues of the Σ 1 etaglemet

19 19 B C B C (a (b Figure C.2: Etaglemet surfaes used i the appliatio of strog sub-additivity to derive Eq. (C12. I (a, is a 3-maifold with multiple geus zero surfaes, B is a 3-ball, C is a 3-ball with small 3-ball removed. I (b, is a ope 3-maifold with multiple geus zero surfaes, B is a 3-ball with a small 3-ball removed ad C is a 3-ball loated exatly i the empty 3-ball iside B. This relates the ostat part of the EE of a give subsystem to that of a system whose boudary has lower geus. pplyig Eq. (C6 repeatedly, we fid TQFT [(0, 0, (1, 1,..., (g, g ] = TQFT [(0, g i=0 i ] + g i=1 i i ( TQFT [(0, 0, (1, 1] TQFT [(0, 1]. (C7 I summary, we a redue the ostat part of the EE of a arbitrary surfae TQFT [(0, 0, (1, 1,..., (g, g ] to a liear ombiatio of TQFT [(0, ] ad TQFT [(0, 0, (1, 1]. 2. Reurree for b 0 We a further simplify TQFT [(0, g i=0 i] i Eq. (C7, by usig TQFT [(0, ] = TQFT [(0, 1]. Here we derive this relatio by makig use of the SS i a maer similar to that of the derivatio above. We osider the ofiguratio show i Fig. C.2(a, where is a 3-maifold with ( 1 geus zero surfaes, B is a 3-ball ad C is a 3-ball with a small 3-ball iside it removed. The ostat parts of the EE for these three maifolds are TQFT ( = TQFT [(0, 1], TQFT (B = TQFT [(0, 1], TQFT (C = TQFT [(0, 2], TQFT (B = TQFT [(0, 1], TQFT (BC = TQFT [(0, 2], S TQFT (BC =S TQFT [(0, ]. (C8 The SS iequality reads S TQFT [(0, 1] + TQFT [(0, 2] TQFT [(0, ] + TQFT [(0, 1]. (C9 We a furthermore osider aother ofiguratio show i Fig. C.2(b, where is a 3-maifold with ( 1 geus-0 surfaes, B is a 3-ball with small 3-ball removed, ad C is a 3-ball loatig exatly i the empty 3-ball iside

20 20 B. The ostat parts of the EE for these three maifolds are TQFT ( = TQFT [(0, 1], TQFT (B = TQFT [(0, 2], TQFT (C = TQFT [(0, 1], S TQFT ( B = S TQFT [(0, ], TQFT (B C = TQFT [(0, 1], TQFT ( B C = TQFT [(0, 1]. (C10 The SS iequality reads S TQFT Combiig Eq. (C9 ad Eq. (C11, oe obtais S TQFT Sie TQFT [(0, 0] = 0, we have Combiig this result with Eq. (C7, we have 114 [(0, ] + TQFT [(0, 2] TQFT [(0, + 1] + TQFT [(0, 1]. (C11 [(0, ] + TQFT [(0, 2] = TQFT [(0, + 1] + TQFT [(0, 1]. (C12 TQFT [(0, ] = TQFT [(0, 1]. (C13 S TQFT [(0, 0,..., (g, g ] = = g i=0 g i=0 i S TQFT [(0, 1] + g i=1 i i ( (1 i i S TQFT [(0, 1] + TQFT g i=1 [(0, 0, (1, 1] TQFT [(0, 1] i i S TQFT [(0, 0, (1, 1] = b 0 TQFT [(0, 0, (1, 1] + χ ( TQFT [(0, 1] TQFT [(0, 0, (1, 1] 2 = b 0 TQFT [T 2 ] + χ ( TQFT [S 2 ] TQFT [T 2 ], 2 (C14 where χ = g i=0 (2 2i i is the Euler harateristi of the etaglemet surfae, whih i the previous examples of this appedix is (BC. This is preisely Eq. (10 i the mai text. I the last lie, we have haged the otatio for larity: S 2 is a 2-sphere ad T 2 is a 2-torus. We emphasize that Eq. (10 gives the ostat part of the EE for a TQFT. I partiular, Eq. (10 shows that the ostat part of the EE aross a arbitrary etaglemet surfae is redued to that aross the sphere S 2 ad that aross the torus T ppedix D: Vaishig of the Mea Curvature Cotributio i KPLW Presriptio I this appedix, we explai why the mea urvature terms ael i the KPLW ombiatio Eq. (14, therefore justifyig Eq. (18 i the mai text. I the mai text, we argued that the KPLW ombiatio of the area law term ad the Euler harateristi term vaish separately, hee we oly eed to osider the topologial term ad the mea urvature term, i.e., S KPLW [T 2 ] = S topo [T 2 ] + 4F 2 d 2 x hh 2. + B+ C B C BC+ BC (D1 Eq. (D1 suggests that the mea urvature term i the KPLW ombiatio is ivariat uder deformatios of the etaglemet surfae sie, as argued i the mai text, both S KPLW [T 2 ] ad S topo [T 2 ] i Eq. (D1 are topologial ivariats. Therefore, we oly eed to show that Eq. (18 vaishes for oe partiular etaglemet surfae that is topologially equivalet to that i Fig. 1 i the