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

Struture of the Etaglemet Etropy of (3+1D Gapped Phases of Matter Classifyig gapped phases of matter has reetly emerged as oe of the etral themes of odesed matter physis 1 5. The groud states of two gapped Hamiltoias are i the same phase if they a be adiabatially oarxiv:1710.01747v2 [od-mat.str-el] 10 May 2018 Yuqi Zheg, 1 Hua He, 1 Barry Bradly, 2 Jeifer Cao, 2 Titus Neupert, 3 1, 4, 5, 6 ad B. drei Berevig 1 Physis Departmet, Prieto Uiversity, Prieto, New Jersey 08544, US 2 Prieto Ceter for Theoretial iee, Prieto Uiversity, Prieto, New Jersey 08544, US 3 Departmet of Physis, Uiversity of Zurih, Witerthurerstrasse 190, 8057 Zurih, Switzerlad 4 Doostia Iteratioal Physis Ceter, P. Mauel de Lardizabal 4, 20018 Doostia-Sa Sebastia, Spai 5 Laboratoire Pierre igrai, Eole Normale Sup erieure-psl Researh Uiversity, CNRS, Uiversit e Pierre et Marie Curie-Sorboe Uiversit es, Uiversit e Paris Diderot-Sorboe Paris Cit e, 24 rue Lhomod, 75231 Paris Cedex 05, Frae 6 Sorboe Uiversit es, UPMC Uiv Paris 06, UMR 7589, LPTHE, F-75005, Paris, Frae (Dated: May 11, 2018 We study the etaglemet etropy of gapped phases of matter i three spatial dimesios. We fous i partiular o size-idepedet otributios to the etropy aross etaglemet surfaes of arbitrary topologies. We show that for low eergy fixed-poit theories, the ostat part of the etaglemet etropy aross ay surfae a be redued to a liear ombiatio of the etropies aross a sphere ad a torus. We first derive our results usig strog sub-additivity iequalities alog with assumptios about the etaglemet etropy of fixed-poit models, ad idetify the topologial otributio by osiderig the reormalizatio group flow; i this way we give a expliit defiitio of topologial etaglemet etropy S topo i (3+1D, whih sharpes previous results. We illustrate our results usig several orete examples ad idepedet alulatios, ad show addig twist terms to the Lagragia a hage S topo i (3+1D. For the geeralized Walker-Wag models, we fid that the groud state degeeray o a 3-torus is give by exp( 3S topo[t 2 ] i terms of the topologial etaglemet etropy aross a 2-torus. We ojeture that a similar relatioship holds for belia theories i (d + 1 dimesioal spaetime, with the groud state degeeray o the d-torus give by exp( ds topo[t d 1 ]. Cotets I. Itrodutio 1 II. Redutio formulas for Etaglemet Etropy 3. Strog Sub-dditivity 3 1. Struture of the EE of Fixed Poit TQFTs 3 2. Redutio to the Costat Part of the EE 4 3. Struture of S ( 4 B. Topologial Etaglemet Etropy 4 1. TQFT [S 2 ] ad TQFT [T 2 ] 5 2. way from the Fixed Poit 5 3. Extratig the TEE 5 III. ppliatio: Etaglemet Etropy of Geeralized Walker-Wag Theories 6. Wave Futio of GWW Models 7 1. BF Theory: (, 0 7 2. Geeral Case: (, p 9 B. Etaglemet Etropy of GWW Models 9 1. EE for the Torus, = 2, p = 1 10 2. EE for the Torus: geeral (, p 12 3. EE for rbitrary Geus 13 B. Loal Cotributios to the Etaglemet Etropy 16 C. Derivatio of the Redutio Formula 17 1. Reurree for Geus 18 2. Reurree for b 0 19 D. Vaishig of the Mea Curvature Cotributio i KPLW Presriptio 20 E. Review of Lattie TQFT 24 F. Surfaes i the dual lattie 25 G. Mutual ad Self-Likig Numbers 27 1. Itersetio ad Likig 27 2. Self-likig Number 28 H. N (C E N (C E is Idepedet of C E 29 I. Case Study of the Cojeture Betwee GSD ad TEE 30 Referees 32 IV. Summary ad Future Diretios 15 kowledgmets 15. Review of Etaglemet Etropy ad Spetrum 15 I. INTRODUCTION

2 eted to oe aother through loal uitary trasformatios, without losig the eergy gap 1. Prior to the disovery of topologial order, the osesus i the physis ommuity was that gapped phases ould be lassified by symmetry breakig order parameters 6,7. The disovery of topologial order 8 10 revealed that two gapped systems a reside i distit phases abset ay global symmetries. The disovery of symmetry proteted topologial (SPT order 2,11 15 further erihed the family of topologial phases of matter: two systems with the same global symmetry a be i differet phases eve with trivial topologial order. The lassifiatio of topologial phases of matter has bee studied systematially from may differet agles. For oiteratig fermioi systems, phases have bee lassified aordig to time reversal symmetry, partile hole symmetry ad hiral symmetry, summarized by the te-fold way 16,17. Reetly this lassifiatio was exteded by osiderig rystal symmetries 18, i partiular osymmorphi symmetries 19,20. For iteratig systems, multi-ompoet Cher Simos theories 21 25, tesor ategory approahes 26 29, various forms of boudary theories 30 32, group ohomology ostrutios 15 ad several additioal methods 33 35 have bee used to lassify topologial phases of matter. Give the groud state of a Hamiltoia, a variety of tehiques have bee developed to determie whih phase it is i. Oe method exploits the aomalous boudary behavior of topologial phases (suh as otrivial propagatig modes if the boudary is gapless, or more exoti fratioalizatio if the boudary is gapped 13,30 32,36 43 by studyig systems with ope boudary oditios. For topologially ordered phases, oe a alteratively study the system o a losed maifold without boudaries, ad examie the braidig ad fusio properties of the gapped exitatios, suh as ayo exitatios i (2+1D ad loop exitatios i (3+1D 35,44 49. dditioally, the etaglemet struture of the groud state a also reveal topologial properties of the system. I partiular, Kitaev ad Preskill 50, as well as Levi ad We 51, realized that i (2+1D the existee of log rage etaglemet of the groud state, haraterized by the topologial etaglemet etropy (TEE, idiates topologial order. mog all approahes for probig topologial order, studyig the etaglemet etropy is oe of the more favorable 52 54, beause it depeds o the groud state oly ad a be omputed with periodi boudary oditios. There have bee may attempts to geeralize this ostrutio to higher dimesios, i partiular to better uderstad topologial order i (3+1D. The first attempt to study the TEE i (3+1D was made i Ref. 55, where the authors omputed the etaglemet etropy (EE for the (3+1D tori ode at fiite temperature. I Ref. 56, the (3+1D etaglemet etropy was omputed for the semio model, whih orrespods to the geeralized Walker Wag (GWW model of type (, p = (2, 1. (See Se. III for the defiitio of the GWW models. I Ref. 57, the authors disussed the tesor ategory represetatio of GWW models, ad the etaglemet etropy was omputed i this framework. We ote that these works oly examie theories at exatly solvable fixed poits. However, to isolate the topologial part of the etaglemet etropy, oe eeds to go beyod exatly solvable models; this is oe of the motivatios for the preset work. The authors of Ref. 58, for the first time, attempted to separate the topologial ad o-topologial ompoets of the etaglemet etropy for a geeri o-fixed-poit system i (3+1D. I partiular, they realized that the ostat (i.e., the otributio idepedet of the area of the etaglemet surfae part of the etaglemet etropy of a geeri gapped system is ot essetially topologial, ad otais a riher struture ompared to that i (2+1D. I this paper, based o previous works (espeially Ref. 58, we preset a more detailed ad omplete aalysis of the struture of the etaglemet etropy (i partiular the topologial etaglemet etropy for gapped phases of matter i (3+1D, whose low eergy desriptios are topologial quatum field theories (TQFT. We first make use of the strog sub-additivity (SS to ostrai the geeral struture of the etaglemet etropy for a TQFT. We fid that the ostat part of the etaglemet etropy (i the groud state of a TQFT aross a geeral etaglemet surfae (whih may otai multiple disoeted ompoets is a liear ombiatio of the ostat part of the EE aross a sphere S 2 ad that aross a torus T 2, with the oeffiiets beig topologial ivariats (Betti umbers of the etaglemet surfae [see Eq.(10]. We further disuss the geeralizatio of this result to geeri o-fixed-poit theories, where we study how the ostat part of the etaglemet etropy gets modified. This allows us to isolate the topologial etaglemet etropy. We also provide expliit alulatios of the etaglemet etropy for a partiular lass of (3+1D models, the GWW models. These alulatios serve as a idepedet hek of the result derived from the SS iequalities, ad also demostrates that the EE a be modified by a topologial twistig term i the atio 96. This pheomea is ew i (3+1D as ompared to (2+1D, beause the topologial twistig term does ot affet the TEE i (2+1D. For example, the Z 2 tori ode ad double semio theories, whih differ by a topologial twistig term, share the same TEE. Our approah has the advatage of simpliity: it starts from a simple-lookig Lagragia ad does ot require workig with disrete lattie Hamiltoias. We olude by ojeturig a formula for the TEE i terms of the groud state degeeray for belia topologial phases i geeral dimesios. We give support to this ojeture by omputig the etaglemet etropy of BF theories i (d + 1 dimesios. The orgaizatio of this paper is as follows: I Se. II, we preset our approah to fid a geeral formula for the ostat part of the EE for TQFTs desribig (3+1D gapped phases of matter. The basi strategy is to use the

3 SS iequality to ostrai the struture of the etaglemet etropy. I the derivatio, we assume a partiular form of the etaglemet etropy. I Se. III, we justify this assumptio through the study of the GWW models. We use a field theoretial approah, ad ompute the etaglemet etropy of these models aross geeral etaglemet surfaes. We summarize our results i Se. IV, ad olude with some ope questios to be addressed i future work. We preset the details of our alulatios i a series of appedies. I ppedix we review the defiitio of the etaglemet etropy ad the etaglemet spetrum. I ppedix B we review existig argumets about the loal otributios to the etaglemet etropy, whih were first disussed i Ref. 58. ppedies C, D, ad H are dediated to derivatios of speifi equatios from the mai text. I ppedix E we review the basis of lattie formulatio of TQFTs. I ppedix F we explai why surfaes i the dual spaetime lattie are otiuous ad losed. I ppedix G we disuss the likig umber itegrals eeded to formulate the GWW wave futio. Fially, i ppedix I we study BF theories i geeral (d + 1-dimesioal spaetime, ad give argumets for the validity of the ojeture that exp( ds topo [T d 1 ] gives the groud state degeeray o the d-dimesioal torus. II. REDUCTION FORMULS FOR ENTNGLEMENT ENTROPY I this setio, we study the geeral struture of the EE for gapped phases of matter i (3+1D. The defiitios of the etaglemet etropy ad the etaglemet spetrum are reviewed i ppedix. We are ispired by the fat that for a (2+1D system, the EE of the groud state of a loal, gapped Hamiltoia obeys the area law. I partiular, if we partitio our system ito two subregios, ad, the EE of subregio with the rest of the system takes the form S( = αl + γ + O(1/l, (1 where αl is the area term, ad l is the legth of the boudary of regio. Importatly the ostat term γ is topologial ad thus dubbed topologial etaglemet etropy 50,51. We would like to uderstad whether a aalogous formula holds for gapped phases of matter i (3+1D. I partiular, we ask how the ostat part of the EE depeds o the topologial properties of both the Hamiltoia ad the etaglemet surfae. Our approah to this questio relies o the SS iequality for the etaglemet etropy. We also make ertai loality assumptios about the form of the etropy, detailed i ppedix B. This allows us to derive a expressio for the ostat part of the EE of a subregio for a TQFT, TQFT (, whih depeds o the topologial properties (e.g. Betti umbers of the etaglemet surfae Σ. 97 We start by reviewig some geeral fats about the EE ad the use SS iequalities to determie the formula for the EE aross a geeral surfae i Se. II. I Se. II B, we disuss the impliatios of our EE formula, espeially regardig models away from a reormalizatio group (RG fixed poit. Our approah is ispired by Ref. 58.. Strog Sub-dditivity 1. Struture of the EE of Fixed Poit TQFTs s reviewed i ppedix B, for a geeri theory with a eergy gap, the EE for a subregio a be deomposed as S( = F 0 Σ + S topo ( 4πF 2 χ(σ +4F 2 d 2 x hh 2 + O(1/ Σ, (2 Σ where the oeffiiets F 0, F 2 ad F 2 are ostats that deped o the system uder study. The first term is the area law term, where Σ is the area of the etaglemet surfae, Σ. The seod term is the topologial etaglemet etropy, whih is idepedet of the details of the etaglemet surfae ad of the details of the Hamiltoia. The third term is proportioal to the Euler harateristi χ(σ of the etaglemet surfae. lthough it oly depeds o the topology of Σ, it is ot uiversal, ad we expet that the oeffiiet, F 2, will flow uder the RG. The fourth term is proportioal to the itegral of the mea urvature, H = (k 1 + k 2 /2, of Σ (see ppedix B for a derivatio of the loal otributios. It depeds o the geometry (i otrast to the topology of Σ, ad its oeffiiet F 2 also flows uder the RG i geeral. The remaiig terms are subleadig i powers of the area Σ, ad vaish whe we take the size of the etaglemet surfae to ifiity. Oe of the mai goals of this paper is to uderstad the struture of the topologial etaglemet etropy, S topo (, ad how it a be isolated from the Euler harateristi term ad the mea urvature term. I this setio, uless otherwise stated, we osider (3+1D TQFTs desribig the low eergy physis of a gapped topologially ordered phase. I this ase the ostat part of the EE depeds oly o the topology of the etaglemet surfae. The reaso is the followig: sie a TQFT does ot deped o the spaetime metri, it is ivariat uder all diffeomorphisms, iludig dilatatios as well as area-preservig diffeomorphisms. Hee, the term related to the mea urvature (whih depeds o the shape of Σ should ot appear. This implies that the oeffiiet F 2 flows to zero at the fixed poit. Whe we regularize the theory o the lattie, we expliitly break the salig symmetry while maitaiig the ivariae uder area preservig diffeomorphisms. Hee the area law term a survive, i.e. F 0 a flow to a o-vaishig

4 value at the fixed poit. (We relegate the explaatio of this subtlety i Se. III B 3. Sie the Euler harateristi is topologial, F 2 a also flow to a o-vaishig value. I summary, the possible form of the EE for a low eergy TQFT (whe regularized o the lattie is S( = F 0 Σ + S topo ( 4πF 2 χ(σ + O(1/ Σ. (3 For the sake of larity, we deote the ostat part of the EE for a geeri theory as S ( = S topo ( 4πF 2 χ(σ + 4F 2 Σ d2 x hh 2, ad the ostat part of the EE for a TQFT as TQFT ( = S topo ( 4πF 2 χ(σ. We poit out that the value of F 2 for a geeral theory ad for a TQFT are ot the same, sie its value flows uder reormalizatio to the oe i the TQFT, whih will be speified i Se. II B 2. Furthermore, the area law part of the EE, F 0 Σ, is deoted as S area (. For ay quatum state, there are several iformatio iequalities relatig EEs betwee differet subsystems that are uiversally valid 59, suh as sub-additivity, strog sub-additivity, the raki-lieb iequality 60 ad weak mootoiity 61. Speial quatum states, suh as quatum error orretig odes 62 ad holographi odes 59,63,64, obey further idepedet iformatio iequalities. The major ostrait o the EE utilized i this paper is the strog sub-additivity iequality, whih is typially used i quatum iformatio theory. Expliitly, the SS iequality is S(B + S(BC S(BC + S(B, (4 where the spae is divided ito four regios, B, C, ad (BC. Here, (BC is the omplemet of BC B C. SS strogly ostrais the struture of the ostat part of S(, i.e., S (, as we will see below. 2. Redutio to the Costat Part of the EE The SS is uiversal, ad hee it is valid for ay hoie of the regios, B ad C. Here we will oly eed to osider the speial ases with C =. This ofiguratio is hose preisely to ael the area law part of the EE o both sides of the SS iequality, thus givig us iformatio about the ostat part S (. Expliitly, whe C =, we have S area (B + S area (BC = S area (BC + S area (B. (5 Equatio (4 the implies S (B + S (BC S (BC + S (B. (6 Whe restrited to a TQFT, we have S TQFT (B+ TQFT (BC TQFT (BC+ TQFT (B. (7 3. Struture of S ( We eed to parametrize TQFT ( i order to proeed. For a TQFT (where F 2 = 0, we see that TQFT ( = S topo ( 4πF 2 χ(σ oly depeds o the topology of the etaglemet surfae Σ through its Euler harateristi. Two-dimesioal orietable surfaes are lassified by a set of umbers { 0, 1, 2,...}, where g is the umber of disoeted ompoets (parts with geus g. 98 We will show that this is a over-omplete labelig for TQFT (, ad that TQFT ( oly depeds o the zeroth ad first Betti umber 65 of Σ defied below i terms of { 0, 1, 2, }. For the time beig, we use the (over-omplete labelig sheme for TQFT ( S TQFT [(0, 0, (1, 1,, (g, g, ], (8 where i eah braket, the first umber deotes the geus, ad the seod umber deotes the umber of disoeted boudary ompoets with the orrespodig geus. The list eds preisely whe g 0 ad g = 0 for ay g > g. I other words, TQFT [(0, 0, (1, 1,..., (g, g ] is the ostat part of the EE of the regio with 0 geus 0 boudaries, 1 geus 1 boudaries, ad g geus g boudaries. We emphasize that the regio a have multiple disoeted boudary ompoets. The set { g } is related to the Betti umbers b i ad the Euler harateristi χ through g g=0 g = b 0, g g=0 g (2 2g = 2b 0 b 1 = χ. (9 These umbers will be useful i the followig alulatios. By applyig the SS iequality to a series of etaglemet surfaes, we derive a expressio for TQFT i terms of the Betti umbers b 0 ad b 1, as well as the etropies TQFT [T 2 ] ad TQFT [S 2 ] aross the torus ad sphere, respetively. Relegatig the details of the derivatio to ppedix C, we fid: S TQFT [(0, 0, (1, 1,, (g, g ] = b 0 TQFT [T 2 ] + χ ( TQFT [S 2 ] TQFT [T 2 ]. (10 2 Notie that Eq. (10 is osistet with the expetatio that disoeted parts of the etaglemet surfae result i additive otributios due to the loal ature of the mutual iformatio. B. Topologial Etaglemet Etropy Our first mai result is Eq. (10, whih larifies two poits. First, as we metioed i the itrodutio (ad as was also disussed i Ref. 58, give a geeral etaglemet surfae [(0, 0, (1, 1,..., (g, g ], we a

5 redue the omputatio of the ostat part of the EE of a TQFT, TQFT [(0, 0, (1, 1,..., (g, g ], to that of TQFT [S 2 ] ad TQFT [T 2 ]. Seod, usig Eq. (10, we a idetify the topologial ad uiversal part of S ( for a geeri theory beyod the TQFT fixed poit. We ow elaborate o these poits. 1. TQFT [S 2 ] ad TQFT [T 2 ] B C For a TQFT, Eq. (10 proves that the ostat part of the EE aross a geeral surfae a be redued to a liear ombiatio of the ostat part of the EE aross S 2 ad T 2. Whether TQFT [S 2 ] ad TQFT [T 2 ] are idepedet of eah other depeds o the type of TQFT. s we show i Se. III, for a BF theory [see Eq. (22] i (3+1D, TQFT [S 2 ] = TQFT [T 2 ]. For the GWW models [see Eq. (19] i (3+1D, we show i Se. III that TQFT [S 2 ] ad TQFT [T 2 ] are differet i geeral. Thus, Eq. (10 is the simplest expressio that is uiversally valid for ay TQFT. 2. way from the Fixed Poit I Se. II 1 ad ppedix B, we revisited the argumets preseted i Ref. 58 that the ostat part of the EE for a theory away from the fixed poit is geerially ot topologial. The struture of the EE of a geeri theory was show i Eq. (2. Combiig Eq. (2 ad Eq. (10, we ow extrat more iformatio about the struture of the EE. First, we argued i Se. II 1 that F 2 0, (11 whe the theory is reormalized to a TQFT fixed poit. Seod, by settig F 2 = 0 i Eq. (2 ad omparig the TEE ad the oeffiiet of the Euler harateristi χ i Eq. (2 ad Eq. (10, we fid that ad S topo [(0, 0,, (g, g ] = b 0 S TQFT [T 2 ] = F 2 1 ( 8π S TQFT ( g i=0 i S TQFT [T 2 ], (12 [S 2 ] TQFT [T 2 ]. (13 Equatio (12 suggests that the TEE aross a arbitrary etaglemet surfae (for a geeri theory is proportioal to TQFT [T 2 ]; i partiular, the TEE aross T 2 (for a geeri theory equals TQFT [T 2 ], i.e., S topo [T 2 ] = TQFT [T 2 ]. Equatio (13 shows that while F 2 a flow whe the theory is ( reormalized, it overges to a otrivial value 1 8π S TQFT [S 2 ] TQFT [T 2 ] at the RG fixed poit. Our idetifiatio of the TEE Eq. (12 Figure 1: KPLW presriptio of etaglemet surfae T 2. The spae iside the two torus is divided ito three regios,, B ad C, eah beig a solid torus. further elaborates o the result from Ref. 58, whih showed that the TEE aross a geus g etaglemet surfae Σ g is S topo [Σ g ] = gs topo [T 2 ] (g 1S topo [S 2 ]. Our result Eq. (12 suggests that S topo [S 2 ] = S topo [T 2 ] ad therefore further simplifies the result of Ref. 58 to S topo [Σ g ] = S topo [T 2 ] for ay g. Our idetifiatio of the TEE also works for etaglemet surfaes with multiple disoeted ompoets. 3. Extratig the TEE Equatio (12 suggests a algorithm to ompute the TEE for a geeri theory: 1 take a groud state wavefutio ψ for a geeri system; 2 reormalize ψ to the fixed poit; 3 ompute the etaglemet etropy for a etaglemet surfae T 2, S TQFT [T 2 ]. The ostat part TQFT [T 2 ] is the TEE aross T 2. Notie that this is osistet with our defiitio TQFT [T 2 ] = S topo [T 2 ] 4πF 2 χ(t 2 sie χ(t 2 = 0. The TEE aross a arbitrary surfae immediately follows from Eq. (12. I this setio, we will explai a more pratial algorithm for extratig the TEE (aross T 2 whih is appliable to the groudstate wavefutio of ay geeri theory, ad does ot require reormalizatio to the TQFT fixed poit. Our algorithm (whih is termed the KPLW presriptio builds upo the study of the topologial etaglemet etropy i (2+1D systems iitiated by Kitaev, Preskill, Levi ad We 50,51 (KPLW ad the proposal i Ref. 58 i (3+1D. We ompute a partiular ombiatio of the EE of differet regios, whih we all S KPLW [T 2 ], ad demostrate that this ombiatio equals S topo [T 2 ]. The same KPLW presriptio was studied i Ref. 58, but here we provide a rigorous proof of the equivalee betwee the etaglemet etropy from the KPLW presriptio Eq. (14 ad the TEE S topo [T 2 ], as we derive i Eq. (17. Via Eq. (12, we a the obtai the TEE aross a geeral surfae. We geeralize the KPLW presriptio to (3+1D by

6 osiderig the ofiguratio of the etaglemet regios show i Fig. 1 ad omputig the ombiatio of EEs S KPLW [T 2 ] S( + S(B + S(C S(B S(C S(BC + S(BC. (14 Followig similar argumets i Ref. 50, it a be show that S KPLW [T 2 ] satisfies two properties: 1. S KPLW [T 2 ] is isesitive to loal deformatios of the etaglemet surfae. 2. S KPLW [T 2 ] is isesitive to loal perturbatios of the Hamiltoia. We first argue that the property 1 holds. If we loally deform the ommo boudary of regio ad B (but away from the ommo boudary of regio, B ad C, whih is a lie, the deformatio of S KPLW [T 2 ] is S KPLW [T 2 ] = [ S( S(C] + [ S(B S(BC]. (15 Beause the deformatio is far away from regio C (farther tha the orrelatio legth ξ 1/m, where m is the eergy gap, S( S(C = 0, ad similarly S(B S(BC = 0. Hee S KPLW [T 2 ] is uhaged uder the deformatio of ommo boudary of ad B, away from the lie whih represets the ommo boudary of, B ad C. If we ow loally deform the ommo boudary of regios, B ad C 99 (the lie B C, S KPLW [T 2 ] S( + S(B + S(C S(B S(C S(BC = [ S(DBC S(BC] + [ S(DC S(C] + [ S(DB S(B], (16 where regio D is the omplemet of the regio BC, i.e., D = (BC, ad we have used = DBC ad S( = S(. Sie the deformatio is far from regio D (farther tha the orrelatio legth ξ as it is atig oly o the lie B C, eah of three square brakets vaishes separately. Hee S KPLW [T 2 ] is uhaged uder the deformatio of the ommo boudary lie of, B ad C. I summary S KPLW [T 2 ] = 0 uder a arbitrary deformatio of the etaglemet surfae. Therefore property 1 holds. We ow argue that property 2 holds. s suggested i Refs. 50,51, whe we loally perturb the Hamiltoia far iside oe regio 100, for istae regio, the fiiteess of the orrelatio legth ξ guaratees that the perturbatio does ot affet the redued desity matrix for the regio. Therefore the etaglemet etropy S( = S( is uhaged. If a perturbatio of the Hamiltoia ours o the ommo boudary of multiple regios, for example regio ad B, oe a deform the etaglemet surfae usig property 1 suh that the perturbatio is o-vaishig i oe regio oly. This shows that S KPLW [T 2 ] is ivariat uder loal deformatios of the Hamiltoia whih does ot lose the gap (i.e., those whih leave ξ <, ad property 2 holds. I summary S KPLW [T 2 ] is a topologial ad uiversal quatity. Lastly we show that the ombiatio S KPLW [T 2 ] equals the TEE, S topo [T 2 ], i.e., S KPLW [T 2 ] = S topo [T 2 ], (17 where S topo [T 2 ] is defied i Eq. (12. We isert the expasio of the EE (2 i the defiitio of S KPLW [T 2 ]. First, it is straightforward to hek that the KPLW ombiatio of the area law terms ael. Seod, the KPLW ombiatio of the Euler harateristi terms vaish sie eah regio i the KPLW ombiatio is topologially a T 2, ad χ(t 2 = 0. Third, as we prove i ppedix D, the KPLW ombiatio of the mea urvature terms vaishes as well, i.e, 4F 2 + B+ C B C BC+ BC d 2 x hh 2 = 0. (18 This was assumed impliitly i Ref. 58, but we demostrate it expliitly here so as to lose the loop i the argumet. Fially, the KPLW ombiatio simplifies to S topo [T 2 ]: it is give by the sum of the TEE aross the four tori, B, C ad BC, mius the TEE aross the three tori B, C ad BC. Therefore, Eq. (17 holds. I summary, we have demostrated that the KPLW presriptio, Eq. (14, gives a orete method to extrat the TEE for a geeri (o-fixed-poit theory. III. PPLICTION: ENTNGLEMENT ENTROPY OF GENERLIZED WLKER-WNG THEORIES I this setio, we ostrut lattie groud state wave futios for a lass of TQFTs kow as the geeralized Walker-Wag (GWW models, whose atios are give by Eq. (19 below. We the ompute the EE aross various two dimesioal etaglemet surfaes. The alulatios i this setio are idepedet of the SS iequality used i Se. II. The alulatios i this setio provide support for our assumptios about the etaglemet etropy for fixed-poit models, ad suggest a ojeture about higher dimesioal topologial phases. The GWW models are desribed by a TQFT with the atio 66 68 S GWW = p B d + B B,, p Z. (19 2π 4π The Walker-Wag models orrespod to the speial ases p = 0 ad p = 1. I Eq. (19 B is a 2-form U(1 gauge field ad is a 1-form U(1 gauge field. (Whe we formulate the theory o a lattie, they will be Z valued.

7 See ppedix E for details. The gauge trasformatios of the gauge fields are + dg pλ, B B + dλ, (20 where λ is a u(1 valued 1-form gauge field (where u(1 is the Lie algebra of U(1 with gauge trasformatio λ λ + df (where f is a salar satisfyig f f + 2π, ad g is a ompat salar (i.e., g g + 2π. The gauge ivariat surfae ad lie operators are respetively ( exp ik B, k {0, 1,..., 1}, Σ 1 ( (21 exp il + ilp, l {0, 1,..., 1}, γ Σ 2 B where Σ 1 is a losed two dimesioal surfae, γ is a losed oe dimesioal loop ad Σ 2 is a ope two dimesioal surfae whose boudary is γ. The gauge ivariae follows from the ompatifiatio of the salar g ad the stadard Dira flux quatizatio oditio of U(1 gauge field λ: γ dg 2πZ ad Σ 1 dλ 2πZ. 101 We will use aoial quatizatio to explai that exp(i Σ 1 B ad exp(i γ + ip Σ 2 B are trivial operators i pp. E.. Wave Futio of GWW Models 1. BF Theory: (, 0 For simpliity, we first disuss the speial ase whe p = 0, whih is referred to as a BF theory. The atio is S BF = B d, (22 2π M 4 where is a 1-form gauge field ad B is a 2-form gauge field. The theory is defied o a spaetime whih is topologially a four ball, M 4 B 4, whose boudary S 3 is a spatial slie, as show i Fig. 2. I the followig, we formulate the theory o a triagulated spaetime lattie. The 1-form gauge field orrespods to 1-ohais (ij 2π Z livig o 1-simplies (ij. The 2-form gauge field B orrespods to 2-ohais B(ijk 2π Z livig o 2-simplies (ijk 102. We defie the Hilbert spae to be H = (ijk H (ijk, where H (ijk is a loal Hilbert spae o the 2-simplex (ijk spaed by the basis B(ijk = 2πq/, q Z. 103 More details about the lattie formulatio of the TQFT are give i ppedix E. We ow disuss the groud state wave futio for this theory. The groud state wave futio is defied o the boudary of the ope spaetime maifold S 3 = M 4 as 69,70 ψ = C C,C C M4 D C M4 DB exp ( i 2π M 4 B d C, (23 l S M 4 Figure 2: shemati figure of the topology of spaetime M 4 ad spae S 3. Iside S 3, we shematially draw a loop l represetig the loop ofiguratios C of the B field i the dual lattie. The dashed surfae S boudig the loop l exteds ito the spaetime bulk M 4, represetig the B field ofiguratio i the dual lattie of spaetime. S represets the B field ofiguratios that form losed surfaes away from the boudary of the spaetime M 4. The boudary oditio i the path itegral Eq. (23 is speified by a fixed B ofiguratio C o S 3. The path itegral should itegrate over all the ofiguratios i the spaetime bulk M 4 with the boudary ofiguratio C o S 3 fixed. where C ad C idiate the boudary ofiguratios for the ad B fields respetively, i.e., the value of ad B fields o M 4. We itegrate over all ad B subjet to the boudary oditios C ad C. C is a ormalizatio fator. Beause ad B are aoially ojugate, the states are speified by the ofiguratio of B oly; C is a speifi state orrespodig to the partiular B field ofiguratio C o M 4. The summatio over C rages over all possible ofiguratios of B-ohai with weights determied by the path itegral. C M4 meas the path itegral is subjet to the fixed boudary oditios C o M 4, ad similarly for C M4. If we take the spaetime M 4 to be a losed maifold, Eq. (23 redues to the partitio futio over M 4. Beause the spaetime is topologially a 4-ball B 4, there is oly oe groud state assoiated with the boudary S 3. 104 We first work out the wavefutio for the BF theory with = 2 expliitly as a geeralizable example. We use B field values as a basis to express C. Itegratig out (otie that we both itegrate over the ofiguratios of the -field with fixed boudary ofiguratios ad also sum over the boudary ofiguratios, i.e., C C M4 D, whih is tatamout to itegratig over all ofiguratios of, we get the ostrait δ(db, ψ = C ( DBδ db C. (24 C C M4 S 0 S 3

8 where the delta futio δ(db ostrais db(ijkl = 0 mod 2π o eah tetrahedro (ijkl i M 4. Coretely, db(ijkl = B(jkl B(ikl + B(ijl B(ijk = 0 mod 2π. (25 y B ofiguratio satisfyig this ostrait is said to be flat (see ppedix E for details. Sie B(ijk {0, π}, i, j, k for the = 2 theory, Eq. (25 meas that for eah tetrahedro, there are a eve umber of 2- simplies where B(ijk = π mod 2π, ad a eve umber of 2-simplies with B(ijk = 0 mod 2π. We refer to the π 2-simplies as oupied ad to the 0 2-simplies as uoupied. It is more trasparet to osider the ofiguratios i the dual lattie of the spatial slie S 3. (I the ext paragraph, we will disuss the dual lattie ofiguratios i the spaetime M 4. s a example, the dual lattie of a tetrahedro is show i Fig. 3. The 2-simplies i the origial lattie are mapped to 1-simplies i the dual lattie. 105 2-ohai B(ijk defied o a 2-simplex i the origial lattie is mapped to a 1-ohai B(ab defied o a 1-simplex i the dual lattie. If B(ijk = π, the we defie the orrespodig B(ab = π i the dual lattie. I the dual lattie, Eq. (25 meas that there are a eve umber of oupied bods (1-simplies assoiated with eah vertex, as well as a eve umber of uoupied bods. If we glue differet tetrahedra together, we fid that the oupied bods i the dual lattie form loops. Pitorially, this is remiiset of the wave futio of the tori ode model i oe lower dimesio 26,71,72. I the (3 + 1D spaetime M 4 [rather tha the 3D spae S 3 ], 2-simplies are dual to the (4 2 = 2-simplies [rather tha the 1-simplies] i the dual lattie. Equatio (25 meas the oupied 2-simplies form otiuous surfaes i the dual spaetime lattie. (Cotiuous meas that the simplies i the dual lattie oet via edges, rather tha via verties. We disuss the otiuity of the dual lattie surfaes i ppedix F. If these surfaes are iside the bulk of the spaetime ad do ot touh M 4 (suh as S i Fig. 2, they are otiuous ad losed surfaes; if the surfaes iterset with the spatial slie M 4 (suh as S i Fig. 2, the itersetios are losed loops i M 4. For the BF theory with a geeral oeffiiet, the wavefutio is also a superpositio of loop ofiguratios. The oly differee is that the loops are formed by 1-simplies i the dual lattie with B = 2π. Whe there is a loop formed by 1-simplies with B = 2πl i the dual lattie, we regard the loop as omposed of l overlappig loops formed by the same 1-simplies with B = 2π. We emphasize that the loop ofiguratio is efored by the flatess oditio Eq. (25. For > 2, we eed to speify the orietatios of the simplies ad keep trat of the sigs i Eq. (25. The orietatio of eah simplex is speified i Fig. 3, where the orietatios of (jkl ad (ijl are poitig ito the tetrahedro, while the orietatio of (ikl ad (ijk are poitig out of the tetrahedro. For example, if the values of j b i a e Figure 3: tetrahedro is draw with solid lies, ad its dual is draw i dash ad gray lies. The 2-simplex (ijk i the origial lattie is dual to the 1-simplex (ab i the dual lattie. Similarily, (ikl is dual to (ad, (ijl is dual to (a ad (jkl is dual to (ea. The olored dash arrows idiate the orietatios of the four 2-simplies, where (ijk ad (ikl share the same orietatio, ad (ijl ad (jkl share the opposite orietatio. The orietatios of the dual-lattie 1-simplies are also idiated by the arrows o the grey/dashed lies. the B-ohais are B = 2πq 1 /, 2πq 2 /, 2πq 3 /, 0 with q 1 q 2 + q 3 = 0 o the 2-simplies (jkl, (ikl, (ijl, (ijk respetively, the dual of (jkl ad (ikl (i.e., (ea ad (ad belog to oe loop i the dual lattie, while the dual of (ijl ad (ikl (i.e., (a ad (ad belog to aother loop i the dual lattie. Note that the two loops share the same dual lattie bod (ad where the value of the B-ohai is the sum of the B values from the two loops B(ad = 2π(q 1 + q 3 / = 2πq 2 /. The gauge trasformatio, B(ijk B(ijk + λ(jk λ(ik + λ(ij, preserves Eq. (25. Hee, although it deforms the positio of loops, it ever turs losed loops ito ope lies. Ope lies i the dual lattie violate the flatess oditio Eq. (25, ad so do ot otribute to the wave futio Eq. (24. Summig over the ofiguratios C esures gauge ivariae of the wave futio. Notie that Eq. (24 implies that the weights assoiated with differet loop ofiguratios C are equal, similar to the tori ode. Thus we see that Eq. (24 redues to ψ = C C L k d C, (26 where the sum is take over the set L of all possible loop ofiguratios C at the spatial slie S 3 = M 4. This is termed loop odesatio, sie the wave futio is the equal weight superpositio of all loop ofiguratios i the dual lattie. l

9 2. Geeral Case: (, p I this setio, we osider GWW models with otrivial p desribed by the atio i Eq. (19, where is still a 1-form ad B a 2-form. Caoial quatizatio of the GWW theories implies that B 2π Z o the lattie (see ppedix E for more details. I order to fid the groud state wave futio, we still use B as the basis to label the ofiguratios C ad the orrespodig states C o the spatial slie. The wave futio is formally give by ψ =C D C,C exp C M4 ( i 2π M 4 C M4 DB p B d + i 4π M 4 B B C. (27 For simpliity, we osider the ase = 2, p = 1 i the followig. s i the BF theory, we first itegrate out the fields, yieldig ψ = C ( DB δ db exp (i 2 B B C. 4π C C M4 M 4 (28 The differee betwee this wave futio ad that of the BF theory, Eq. (24, is that whe the flatess oditio δ(db is satisfied, the states with differet ofiguratios C are assoiated with differet weights. The weights are determied by the itegral ( exp i 2 4π B B, (29 M 4 where B must satisfy the flatess oditio db = 0 with the boudary oditio labeled by C. We proeed to evaluate the itegral i Eq. (29. Notie that the flatess oditio, Eq. (25, implies that the 2-simplies at whih B = π form two-dimesioal spaetime surfaes i the dual lattie of M 4 whose boudaries o the spatial slie S 3 are losed loops belogig to C. Relegatig the details of the derivatio to ppedix G, we show that whe B = π oly at two dual lattie surfaes S 1, S 2, whose boudaries are dual lattie loops l 1 = S 1, l 2 = S 2 i C, it follows that ( exp i 2 B B 4π M ( 4 = exp iπlik(l 1, l 2 + i π 2 lik(l 1, l 1 + i π 2 lik(l 2, l 2. (30 The first term is assoiated with the mutual likig umber, lik(l 1, l 2, betwee differet loops, while the seod ad the third terms are assoiated with the self-likig umber, lik(l i, l i, of oe loop, l i, with itself, defied i ppedix G. Equatio (30 a be geeralized to ofiguratios with may loops, ad the weights of differet ofiguratios are determied by the likig umbers of the loops. I summary, the groud state wave futio for the (, p = (2, 1 theory is: ψ = C C L( 1 #(Mutual liks i #(Self liks C. (31 For geeral (, p, a similar argumet a be made. B a ow take differet values 2πk, k = 0, 1,, 1 o eah 2-simplex i the lattie, or o eah 1-simplex i the dual lattie. Due to the ostrait of Eq. (25, the 1- simplies where B = 2π/ form loops i the dual lattie. Similar to the disussio of the ase p = 0 ad geeral, two dual-lattie loops a touh i oe tetrahedro. We also regard a loop with B = 2πq/ to be q overlappig loops with B = 2π/. If there are q 1 loops with B = 2π/ that are overlappig o l 1 (whih is equivalet to oe loop with B = 2πq 1 / o l 1 ad q 2 loops with B = 2π/ that are overlappig o l 2 (whih is equivalet to oe loop with B = 2πq 2 / o l 2, the ( exp i p B B 4π M 4 [ = exp 2i p(2π2 q 1 q 2 4π 2 lik(l 1, l 2 + i p(2π2 q1 2 4π 2 lik(l 1, l 1 + i p(2π2 q2 2 ] 4π 2 lik(l 2, l 2 [ = exp i 2πpq 1q 2 lik(l 1, l 2 + i πpq2 1 lik(l 1, l 1 + i πpq2 2 lik(l 2, l 2 ]. (32 Therefore after evaluatig these weights, the wave futio Eq. (27 redues to ψ = C C L e i 2πp #(Mutual liks e i πp #(Self liks C, (33 where the mutual-likig ad self-likig umbers are outed with multipliities q 1 ad q 2 as give i Eq. (32. The sum over C L otais ofiguratios with all possible q 1 ad q 2. B. Etaglemet Etropy of GWW Models I this setio, we show that the ostat part of the EE of GWW theories depeds o the topology of the etaglemet surfae i a otrivial way. I partiular, S [S 2 ] S [T 2 ] i geeral. Hee, S [S 2 ] ad S [T 2 ] are truly idepedet quatities. This setio is divided ito two parts: I Se. III B 1, we alulate the EE for GWW models with arbitrary (, p aross the etaglemet surfae T 2. I Se. III B 3, we ompute the EE for GWW models aross losed surfaes with arbitrary geus ad a arbitrary umber of disoeted ompoets. These idepedet alulatios ofirm Eq. (10.

10 B=π simplies o the etaglemet surfae Figure 4: example of the lattie struture of a etaglemet ut i (2 + 1D. The gree simplies form the etaglemet ut Σ, whih partitios the lattie ito regio ad regio. We ilude Σ as part of regio. B = π o the red simplies, while B = 0 elsewhere. The dotted loop is the dual lattie ofiguratio of the red simplies. I this example, the ofiguratio C E otais two B = π 1-simplies at the etaglemet ut Σ, whih are the fourth ad eighth 1-simplies of Σ (outig from the left side as show i the figure. the fourth ad the eighth gree 1-simplies (outig from the left side are oupied o the etaglemet surfae Σ, whih also belog to regio aordig to our partitio. We deote by N (C E the umber of ofiguratios i the regio (but ot iludig Σ osistet with the hoie of C E. We label suh ofiguratios by (a, α, where α is the parity (eve e or odd o of the umber of oupied loops widig aroud the otrivial spatial yle iside the regio i the dual lattie, ad the ofiguratios of either parity are eumerated by a = 1,..., N (C E /2. 107 Similarly, (b, β labels the N (C E ofiguratios i regio. Figure 5 presets a partiular ofiguratio where, besides two otratible dual-lattie loops, there is oe dual lattie loop wrappig the o-otratible yle i the dual lattie of regio ad oe dual lattie loop wrappig the o-otratible yle i the dual lattie of regio, whih orrespods to α = o ad β = o. Note that two o-otratible yles are i differet regios ad. To be illustrative, we also draw 2-simplies i the real lattie where B = π whose dual ofiguratios form loops i the spae. Hee the summatio over C splits as: C = N (C E/2 C E a=1 N (C E/2 b=1 α=e,o β=e,o. (36 1. EE for the Torus, = 2, p = 1 I this subsetio, we ompute the EE of GWW models aross Σ = T 2. For simpliity, we first osider the ase = 2, p = 1, ad the geeralize to models with arbitrary ad p. We start with the wave futio obtaied i the last setio, Eq. (31: ψ = C C ( 1 #(Mutual liks i #(Self liks C. (34 4 3 2 1 We hoose the subregio to be a solid torus whose surfae is T 2, ad to be the omplemet of. We illustrate the mirosopi struture of the spatial partitioig i Fig. 4 via a lower-dimesioal example. The etaglemet surfae Σ is hose to be a smooth surfae i the real spatial lattie (gree simplies i Fig. 4. The real spae simplies that form the etaglemet surfae Σ are outed as part of regio. 106 We will fid the hmidt deompositio of the wavefutio orrespodig to this spatial partitioig i order to alulate the EE. To do so, we first parametrize the ofiguratios C appearig i Eq. (34 as: C {C E, (a, α, (b, β}, (35 whih we ow explai. C E labels the real spae B-ohai ofiguratio at the etaglemet surfae Σ. (I Fig. 4, Figure 5: partiular spatial ofiguratio with oe loop γ 1 (dashed lie threadig through the hole (the hole itself belogs to regio iside the regio ad oe loop γ 2 (grey lie threadig through the hole iside the regio. γ 3 ad γ 4 are two liked otratible loops, where γ 3 loates iside regio, ad γ 4 loates both i regio ad. The two blue poits are the itersetio of l 4 with Σ. The simplies (gray triagles are livig i the real lattie where B = π. The lies perpediular to the simplies are livig i the dual lattie where B = π ad they form loops i the dual lattie. This ofiguratio orrespods to α = o, β = o. For oveiee we also itrodue the otatio

11 l CE a,e =( 1 #(Mutual liks with fixed CE ofiguratio of regio i eve setor, l CE a,o =( 1 #(Mutual liks with fixed CE ofiguratio of regio i odd setor, s CE a,e =i #(Self liks with fixed CE ofiguratio of regio i eve setor, s CE a,o =i #(Self liks with fixed CE ofiguratio of regio i odd setor, (37 where eve/odd setor refers to the set of states with a eve/odd umber of loops i the dual lattie threadig the o-otratible yle i regio. Similar defiitios apply to regio. See Fig. 5 for a illustratio. We further defie ÃCE a α to be a state assoiated with oe partiular ofiguratio i regio, whih is labeled by {C E, a, α}, ad defie ÃCE b β likewise i regio. There is a subtlety: we also eed to speify the mutual-likig/self-likig umber of loops whih ross the etaglemet surfae. We speify that whe two loops (amog whih at least oe of them rosses the etaglemet surfae are liked, suh as γ 3 ad γ 4 i Fig. 5, the mutual-likig umber is outed as part of the side, i.e., la,e CE ad la,o. CE dditioally, whe a loop rosses the etaglemet surfae, the self-likig umber of the loop is outed as part of the side, i.e., s CE a,e ad s CE a,o. We are able to make suh a hoie beause there is a phase ambiguity i the hmidt deompositio, ad phases a be shuffled betwee ad by redefiig the basis ÃCE a e/o ad ÃCE b e/o. (For example, we a defie aother set of states via ÂCE a e/o = s CE 1 a,e/o ÃCE a e/o, ad ÂCE b e/o = s CE a,e/o ÃCE b e/o. s we will see, the redued desity matrix Eq. (39 does ot deped o the hoie of phase assigmet. Combiig the above, we get ψ =C C E N (C E/2 a=1 N (C E/2 b=1 ( 1 αβ la,αl CE CE b,β sce a,αs CE b,β ÃCE a α=e/o β=e/o α ÃCE b β. (38 The fator ( 1 αβ, whih equals 1 whe α = β = o ad 1 otherwise, reflets the mutual-likig betwee the o-otratible loops i regio (suh as γ 1 i Fig. 5 ad the o-otratible loops i regio (suh as γ 2 i Fig. 5. Figure 5 shows a speial ofiguratio where there is oe o-otratible loop i regio ad oe o-otratible loop i regio. From this we easily obtai the redued desity matrix for regio by traig over the Hilbert spae i regio, ρ = C 2 C E N (C E 2 ( 1 (α αγ CE a N (C E/2 a,ã=1 α CE ã α = C N (C E/2 2 N (C E C E ( CE a e CE ã a,ã=1 e + CE a α, α,γ=e,o o CE ã o, (39 where we have performed uitary trasformatios o the bases ÃCE a e/o ad ÃCE b e/o to absorb the mutuallikig ad self-likig fators withi regio ad regio respetively. The trasformed bases are deoted CE a α = la,αs CE a,α ÃCE a α ad CE b β = l CE b,β sce b,β ÃCE b β. Furthermore, the ostrait Tr H (ρ = C 2 C E N (C E N (C E = 1 (40 fixes the ormalizatio ostat C. For eah fixed ofiguratio C E o the etaglemet surfae, the produt of the umber of ofiguratios i the regio ad the umber of ofiguratios i regio, i.e., N (C E N (C E, is idepedet of C E (see ppedix H for details. Thus, to ompute C we eed oly to out the umber of differet hoies of C E. There are i total 2 Σ 1 differet boudary ofiguratios, where the 1 omes from the ostrait that losed dual lattie loops always iterset the etaglemet surfae twie (hee the umber of oupied 1-simplies o Σ is eve, ad Σ is the umber of 2-simplies o the etaglemet surfae. Sie C 2 N (C E N (C E is idepedet of C E, ad there are 2 Σ 1 hoies of C E, C 2 N (C E N (C E = 1. (41 2 Σ 1 We give a more detailed derivatio of this formula i ppedix H. From the redued desity matrix ρ, we a alulate the etaglemet etropy of the groud state ψ assoiated with the torus etaglemet surfae by the replia trik, S( = Tr H ρ log ρ = d dn ( TrH ρ N (Tr H ρ N N=1 (42

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

[ where ρ CE ] a,α;ã α are matrix elemets give by [ ρ CE ] a,α;ã α [ ] [ = C 2 N (C E 11 J gd(,p gd(,p α α J N (C E ] aã 13. (50 Here, 11 m is the m m idetity matrix, ad J l is a l l matrix of oes (whih has oe ozero eigevalue equal to l. The first term i this expressio origiates from the periodi delta futio i Eq. (50a, ad the seod term omes from the sum over a, ã i the outer produt. Notig that eah J m is a rak oe matrix with ozero eigevalue m, we see immediately that ρ CE a be put i diagoal form ρ CE = C 2 N (C E N (C E gd(, p(11 gd(,p 0 N (C E /gd(,p. (51 The matrix i Eq. (51 is 1 1 gd(,p 1 s N (C E... 1 0 0... gd(,p 0 s 0 0 0 (52 Fially, usig Eq. (47, we fid that the ozero etaglemet eigevalues are give by e ξ C E,r = gd(, p Σ, (53 where r = 1,, Σ /gd(, p. With this spetrum, it is straightforward to evaluate Eq. (48 to obtai the etaglemet etropy as S( = Σ log log gd(, p. (54 The first term is proportioal to the area of the etaglemet surfae. The seod ostat term is the TEE 108 : S TQFT ( = S topo ( = log gd(, p. (55 We see that the TEE depeds otrivially o the parameters ad p. If ad p are oprime, i.e., gd(, p = 1, the TEE vaishes. If p = 0, usig the defiitio gd(, 0 =, the ostat part of the EE redues to log. lteratively, we a also ompute the EE of the BF theory usig the wave futio Eq. (26, ad we fid the ostat part to be log. Note that this result is osistet with Refs. 67 ad 73 where the groud state degeeray (GSD o T 3 was omputed to be gd(, p 3. The groud state degeeray suggests that the GWW models a be topologially ordered, whih, i our otext, is refleted by the ozero TEE, log gd(, p. Whe gd(, p = 1, the groud state o T 3 is o-degeerate, ad the TEE vaishes. I partiular, for the ase of the Walker-Wag model = 2, p = 1, we obtai S( = Σ log 2, (56 ad there is o topologial order. We otie the relatio betwee the GSD o T 3 ad the TEE aross the torus T 2, exp( 3S topo [T 2 ] = GSD[T 3 ], (57 whih should be ompared to the similar relatio, exp( 2S topo [T 1 ] = GSD[T 2 ], for the (2+1D belia theories. For a belia theory i (d + 1D, our omputatio leads us to ojeture that exp( ds topo [T d 1 ] = GSD[T d ]. (58 For (d + 1D BF theory with level, we have omputed both the TEE ad the GSD[T d ], ad we foud S topo [T d 1 ] = log ad GSD = d. This is osistet with our ojeture. (See ppedix I for details. We ojeture that this relatioship is true for more geeral theories suh as Dijkgraaf-Witte models, ad higher dimesioal Cher-Simos theories as well. For a geeri (2 + 1 dimesioal oabelia Cher- Simos theory, Eq. (58 may ot hold. For example, the TEE of the SU(2 3 Cher-Simos theory is S topo [T 1 ] = log( 5/(2 si(π/5 74, ad exp( 2S topo [T 1 ] is ot a iteger. Hee Eq. (58 a ot hold beause the GSD should be a iteger. However, we ote that for some oabelia theories, the ojeture still holds. For example, for the bosoi Moore-Read quatum Hall state i (2 + 1D, GSD[T 2 ] = 4 (whih osists of 3 states from the eve parity setor ad 1 state from the odd parity setor, ad S topo [T 1 ] = log 2, hee Eq. (58 holds i this ase. 3. EE for rbitrary Geus Followig the same proedure used for the torus, we alulate the EE aross a geeral etaglemet surfae

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=1 1 1 1 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

15 spetrum are give by 1/N,p,g; Σ, where N,p,g; Σ Σ χ/2, χ = 2 2g. (62 gd(, p g χ is the Euler harateristi of Σ. Thus, the EE aross a geeral surfae of geus g is: S[(0, 0,(1, 0,..., (g 1, 0, (g, 1] = Σ log g log gd(, p (1 g log = Σ log χ (63 2 log log gd(, p. gd(, p Equatio (63 is osistet with Eq. (10. We summarize S topo ( ad TQFT ( for various systems ad various etaglemet surfaes i Table I. We ote that although Eq. (63 is the EE for a low eergy TQFT, there is still a area law term. Sie the TQFT is idepedet of the metri of the etaglemet surfae, oe may aively expet that the area law term should vaish. The reaso that the area law term appears i Eq. (63 is that we formulated our theory o a lattie, whih expliitly broke the salig symmetry (i.e., hagig the area of the ut hages the umber of liks passig through Σ. However symmetry uder area-preservig diffeomorphisms was uaffeted by the lattie regularizatio (hagig the shape of the ut does ot hage the umber of liks passig through Σ. Beause of this, we get terms that sale like the area of the ut (area law term, but o further shape-depedet terms. Therefore, we expet, ad ideed fid, that the mea urvature term vaishes for the TQFT (F 2 0. IV. SUMMRY ND FUTURE DIRECTIONS I this paper, we have aalyzed the geeral struture of the EE for gapped phases of matter whose low eergy physis is desribed by TQFTs i (3+1D. The EE for gapped phases geerally obeys the area law. The area law part of the EE is ot uiversal, while the ostat part of EE otais topologial iformatio. Hee we have foused o the ostat part of the EE. For TQFTs, our aalysis relied o the SS iequalities. We foud that they are strog eough to ostrai the possible expressios for the EE. Oe of our mai results is Eq. (10: the EE aross a geeral surfae a be redued to a liear ombiatio of TQFT [S 2 ] ad TQFT [T 2 ]. We have idetified the topologial ad uiversal otributio to the etaglemet etropy, i.e., the topologial etaglemet etropy (TEE. We also aalyzed the behavior of various terms i the EE whe the theory was deformed away from the fixed poit, ad argued that a geeralizatio of the KPLW presriptio allows to extrat the TEE. We the provided idepedet alulatios of the etaglemet etropy for the GWW lass of TQFTs. We determied the groud state wave futios of the GWW models, whih allowed us to alulate the EE. The results ofirm our more geeral aalysis of the EE. We showed that twistig terms i the Lagragia a i geeral hage the topologial etaglemet etropy. We the ojetured a relatioship betwee the topologial etaglemet etropy ad the groud state degeeray of belia theories i (d + 1 dimesios. Sie we have oly osidered gapped systems without global symmetry, oe atural questio for future work is whether oe a use the etaglemet of the groud state wavefutio to probe topologial phases with global symmetries, suh as SPT order ad symmetry erihed topologial order i higher dimesios. I partiular, for systems with SPT order, there is o itrisi topologial order ad the groud state wavefutio is oly short rage etagled, hee the TEE is trivial. However, it has bee realized that the etaglemet spetrum serves as a useful tool to probe SPT order. I Refs. 75 ad 76, the etaglemet spetra of oe dimesioal spi ad fermio systems were studied, where the otrivial degeeray of the spetra revealed otrivial SPT order. I Refs. 77 ad 78, the existee of i-gap states i the sigle body etaglemet spetrum was prove to reveal the otrivial topology of a topologial bad isulator. Furthermore, there are extesive theoretial ad umerial studies o the etaglemet spetrum of quatum Hall systems 79 85 ad fratioal Cher isulators 86. It would be beefiial to omplemet this with a more systemati ivestigatio of the etaglemet spetrum as a probe of SPT order i higher dimesios i the future. kowledgmets We thak F. Burell, M. Mezei, S. Pufu ad S. Sodhi for useful ommets. B.. Berevig wishes to thak Eole Normale Superieure, UPMC Paris, ad the Doostia Iteratioal Physis Ceter for their geerous sabbatial hostig durig some of the stages of this work. BB akowledges support for the aalyti work from NSF EGER grat DMR 1643312, ONR - N00014-14-1-0330, NSF-MRSEC DMR-1420541. The omputatioal part of the Prieto work was performed uder departmet of Eergy de-s0016239, Simos Ivestigator ward, the Pakard Foudatio, ad the hmidt Fud for Iovative Researh. ppedix : Review of Etaglemet Etropy ad Spetrum I this appedix, we review the defiitio of the etaglemet etropy, ad review the otatio that we use i this work. To defie the etaglemet etropy, we first partitio the spae ito two parts,, ad its omplemet, B, via a etaglemet surfae Σ. 109 For a give pure quatum

16 state ψ, the wave futio a be deomposed as ψ = ab W ab a b, (1 where a labels ormalized basis states of the Hilbert spae H loalized i regio ad b labels ormalized basis states of the Hilbert spae H loalized i regio. We perform a sigular value deompositio (SVD of the matrix W as W ab = U a D d V db ad defie ew bases = U a a ad d = V db b. D d is a diagoal matrix with positive etries, but ot all the diagoal elemets eed be ozero. The umber of ozero elemets is the rak of W, ad the ozero sigular values are deoted as e ξλ/2. ξ λ are termed the etaglemet eergies, ad the whole set of etaglemet eergies is the etaglemet spetrum {ξ λ } λ=1,,rak(w. Zero sigular values orrespod to ifiite etaglemet eergies. Thus, ψ = Rak(W λ=1 e ξ λ/2 λ λ. (2 To ompute the etaglemet etropy, we trae over the states i regio to obtai a redued desity matrix of regio, ρ = Tr H ψ ψ = Rak(W λ=1 e ξ λ λ λ. (3 The etaglemet etropy is defied as the vo Neuma etropy of the redued desity matrix ρ (see Refs. 4 ad 87 for a review, Rak(W S( = Tr H ρ log ρ = e ξ λ log e ξ λ. λ=1 (4 Heuristially, the etaglemet etropy measures how muh the degrees of freedom i the two regios ad B are orrelated. I this paper, we deote the etaglemet etropy of subregio (whose boudary is Σ as either S( or S[Σ], usig either paretheses or square brakets to highlight the sub regio or the etaglemet surfae, respetively. ppedix B: Loal Cotributios to the Etaglemet Etropy I this appedix, we review the geeral properties of the etaglemet etropy. Followig the disussios i Ref. 58, we provide some detailed ad quatitative aalyses o how the o-uiversal ad shape depedet terms a eter ito the ostat part of the EE. The simplest property of the EE is S( = S(, whih says the etropy omputed for regio is equal to the etropy omputed for its omplemet. This is also true for the full etaglemet spetrum, ad follows diretly from Eq. (2. We assume that i a gapped system with fiite orrelatio legth, the EE a be deomposed ito a loal part ad a topologial part, S( = S loal ( + S topo (. (B1 The loal part S loal ( oly depeds o the loal degrees of freedom ear the etaglemet surfae, ad therefore a be writte i the form of a itegral over loal variables. Sie the oly loal futios o Σ are the metri h µν, the extrisi urvature (seod fudametal form K µν, ad the ovariat derivatives of K µν (ovariat derivatives of h µν are zero by defiitio, Refs. 58,88,89 argued that S loal should be expressible i terms of loal geometri quatities of the etaglemet surfae Σ, i.e., S loal ( = d 2 x hf (K µν, ρ K µν,..., h µν, (B2 Σ where F is a loal futio of K µν ad h µν ad their ovariat derivatives. 110 I otrast, the topologial part of the EE, S topo (, is preisely the otributio that aot be writte as a itegral of loal variables ear the etaglemet surfae. (I partiular, the Euler harateristi term does ot otribute to S topo (. S topo ( should be ivariat uder smooth deformatios of the etaglemet surfae, ad should also be ivariat uder smooth deformatios of the Hamiltoia of the system (provided the gap does ot lose. Therefore, remiiset of two-dimesioal systems, S topo ( is expeted to be the ostat part of the EE. However, i three spatial dimesios, there are subtleties as we will explore below. Before movig o, it is importat for us to first speify for whih systems the EE separates ito a loal ad a topologial part. Systems suh as the tori ode ad its geeralizatios (e.g. Dijkgraaf Witte models, as well as the Walker-Wag models 66 ad their geeralizatios (e.g., the geeralized Walker-Wag models whih we study i Se. III satisfy this deompositio. There are some systems for whih this deompositio is obviously ot valid. For istae, the systems ostruted by layer stakig of two-dimesioal systems do ot satisfy Eq. (B1. The ostat part of etropy depeds o the thikess L z of the layered diretio, i.e., γ 2D L z, where γ 2D is the topologial etropy of a two-dimesioal layer. other lass of systems beyod our disussio are frato models 90, whose etaglemet etropy does ot satisfy Eq. (B1. part from the area law term ad the ostat term, the etaglemet etropies of these model geerially otai a term liearly proportioal to the size of the subregio 91,92. Sie the deompositio Eq. (B1 does ot lead to a liear subleadig term, its presee i the layered models ad the frato models suggest the deompositio Eq. (B1 does ot hold. Sie the defiitio of the EE ditates that S( = S(, this should also be true of the loal part

17 of the EE. To ompute S(, oe a expad F (K µν, ρ K µν,..., h µν as F (K µν, ρ K µν,..., h µν = F 0 + F 1 K µ µ + F 2 [K µν K µν (K µ µ 2 ] + F 2(K µ µ 2 + F 3 µ ν K µν +..., (B3 where µ is the ovariat derivative idued from h µν, ad the idies are raised ad lowered via h µν ad its iverse h µν. ll idies are otrated so that the formula Eq. (B3 is idepedet of the hoie of the oordiates. Demadig that S( = S( ostrais the form of the futio F. To see this, we may simply trasform x 1 x 1 ad x 2 x 2, uder whih K µν K µν ad h µν h µν. 111 The S( = S( implies F (K µν, ρ K µν,..., h µν = F ( K µν, ρ K µν,..., h µν. (B4 fter itegratio, keepig oly those terms eve uder refletio, we fid that the loal part of the EE has the form S loal ( = F 0 Σ F 2 4πχ+4F 2 d 2 x hh 2 +..., (B5 where Σ is the area of the etaglemet surfae. The part proportioal to F 2 gives the Euler harateristi χ(σ of the surfae Σ, defied by Σ d2 x h[k µν K µν (K µ µ 2 ] = 4πχ(Σ. This term is ivariat uder ay smooth deformatio of the etaglemet surfae beause the Euler harateristi is a topologial ivariat of Σ. The part proportioal to F 2 gives the itegral of the square of the mea urvature H = (k 1 + k 2 /2 (sie 2H = K µ, µ where k 1, k 2 are the two priipal urvatures of Σ, i.e., the eigevalues of K µν. This term, though idepedet of the size of Σ, depeds o its shape. This shows that the loal part of the EE has ostat terms, whih otrasts with the familiar ase i (2+1D. Therefore, omputig the EE ad extratig the ostat part is ot a promisig way to extrat topologial iformatio about the uderlyig theory. 112 The above aalysis shows that for a geeri gapped system (whih is ot at a RG fixed poit, the struture of the etaglemet etropy is S( = F 0 Σ + S topo ( 4πF 2 χ(σ +4F 2 d 2 x hh 2 + O(1/ Σ. Σ Σ (B6 I the mai text, we deote the ostat part of the EE as S ( = S topo ( 4πF 2 χ(σ + 4F 2 Σ d2 x hh 2. The above aalysis gives all the possible terms that a exist, but does ot require that they are o-vaishig for a give theory. I Ref. 93, the authors omputed the etaglemet etropy for massive bosos ad massive fermios i (3+1D aross S 2. Their results show a ostat term i the etaglemet etropy. For a massive salar with mass m ad urvature ouplig term 1 2 ξrφ2, S ( = (ξ 1 6 log(mδ, where δ is the ut off. For a massive Dira fermio with mass m, S ( = 1 18 log(mδ. Obviously, these etropies are ot topologial (they deped o the utoff ad o mass parameters, whih shows that o-uiversal otributios to the loal term i fat do exist. ppedix C: Derivatio of the Redutio Formula B (a (b Figure C.1: Etaglemet surfaes used i the appliatio of strog sub-additivity to derive the reurree relatio Eq. (C7. I (a, is a geeral 3-maifold (as a example, we draw with 1 geus 3 surfae ad 2 geus 0 surfaes, B is 3-ball ad C is a solid torus. I (b, is a geeral 3- maifold (as a example, we draw with 1 geus 3 surfae ad 2 geus 0 surfaes, B is a solid torus, ad C is a 3-ball, whih is loated exatly at the hole of B. I this appedix we preset the omplete derivatio of the etropy redutio formula Eq. (10. We will use the SS iequality i two steps. First, i Subsetio C 1 we derive ad solve a reurree relatio for the depedee of TQFT o the geus of the etaglemet ut. Seod, i Subsetio C 2 we derive a additioal reurree relatio for the depedee of TQFT o the umber of disoeted ompoets of the etaglemet surfae. We solve this reurree relatio to obtai our mai result Eq. (10. Our derivatio expads upo the disussio i Ref. 58 i that we obtai expliit formulas for the etropy of arbitrary multiply-oeted etaglemet surfaes. B C C

18 1. Reurree for Geus I order to fid the depedee of the TEE o the data { g }, we eed to osider the ofiguratio of etaglemet surfaes as show i Fig. C.1(a: We start with a geeral oeted 3-maifold with boudary speified by [(0, 0,..., (g, g ]. The 3-maifold is ut ito three regios, B ad C. B is a 3-ball, C is a solid torus ad oupies the remaider of the maifold. is oeted to B ad disoeted from C. Suppose oets with B via a disk (show as a shaded regio whih belogs to a geus (g 1 113 boudary of ad also belogs to the geus 0 boudary of B. The the boudary of regio is speified by [(0, 0,..., (g 1, g 1 + 1, (g, g 1], where we adopt the labelig sheme defied i Se. II 3. We list the ostat part of the EE of all regios by their topologies as follows: TQFT ( = TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1], TQFT (B = TQFT [(0, 1], TQFT (C = TQFT [(0, 0, (1, 1], TQFT (B = TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1], TQFT (BC = TQFT [(0, 0, (1, 1], S TQFT (BC = S TQFT [(0, 0,..., (g 1, g 1, (g, g ]. (C1 The the SS iequality for regios, B, ad C i Eq. (4 reads S TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1] S TQFT [(0, 0,..., (g, g ] + TQFT [(0, 1] TQFT [(0, 0, (1, 1]. We ould have take ad B to be oeted via a disk whih belogs to a geus i (i g 1 boudary of ad also belogs to the geus 0 boudary of B. Followig a idetial proedure, we olude: TQFT [(0, 0,..., (i, i + 1, (i + 1, i+1 1,..., (g, g ] + TQFT [(0, 0, (1, 1] TQFT [(0, 0,..., (i, i, (i + 1, i+1,..., (g, g ] + TQFT [(0, 1]. For simpliity, we will oly eed to adopt the hoie where i = g 1. We proeed to osider aother ofiguratio illustrated i Fig. C.1(b: We start with a geeral 3-maifold with boudary speified by [(0, 0,..., (g, g ]. The 3-maifold is ut ito two regios, ad B. B is a solid torus, ad is the rest of the maifold. We assume oets with B via a disk (show as a shaded regio i the geus (g 1 boudary of ad the geus 1 boudary of B. Hee the boudary of is labeled by [(0, 0,..., (g 1, g 1 + 1, (g, g 1]. I additio, we deote the 3-ball loated i the hole of B as C. We list the ostat part of the EE of all regios as follows: (C2 (C3 TQFT ( = TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1], TQFT (B = TQFT [(0, 0, (1, 1], TQFT (C = TQFT [(0, 1], S TQFT ( B =S TQFT [(0, 0,..., (g, g ], TQFT (B C = TQFT [(0, 1], TQFT ( B C = TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1]. The SS for, B ad C i Fig. C.1(b reads i this ase: S TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1] S TQFT [(0, 0,..., (g, g ] + TQFT [(0, 1] TQFT [(0, 0, (1, 1]. Combiig iequalities Eq. (C2 ad Eq. (C5, we fid the followig equality TQFT [(0, 0,..., (g 1, g 1 + 1, (g, g 1] = TQFT [(0, 0,..., (g, g ] + TQFT [(0, 1] TQFT [(0, 0, (1, 1]. (C4 (C5 (C6

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 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 2. 115 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

21 mai text, suh as Fig. D.1. The by topologial ivariae, Eq. (18 vaishes for geeral ofiguratios. B C h1 h2 r2 Figure D.1: KPLW presriptio of regularized etaglemet surfae T 2. For the ofiguratio i Fig. D.1, we a ompute the mea urvature straightforwardly. The mea urvature is H = (k 1 + k 2 /2, where k 1 ad k 2 are the two priipal urvatures at eah poit of the etaglemet surfae. We distiguish three types of poits o the ylider i Fig. D.1. Poits o the top/bottom of a ylider: the surfae is loally flat, k 1 = k 2 = 0. Hee, H = (k 1 + k 2 /2 = 0. Poits o the side of a ylider: k 1 = ±1/r, k 2 = 0, where r is the radius of the ylider, ad the ± sig depeds o whether it is ier or outer side surfae. Hee, H = (k 1 + k 2 /2 = ±1/2r. I the followig, we will pik the + sig. Poits o the hige of a ylider: Oe of the higes of the regular yliders i Fig. D.1 is show as the thik gree loop. O every poit of the hige, the Gauss urvature is the same. To fid it, we apply the Gauss-Boet theorem to a ylider. Beause the Gauss urvature o the side ad top/bottom of the ylider vaishes, itegratio over the etire surfae of the ylider is redued to the itegratio over the hige. Hee the Gauss-Boet theorem ditates 2 hige r3 r1 1 r 3 kdσ = 2πχ[C] = 4π, (D2 where C is the full ylider, r 3 is the radius of the ylider. 1/r 3 is the priiple urvature alog the hige ad k is the priipal urvature alog the diretio perpediular to the hige. I order to perform the twodimesioal surfae itegral, we eed to regularize the oe-dimesioal hige by smoothig it ito a ar of ifiitesimal radius, as show i Fig. D.2. ssumig the legth of the ar is l 0, Eq. (D2 implies l 0 kdl = 1, whih 0 redues to k = 1/l 0. The priipal urvature for a ideal hige (whih orrespods to l 0 0 is ifiite, ad we regularize it with the small parameter l 0 to hadle the omputatio. To ompute the itegral of the mea urvature squared over various surfaes i Fig. D.1, we first itrodue some otatio. Let r 1 be the ier radius of regio B/C, r 2 be the outer radius of regio B/C, r 3 be the outer radius of regio, h 1 be the height of regio B, ad h 2 be the height of regio C. We adopt the same fiite regularizatio for every hige, although this is ot essetial. For regio, the itegratio H2 splits ito three parts: the top/bottom, the side ad the higes. Sie the top/bottom surfae are flat, they do ot otribute to the mea urvature itegral. The mea urvature of the outer side surfae is 1/2r 3, ad that of the ier side surfae is 1/2r 2. The itegratio of the mea urvature over the outer ad ier side of is l 0 ˆr ˆ r 3 1 3 5 4 8 Figure D.2: Left: Regularizatio of a retagular hige with small ars. Right: Oe hoie of regularizatio of eah hige i Fig. D.1. The umbers label various higes. 2 r 2 6 7 B C r 1 9 10 11 12 ( 2 1 2πr 3 (h 1 + h 2 + 2πr 2 (h 1 + h 2 2r 3 ( 1 2r 2 2 = π(h 1 + h 2 2r 3 + π(h 1 + h 2 2r 2. (D3 The mea urvature of the outer hige is (1/r 3 + 1/l 0 /2, while aordig to our hoie of regularizatio i Fig. D.2,

the mea urvature of the ier hige is (1/l 0 1/r 2 /2 beause the priiple urvature alog the ˆθ diretio (the meaig of ˆθ ad ˆr are speified i Fig. D.2 is 1/r 2 ad the priiple urvature alog the ˆr diretio is 1/l 0 (beause we evaluate the urvature from the iside. The itegratio of the mea urvature over the higes is ( 1 2 2πr 3 l 0 + 1 2 ( 1 + 2 2πr 2 l 0 + 1 2, (D4 2r 3 2l 0 2r 2 2l 0 where the fator of 2 i the frot omes from equal otributio of the higes from the top ad bottom respetively. Colletig the above results, we have H 2 = π(h 1 + h 2 + π(h 1 + h 2 + π(r 3 + l 0 2 + π(r 2 l 0 2. (D5 2r 3 2r 2 r 3 l 0 r 2 l 0 For oveiee, we list the mea urvature of eah hige i the followig table. Hige Mea urvature 1 1/2r 3 + 1/2l 0 2 1/2r 3 + 1/2l 0 3 1/2r 2 + 1/2l 0 4 1/2r 2 + 1/2l 0 5 1/2r 2 + 1/2l 0 6 1/2r 2 + 1/2l 0 7 1/2r 2 + 1/2l 0 8 1/2r 2 + 1/2l 0 9 1/2r 1 + 1/2l 0 10 1/2r 1 + 1/2l 0 11 1/2r 1 + 1/2l 0 12 1/2r 1 + 1/2l 0 where the labels of higes are show i Fig. D.2. For regio B, the side surfae otributio is 22 The hige otributio is 2πr 2 h 1 ( 1 2r 2 2 + 2πr 1 h 1 ( 1 2r 1 2 = πh 1 2r 2 + πh 1 2r 1 (D6 ( 1 2 2πr 2 l 0 + 1 2 ( 1 + 2 2πr 1 l 0 + 1 2 = π(r 2 + l 0 2 + π(r 1 l 0 2 2r 2 2l 0 2r 1 2l 0 r 2 l 0 r 1 l 0 Hee the total otributio from regio B is H 2 = πh 1 + πh 1 + π(r 2 + l 0 2 + π(r 1 l 0 2 2r 2 2r 1 r 2 l 0 r 1 l 0 B For regio C, the side surfae otributio is (D7 (D8 The hige otributio is 2πr 2 h 2 ( 1 2r 2 2 + 2πr 1 h 2 ( 1 2r 1 2 = πh 2 2r 2 + πh 2 2r 1 (D9 ( 1 2 2πr 2 l 0 + 1 2 ( 1 + 2 2πr 1 l 0 + 1 2 = π(r 2 + l 0 2 + π(r 1 l 0 2 2r 2 2l 0 2r 1 2l 0 r 2 l 0 r 1 l 0 Hee the total otributio from regio C is H 2 = πh 2 + πh 2 + π(r 2 + l 0 2 + π(r 1 l 0 2 2r 2 2r 1 r 2 l 0 r 1 l 0 C For regio B, the side surfae otributio is ( 2 ( 2 ( 2 1 1 1 2πr 3 (h 1 + h 2 + 2πr 1 h 1 + 2πr 2 h 2 = π(h 1 + h 2 + πh 1 + πh 2 2r 3 2r 1 2r 2 2r 3 2r 1 2r 2 (D10 (D11 (D12

The hige otributio is ( 1 2 2πr 3 l 0 + 1 2 ( + 2 2πr 1 l 0 1 + 1 2 ( + 2πr 2 l 0 1 1 2 ( + 2πr 2 l 0 1 + 1 2 (D13 2r 3 2l 0 2r 1 2l 0 2r 2 2l 0 2r 2 2l 0 Notie that the third term orrespods to the opposite of hige 7 (whih is ot hige 6. Hee the total otributio from regio B is B H 2 = π(h 1 + h 2 2r 3 + πh 1 2r 1 + πh 2 2r 2 + π(r 3 + l 0 2 r 3 l 0 + π(r 1 l 0 2 r 1 l 0 + π(r 2 + l 0 2 2r 2 l 0 + π(r 2 l 0 2 2r 2 l 0 For regio C, the side surfae otributio is 23 (D14 ( 2 ( 1 1 2πr 3 (h 1 + h 2 + 2πr 2 h 1 2r 3 2r 2 2 + 2πr 1 h 2 ( 1 2r 1 2 = π(h 1 + h 2 2r 3 + πh 1 2r 2 + πh 2 2r 1 (D15 The hige otributio is ( 1 2 2πr 3 l 0 + 1 2 ( + 2 2πr 1 l 0 1 + 1 2 ( + 2πr 2 l 0 1 1 2 ( + 2πr 2 l 0 1 + 1 2 (D16 2r 3 2l 0 2r 1 2l 0 2r 2 2l 0 2r 2 2l 0 Hee the total otributio from regio C is C H 2 = π(h 1 + h 2 2r 3 + πh 2 2r 1 + πh 1 2r 2 + π(r 3 + l 0 2 r 3 l 0 + π(r 1 l 0 2 r 1 l 0 + π(r 2 + l 0 2 2r 2 l 0 + π(r 2 l 0 2 2r 2 l 0 (D17 For regio BC, the side surfae otributio is ( 2 1 2πr 2 (h 1 + h 2 + 2πr 1 (h 1 + h 2 2r 2 ( 1 2r 1 2 = π(h 1 + h 2 2r 2 + π(h 1 + h 2 2r 1 (D18 The hige otributio is ( 1 2 2πr 2 l 0 + 1 2 ( 1 + 2 2πr 1 l 0 + 1 2 = π(r 2 + l 0 2 + π(r 1 l 0 2 2r 2 2l 0 2r 1 2l 0 r 2 l 0 r 1 l 0 Hee the total otributio from regio BC is BC H 2 = π(h 1 + h 2 2r 2 + π(h 1 + h 2 2r 1 + π(r 2 + l 0 2 r 2 l 0 + π(r 1 l 0 2 r 1 l 0 (D19 (D20 Fially, for regio BC, the side surfae otributio is ( 2 1 2πr 3 (h 1 + h 2 + 2πr 1 (h 1 + h 2 2r 3 ( 1 2r 1 2 = π(h 1 + h 2 2r 3 + π(h 1 + h 2 2r 1 (D21 The hige otributio is ( 1 2 2πr 3 l 0 + 1 2 ( 1 + 2 2πr 1 l 0 + 1 2 = π(r 3 + l 0 2 + π(r 1 l 0 2 2r 3 2l 0 2r 1 2l 0 r 3 l 0 r 1 l 0 Hee the total otributio from regio BC is BC H 2 = π(h 1 + h 2 2r 3 + π(h 1 + h 2 2r 1 + π(r 3 + l 0 2 r 3 l 0 + π(r 1 l 0 2 r 1 l 0 (D22 (D23

I summary, we obtai the otributio of mea urvature squared of seve regios as follows. H 2 = π(h 1 + h 2 + π(h 1 + h 2 + π(r 3 + l 0 2 + π(r 2 l 0 2., 2r 3 2r 2 r 3 l 0 r 2 l 0 H 2 = πh 1 + πh 1 + π(r 2 + l 0 2 + π(r 1 l 0 2, B 2r 2 2r 1 r 2 l 0 r 1 l 0 H 2 = πh 2 + πh 2 + π(r 2 + l 0 2 + π(r 1 l 0 2, C 2r 2 2r 1 r 2 l 0 r 1 l 0 H 2 = π(h 1 + h 2 + πh 1 + πh 2 + π(r 3 + l 0 2 + π(r 1 l 0 2 + π(r 2 + l 0 2 + π(r 2 l 0 2, B 2r 3 2r 1 2r 2 r 3 l 0 r 1 l 0 2r 2 l 0 2r 2 l 0 H 2 = π(h 1 + h 2 + πh 2 + πh 1 + π(r 3 + l 0 2 + π(r 1 l 0 2 + π(r 2 + l 0 2 + π(r 2 l 0 2, C 2r 3 2r 1 2r 2 r 3 l 0 r 1 l 0 2r 2 l 0 2r 2 l 0 H 2 = π(h 1 + h 2 + π(h 1 + h 2 + π(r 2 + l 0 2 + π(r 1 l 0 2, BC 2r 2 2r 1 r 2 l 0 r 1 l 0 H 2 = π(h 1 + h 2 + π(h 1 + h 2 + π(r 3 + l 0 2 + π(r 1 l 0 2. 2r 3 2r 1 r 3 l 0 r 1 l 0 BC It is straightforward to hek that the ombiatio Eq. (18 vaishes. Hee the relatio Eq. (17 i the mai text holds. 24 (D24 ppedix E: Review of Lattie TQFT I this setio, we briefly review the lattie formulatio of TQFTs. We begi with a triagulatio of spaetime. The letters i, j, k et. label the verties of a spaetime lattie. Combiatios of verties deote the simpliies of the lattie. For istae, (ij is the 1-simplex (bod whose eds are verties i ad j. (ijk is a 2-simplex (triagle whose verties are i, j ad k. Gauge fields live o these simpliies. I our paper, 1-form gauge fields live o 1-simpliies; 2-form gauge fields B live o 2-simpliies; et. I the laguage of disrete theories, (ij, B(ijk are the 1-ohai ad 2-ohai assoiated with the idiated 1-simplex ad 2-simplex, respetively. Exterior derivatives are defied by: d(ijk =(jk (ik + (ij, db(ijkl =B(jkl B(ikl + B(ijl B(ijk. (E1 Note that the verties are ordered suh that i < j < k < l. We further illustrate the values that the ohais (ij ad B(ijk a take usig aoial quatizatio. Let us first osider the GWW model desribed by Eq. (19 o a otiuous spaetime with U(1 gauge group. It is kow that there are surfae operators exp(is Σ B, s = 0, 1,, 1 67,68, ad exp(i B = 1 is a trivial operator for a arbitrary losed surfae Σ. Hee Σ B = 2πq, Σ where q Z ad Σ is ay losed surfae. The fat that exp(i B is a trivial operator a be verified via Σ aoial quatizatio. To perform aoial quatizatio, we first use the gauge trasformatio Eq. (20 to fix the gauge t = 0, B tx = 0, B ty = 0, B tz = 0. The ommutatio relatios from aoial quatizatio are [ x (t, x, y, z,b yz (t, x, y, z ] = i 2π δ(x x δ(y y δ(z z. (E2 ad similarly for other ompoets. Usig Eq. (E2, we fid that exp(i B ommutes with all other gauge ivariat operators. Speifially, we ompute the ommu- Σ tatio relatio betwee the surfae operator exp(i Σ B ad the lie operator exp(il γ +ilp Σ 2 B. Here Σ is a losed surfae i a spatial slie, ad Σ 2 is a ope surfae with boudary γ. Both Σ 2 ad γ are livig i the spatial slie. We fid e i Σ B e il γ +ilp Σ 2 B = e i 2π lnσ,γ e il γ +ilp Σ 2 B e i Σ B = e il γ +ilp Σ 2 B e i Σ B, (E3 where N Σ,γ is the itersetio umber of the surfae Σ ad the loop γ. Sie the phase fator omig from the ommutatio relatio is always 1, exp(i B ommutes with all lie operators. Sie it also ommutes Σ with exp(il B for ay l ad Σ, we olude that Σ exp(i B ommutes with all the gauge ivariat operators. Therefore, it must be a ostat operator, Σ e i Σ B = e iθ where θ is a ostat umber. We further show that e i Σ B = 1. To show this, we at e i Σ B o a state 0 where B = 0 everywhere (more oretely, if the spaetime is disrete, B = 0 o every 2-simplex. Sie e i Σ B measures the value of B-field of the state, ad B-field is zero everywhere, e iθ 0 = e i Σ B 0 = 0 (E4

25 Hee the ostat umber e iθ = 1 everywhere. This proves that e i Σ B = 1. Similarly, exp(i γ + ip Σ 2 B ommutes with all other operators as well. ad e i γ +ip Σ 2 B e il Σ B = e i 2π lnσ,γ e il Σ B e i γ +ip Σ 2 B = e il Σ B e i γ +ip Σ 2 B. e i γ +ip Σ B 2 e il γ +ilp Σ B 2 (E5 = e i 2π lp(n γ,σ N γ 2,Σ 2 il e γ +ilp Σ B i 2 e γ +ip B Σ 2 = e il γ +ilp Σ 2 B e i γ +ip Σ 2 B. e i γ +ip Σ 2 B (E6 Therefore ommutes with all gauge ivariat operators as well, whih implies e i γ +ip Σ B 2 = e iη where e iη is a ostat. Usig the same aalysis for the operator e i Σ B, we fid e i γ +ip Σ B 2 = 1. O a triagulated lattie, sie Σ is ay two dimesioal surfae, exp(i B = 1 implies that Σ exp(i B = 1 for ay 3-simplex (ijkl. Usig the (ijkl Stokes formula, (ijkl B = db = (db(ijkl = (ijkl B(ijk B(ijl + B(ikl B(jkl where we used the fat that itegratig db over the volume of 3-simplex (ijkl is just evaluatig the db o (ijkl itself. Hee exp(i B = 1 implies that B(ijk B(ijl + (ijkl B(ikl B(jkl 2π Z for ay 3-simplex (ijkl. Sie the hoie of (ijkl is arbitrary, we olude that o eah 2-simplex (ijk, B(ijk takes values i 2π Z. Similarly, o eah 1-simplex (ij, (ij takes values i 2π Z for ay i, j. Next, we ommet o the delta futios obtaied from itegratig out the fields as i Eq. (24. For simpliity, we work with a level = 2 BF/GWW theory. O eah 4-simplex with verties labeled by (i, j, k, l, s, the atio is 2 2π (db(ijkls = 2 2π (ijdb(jkls. (E7 Itegratig over meas summig over all ofiguratios of (ij = 0, π. Hee the path itegral is 1 [ exp i 2 ] 2 2π (ijdb(jkls (ij=0,π = 1 { 1 + exp [ idb(jkls ]} δ [ db(jkls ]. (E8 2 This explais the meaig of the delta futio i the disrete theory, ad we refer to the B field as flat if the above delta futio ostrait is satisfied, i.e. if db(jkls = 0 mod 2π. lthough we write TQFT atios as itegrals i the otiuum i the mai text, they a atually be traslated ito lattie atios usig the ovetios we have itrodued i this appedix. The wave futios defied via the path itegral i Eqs. (23 ad (27 are the wave futios o the lattie. ppedix F: Surfaes i the dual lattie I this appedix, we argue that the simplies o whih B = π i the dual lattie form otiuous surfaes. Cotiuous meas that oeted simplies i the dual lattie joi via edges, rather tha via verties. Speifially, 1. I three-dimesioal spae, if a real spae 2-ohai B(ijk satisfies the flatess oditio db(ijkl = B(jkl B(ikl + B(ijl B(ijk = 0 mod 2π the its dual B = π o a losed loop i the dual lattie. 2. I (3 + 1-dimesioal spaetime, if a real spae 2-ohai B(ijk satisfies the flatess oditio db(ijkl = B(jkl B(ikl + B(ijl B(ijk = 0 mod 2π the its dual B = π o a otiuous ad losed surfae i the dual lattie. The first statemet is prove i the mai text. I the followig, we will preset a more algebrai proof of the first statemet, whih is easier to geeralize to (3 + 1- dimesios, allowig for a proof of the seod statemet. p j b i s a k e Figure F.1: Dual lattie of a tetrahedro (ijkl. (ijkp, (ijlq, (iklr, (jkls are four adjaet tetrahedra to (ijkl, whih are dual to (b, (, (d, (e, (a respetively. The red dots are the itersetio betwee 2-simplies i the real lattie ad the 1-simplies i the dual lattie. For example, the red dot o (ab is the itersetio poit of (ab ad (ijk. We first redraw the simplex i Fig. 3 with some additioal details, as show i Fig. F.1. To ostrut the q d l r

26 duals of simplies i three-dimesioal spae, we begi by osiderig the tetrahedro (ijkl, i additio to its eighbors (ijkp, (ijlq, (iklr, ad (jkls. 3-simplies i the real lattie are dual to poits i the dual lattie: for example (ijkl is dual to the poit (a, ad similarly (ijkp is dual to (b, (ijlq is dual to (, (iklr is dual to (d, ad (jkls is dual to (e. 2-simplies i the real lattie are dual to 1-simplies (bods. For example, (ijk is the itersetio of (ijkl ad (ijkp, i.e., (ijk = (ijkl (ijkp. Therefore, the dual of (ijk is the bod (ab, joiig the dual of (ijkl ad (ijkp. Similarly, we are able to idetify the duals of all other simplies. We list the result i the followig table: Real Dual (ijkl (a (ijkp (b (ijlq ( (iklr (d (jkls (e Real Dual (ijk (ab (ijl (a (ikl (ad (jkl (ae The flatess oditio implies that there are eve umber of 2-simplies amog the four faes of the tetrahedro (ijkl o whih B = π. It follows that there are a eve umber B = π bods amog the four dual lattie bods (ab, (a, (ad, (ae. Thus these form losed loops i the dual lattie. This proves the first statemet. We proeed to prove the seod statemet. I (3 + 1 dimesios, spaetime is triagulated ito 4-simplies. Let us osider a 4-simplex labeled by the five verties (ijklm where m is i the extra dimesio ompared with 3D ase show i Fig. F.1. To fid the dual of 2-simplies, we will begi as above by osiderig the 4-simplies adjaet to (ijklm whih share oe 3- simplex with (ijklm. Itroduig the additioal verties p, q, r, s, ad t 116, these 4-simplies are: (ijkmp, (ijlmq, (iklmr, (jklms, ad (ijklt. Dual simplies i (3 + 1 dimesioal spaetime are determied as follows: 4-simplies i the real lattie are dual to poits i the dual lattie; (ijklm is dual to a poit (a, (ijkmp is dual to (b, (ijlmq is dual to (, (iklmr is dual to (d, (jklms is dual to (e, ad (ijklt is dual to (f 117. 3- simplies i the real lattie are dual to bods i the dual lattie. For istae, sie (ijkm is the itersetio of (ijklm ad (ijkmp, i.e., (ijkm = (ijklm (ijkmp, the dual of (ijkl is the bod (ab, joiig the dual of (ijklm ad (ijkmp. Similarly, (ijlm is dual to (a, (iklm is dual to (ad, (jklm is dual to (ae, ad (ijkl is dual to (af. We further proeed to osider the dual of 2-simplies, applyig the same method. For istae, sie the 2-simplex (ijk is the ommo simplex of (ijkm ad (ijkl, i.e., (ijk = (ijkl (ijkm, the dual of (ijk is the surfae (abf joiig the dual of (ijkl ad (ijkm. Similarly, we a idetify the duals of the remaiig 2-simplies. We list all the results i the followig table: Real Dual (ijklm (a (ijkmp (b (ijlmq ( (iklmr (d (jklms (e (ijklt (f Real Dual (ijkm (ab (ijlm (a (iklm (ad (jklm (ae (ijkl (af Real Dual (ijk (abf (ijl (af (ijm (ab (ikl (adf (ikm (abd (ilm (ad (jkl (aef (jkm (abe (jlm (ae (klm (ade The four surfaes (abf, (af, (adf, (aef are dual to the four faes (ijk, (ijl, (ikl, (jkl of the tetrahedro (ijkl. ll of these dual surfaes share a ommo lik (af. The flatess oditio db(ijkl = B(jkl B(ikl + B(ijl B(ijk = 0 mod 2π implies that a eve umber of faes of the tetrahedro (ijkl are oupied. Thus, there are a eve umber of surfaes amog (abf, (af, (adf, (aef oupied i the dual lattie. Sie all these oupied surfaes i the dual lattie share a ommo edge (af, it follows from our defiitio of otiuity (at the begiig of this appedix that surfaes i the dual lattie are otiuous. Furthermore, the otiuous surfaes formed by the oupied simplies i the dual lattie are losed, beause for ay bod i the dual lattie, for example (af, there exist eve (amog four umber of oupied dual-lattie 2-simplies adjaet to it. While for a ope dual-lattie surfae, there exist at least oe dual-lattie bod suh that there are oly odd umber of the adjaet dual-lattie 2-simplies oupied, whih violate the flatess oditio for the B- ohai. Hee the dual-lattie surfae is losed. This proves the seod statemet. For ompleteess, we ommet o how two loops a iterset i the dual spae lattie, ad how two surfaes a iterset i the dual spaetime lattie. We first prove by ostrutio that two loops i the dual spatial lattie a iterset at a vertex: suppose oe dual lattie loop iludes the oupied bods (ab, (a, ad the other dual lattie loop iludes the oupied bods (ad, (ae. Hee these two loops iterset at the vertex (a. We ow argue that if two surfaes i the dual spaetime lattie otai the same poit, the they must share a bod. Let us assume two surfaes iterset (at least at (a. Sie all the 2- simplies i the dual lattie iludig the vertex (a are (ab, (abd, (ad, (abe, (ae, (ade, (abf, (af, (adf ad (aef, by eumeratig all possibilities, we fid the two surfaes must share at least oe bod. Without loss of geerality, suppose oe surfae iludes the 2-simplies (ab ad (abd (otie that (ab ad (abd joi via the bod (ab ad therefore form a otiuous surfae i the dual lattie. The surfae thus iludes the three bods (ab, (a, ad (ad emaatig from (a. y other surfae that otais (a, would ilude, just like this surfae, three of bods emaatig from (a. Thus, as (a is the oly shared part of five bods

27 (ab, (a, (ad, (ae, ad (af, two surfaes that ilude (a have to share at least oe of these bods, as they oupy three bods eah. I summary, two loops a iterset at verties i the dual spae lattie, ad two surfaes a iterset at bods (but ot verties i the dual spaetime lattie. these otios with a example i lower dimesios. Returig to the itegral i the wavefutio Eq. (29, we thus have l 1 ppedix G: Mutual ad Self-Likig Numbers I this setio, we provide all details eeded to evaluate the itegral Eq. (29. s a simple ase, we assume a ofiguratio where B = π oly at two surfaes S 1, S 2 i the dual lattie of M 4, with their boudaries give by the loops l 1 = S 1, l 2 = S 2 o the dual lattie of M 4. We a write this suitly as B = π 4 Σ(S 1 + π 4 Σ(S 2, (G1 where 4 is the disretized versio of Hodge star i four spaetime dimesios; its meaig is explaied pitorially i Fig. G.1. Let us ommet o Eq. (G1 i detail. O M 4, B is a 2-ohai, whih a be 0 or π; while o the dual lattie of M 4, the π-valued 1-ohais Σ(l i (whih are the dual of real-spae 2-ohais form loops l i, i = 1, 2. Moreover, o the spaetime M 4, B is still a 2-ohai valued i 0 or π; while o the dual lattie of M 4, the π-valued 2-ohais Σ(S i (whih are the dual of the real spaetime 2-ohais form surfaes S i, i = 1, 2 whose boudaries are l i, i = 1, 2. Notie that the losed dual-lattie surfaes whih do ot iterset with the spatial slie do ot otribute to the wavefutio. Further 4 Σ(S i is a 2-ohai o the origial lattie (dual to S i, whih is 1 o the dual of S i, ad 0 elsewhere. Hee, the role of the Hodge star is to trasform the ohai defied o the dual lattie to the ohai defied o the real lattie. I Fig. G.1 we illustrate the geometri meaig of Figure G.1: We illustrate the geometri meaig of the Hodge dual i a two-dimesioal spae example. Suppose is a 1- ohai, whih equals π o 1-simplies i the dual lattie ad 0 elsewhere. = π 2 Σ(l 1 + π 2 Σ(l 2, where l 1 ad l 2 are loops i the dual lattie draw i dashed lies. Σ(l 1 ad Σ(l 2 are 1-ohais livig o the 1-simplies i the dual lattie. 2 is a lattie versio of Hodge star, whih trasforms the 1-ohai livig o the dual lattie (dashed lies to a 1-ohai livig o the lattie (gree ad purple bold lies. Correspodigly, = π 2 Σ(l 1 + π 2 Σ(l 2 is a 1-ohai livig o the gree ad purple bold lies. We use the dual lattie ofiguratio S i, l i to label the B, -ohais beause the dual lattie ofiguratios are easier to visualize. The iterpretatio of the 2-ohai B a be straightforwardly geeralized to three spatial dimesios. l 2 ( ( B B =π 2 4 Σ(S 1 + 4 Σ(S 2 4 Σ(S 1 + 4 Σ(S 2 M 4 M 4 =2π 2 4 Σ(S 1 4 Σ(S 2 + π 2 4 Σ(S 1 4 Σ(S 1 + π 2 M 4 M 4 =2π 2 lik(l 1, l 2 + π 2 lik(l 1, l 1 + π 2 lik(l 2, l 2, M 4 4 Σ(S 2 4 Σ(S 2 (G2 where lik(l 1, l 2 is the likig umber betwee two loops l 1 ad l 2. This leads to Eq. (30 i the mai text. We will derive the last equality of Eq. (G2 i ppedix G 1, ad provide a detailed disussio of the selflikig umbers of oe sigle loop i ppedix G 2. 1. Itersetio ad Likig We prove a statemet relatig the itersetio form i the bulk ad the likig umber o the boudary, whih i tur explais the last equality i Eq. (G2. s explaied below Eq. (G1, 4 Σ(S i is a 2-ohai i the real spaetime, whih equals 1 if it is evaluated o ay triagulatio of S i (i the dual spaetime lattie ad 0 if