P u b l i s h e d f o r S IS S A b y S p r i n g e r R eceiv ed : June 17, 2019
A c c e p te d : July 9, 2019 P u b lish e d : July 29, 2019
Towards an UV fixed point in C D T gravity
J . A m b j 0 r n , b,c J . G i z b e r t - S tu d n i c k i, " A. G orlich," J. J u rk ie w ic z " a n d D. N e m e t h "
a The M. Sm oluchow ski In stitu te o f Physics, Jagiellonian U niversity, Lojasiew icza 11, Kraków, P L 30-348, Poland
bThe N iels B o h r In stitu te, Copenhagen U niversity, B legdam svej 17, D K 2100 Copenhagen, D enm ark cIM A A P , Radboud U niversity,
P O B o x 9010, N ijm egen, The N etherlands
E -m a il: ambjorn@nbi.dk, jakub.gizbert-studnicki@uj.edu.pl, andrzej.goerlich@uj.edu.pl, jerzy.jurkiewicz@uj.edu.pl, dnemeth@th.if.uj.edu.pl
A b s t r a c t : C D T is a n a tte m p t to fo rm u la te a n o n -p e rtu rb a tiv e la ttic e th e o ry o f q u a n tu m g ravity. W e d esc rib e th e p h a se d ia g ra m a n d a n a ly se th e p h a se tra n s itio n b etw e en p h ase B a n d p h a se C (w hich is th e an a lo g u e of th e d e S itte r p h a se o b serv ed for th e sp h eric al s p a tia l to p o lo g y ). T h is tra n s itio n is accessible to o rd in a ry M o n te C arlo sim u la tio n s w hen th e to p o lo g y of space is to ro id a l. W e find t h a t th e tra n s itio n is m o st likely first o rd e r, b u t w ith u n u su a l p ro p e rtie s. T h e en d p o in ts of th e tra n s itio n line a re c a n d id a te s for second o rd e r p h a se tra n s itio n p o in ts w h ere a n U V c o n tin u u m lim it m ig h t exist.
K e y w o r d s : L a ttic e M odels o f G rav ity , L a ttic e Q u a n tu m F ie ld T heo ry, M odels of Q u a n tu m G ra v ity
A r X i v e P r i n t : 1906.04557
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
C o n te n ts
1 I n tr o d u c tio n 1
2 T h e p h a s e s tr u c tu r e o f C D T 5
3 O rd er p a r a m e te r s 9
4 C o n c lu s io n a n d d is c u ss io n 13
1 In tr o d u c tio n
Since th e m id d le o f la st c e n tu ry p h y sicists have b een p u rs u in g th e id ea of u nify in g th e four fu n d a m e n ta l in te ra c tio n s, th e stro n g , th e w eak, th e e le c tro m a g n e tic a n d th e g ra v ita tio n a l in te ra c tio n s. T h e fram ew o rk o f Q u a n tu m F ie ld T h e o ry (Q F T ) unified th e first th re e of th e m in th e so-called S ta n d a rd M odel. In c lu d in g g ra v ity re m a in s an u n so lv ed p ro b lem in a Q F T c o n te x t.1 D ifficulties a p p e a r w h en on e trie s to fo rm u la te a q u a n tu m v ersio n of E in s te in 's th e o ry of G e n e ra l R e la tiv ity . T h e naive q u a n tiz a tio n lead s to a p e rtu r b a tiv e ly n o n -re n o rm a liz a b le th e o ry w hich c a n n o t b e sim p ly in clu d ed in th e unified m odel of all in te ra c tio n s. T h e id ea of a s y m p to tic safety in tro d u c e d by W ein b erg [1] is a n a tte m p t to fo rm u la te a n o n -p e rtu rb a tiv e Q F T o f g ravity. I t assum es t h a t th e re n o rm a liz a tio n g ro u p flow in th e b a re co u p lin g c o n s ta n t space leads to a n o n -triv ia l fin ite-d im en sio n al u ltra v io le t fixed p o in t a ro u n d w hich a new p e r tu r b a tiv e e x p a n sio n c a n be c o n s tru c te d w hich leads to a p re d ic tiv e q u a n tu m th e o ry of g rav ity . T h e so-called E x a c t R e n o rm a liz a tio n G ro u p p ro g ra m [2- 6] h as trie d to e s ta b lish th e ex isten c e of such a fixed p o in t w ith a fa ir a m o u n t o f success, b u t relies in th e end, d e sp ite th e n am e, on tr u n c a tio n o f th e re n o rm a liz a tio n g ro u p eq u a tio n s. T h u s it w ould b e re a ssu rin g if o th e r n o n -p e rtu rb a tiv e Q F T ap p ro ach e s cou ld confirm th e e x a c t re n o rm a liz a tio n g ro u p resu lts.
L a ttic e Q F T is such a n o n -p e rtu rb a tiv e fram ew o rk a n d it is well su ite d to d ea l p re cisely w ith th e s itu a tio n w h ere one id entifies fixed p o in ts, since th e s e a re w h ere o ne w a n ts to re ach c o n tin u u m physics by scaling th e la ttic e sp ac in g to zero in a w ay w h ich keeps physics fixed. It h as b ee n very successful p ro v id in g us w ith re su lts for Q C D w h ich are n o t accessible v ia p e r tu r b a tio n th eo ry . T h e re e x ists a n u m b e r of la ttic e Q F T of g ravity. O n e of th e m , th e so-called D y n a m ic a l T ria n g u la tio n (D T ) fo rm alism [7- 12] h as p ro v id ed us w ith a
“p ro o f o f c o n c e p t” , in th e sense t h a t it h as show n us, in th e case o f tw o -d im en sio n al q u a n tu m g ra v ity [13- 16], t h a t th e c o n tin u u m lim it of th e la ttic e th e o ry of g ra v ity co u p led to co n fo rm al field th e o rie s agree w ith th e co rre sp o n d in g c o n tin u u m th e o rie s. O f co u rse th e re a re no p ro p a g a tin g g ra v ita tio n a l degrees o f freedo m in tw o d im en sio n s, b u t th e m a in issue
1 Going beyond conventional Q FT, string theory provides us with a theory unifying the interaction of m atter and gravity. Likewise loop quantum gravity uses concepts beyond conventional QFT.
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
w ith th e la ttic e re g u la riz a tio n is w h e th e r o r n o t d iffeo m o rp h ism in v aria n ce is recovered w h en th e la ttic e sp ac in g goes to zero. T h a t is th e case in th e D T fo rm alism , a n d for th e
scaling d im en sio n s o b ta in e d also in th e co n tin u u m , i.e. scaling d im en sio n s w hich are differ
e n t from th e ones in flat s p a c e tim e (th e so-called K P Z scalin g [17- 19]). T h e D T fo rm alism w as e x te n d e d to h ig h er d im e n sio n a l g ra v ity [20- 27], b u t th e re it w as less successful [2 8 , 29].
I t is n o t ru led o u t t h a t th e th e o ry c a n p ro v id e us w ith a successful v ersio n of q u a n tu m g ravity, b u t if so th e fo rm u la tio n h as to be m o re e la b o ra te th a n th e first m o dels (see [30- 33]
for re cen t a tte m p ts ) . H ow ever, th e re is on e m o d ificatio n o f D T w hich seem s to w ork in th e sense t h a t la ttic e th e o ry m ig h t have a n o n -triv ia l c o n tin u u m lim it, th e so-called C a u sa l D y n a m ic a l T ria n g u la tio n s m od el (C D T ). T h e m od el is m o re c o n s tra in e d th a n th e D T m odels b ec au se one assum es g lo bal h y p erb o lic ity , i.e. th e ex isten c e of a g lo bal tim e fo liatio n .
T h e C D T m o d el o f fo u r-d im e n sio n a l q u a n tu m g ra v ity is realized by co n sid erin g piece
w ise lin e a r sim plicial d is c re tiz a tio n s o f sp ac e-tim e. T h e sim p licial b u ild in g blocks c a n be glu ed to g e th e r, sa tisfy in g th e b asic to p o lo g ic al c o n s tra in ts of g lo bal h y p e rb o lic ity (as m en tio n e d ) a n d a sim plicial m an ifo ld s tru c tu re . T h e q u a n tu m m od el is now defined usin g th e F e y n m a n p a th in te g ra l fo rm alism , su m m in g over all such g eo m etrie s w ith a s u ita b le a c tio n to b e defined below . T h e s p a tia l U niv erse w ith a fixed to p o lo g y evolves in p ro p e r tim e.
G e o m e tric s ta te s a t a fixed valu e of th e (d iscrete) tim e a re tria n g u la te d , u sin g re g u la r th re e -d im e n sio n a l sim plices ( te tra h e d ra ) g lu ed alon g tria n g u la r faces in all p o ssib le w ays, c o n s iste n t w ith topology. T h e co m m o n le n g th of th e edges o f s p a tia l links is a ssu m ed to b e a s . T e tra h e d ra a re th e bases of fo u r-d im e n sio n a l { 4 ,1 } a n d {1 ,4 } sim plices w ith fo u r v ertic es a t tim e t co n n e c te d by tim e links to a v e rte x a t t ± 1. All tim e edges a re assu m ed to h ave a u n iv ersa l le n g th a t . To c o n s tru c t a fo u r-d im e n sio n a l m an ifold o ne need s tw o a d d itio n a l ty p e s o f four-sim plices: {3 ,2 } a n d {2 ,3 } (h a v in g th re e v ertic es a t tim e t an d tw o v ertices a t t ± 1). T h e s tr u c tu r e d e sc rib e d ab o v e p e rm its for every c o n fig u ra tio n th e a n a ly tic c o n tin u a tio n b etw e en im a g in a ry at (L o re n tz ia n sig n a tu re ) a n d re al at (E u c lid e a n sig n a tu re ). E v en a fte r W ick ro ta tio n th e o rie n ta tio n of th e tim e axis is re m e m b ered . T h e sp a tia l a n d tim e links m ay have a d ifferen t le n g th , a n d a re re la te d by a a 2 = a 2. T h e q u a n tu m a m p litu d e b etw e en th e in itia l an d final g eo m etric s ta te s s e p a ra te d by th e in teg er tim e T is a w eig h ted sum over all sim plicial m an ifo ld s c o n n e c tin g th e tw o s ta te s . In th e L o re n tz ia n fo rm u la tio n th e w eight is a ssu m ed to b e given by a d iscre tiz ed versio n of th e H ilb e rt-E in s te in ac tio n .
w h ere [g] d e n o te s a n eq u iv ale n t class o f m e tric s a n d Dm [g] is th e in te g ra tio n m e a su re over n o n eq u iv ale n t classes o f m etrics. A piecew ise lin e a r m anifo ld w h ere we have specified th e le n g th of links defines a g e o m e try w ith o u t th e need to in tro d u c e c o o rd in a te s. In th e C D T a p p ro a c h th e in te g ra tio n over eq u iv ale n t classes of m e tric s is th u s re p la ced by a su m m a tio n over all tria n g u la tio n s T satisfy in g th e c o n s tra in ts . A fte r a W ick ro ta tio n th e a m p litu d e becom es a p a r titio n fu n c tio n
co n fo rm al field th e o rie s living o n th e la ttic e o ne o b ta in s p recisely th e n o n -triv ia l critic a l
( 1 . 1)
Zc d t = ^ e ,
T
( 1 .2)
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
w h ere Sr is a s u ita b le form o f th e E in s te in -H ilb e rt a c tio n o n piecew ise lin e a r geo m etries.
T h e re e x ists such a n ac tio n , w hich even h as a nice g eo m etric in te rp re ta tio n , th e so-called R egge a c tio n Sr for piecew ise lin e a r g eo m etrie s [34]. In o u r case it becom es v ery sim ple b ec au se we h ave o n ly tw o k in d s of fo ur-sim plices w hich we glu e to g e th e r to form o u r piecew ise lin e a r fo u r-m an ifo ld :
Sr = - ( K o + 6A ) ■ No + K4 ■ (N41 + N32) + A ■ N4 1, (1.3) w h ere N 0 is th e n u m b e r o f v ertic es in a tria n g u la tio n T , N 41 a n d N 32 a re th e n u m b e rs of {4 ,1 } p lu s { 1 ,4 } a n d { 3 ,2 } plus { 2 ,3 } sim plices, resp ectiv ely . T h e a c tio n is p a ra m e triz e d by a set of th re e dim en sio n less b a re co u p lin g c o n s ta n ts , K 0, re la te d to th e inverse g ra v ita tio n a l c o n s ta n t, K 4 — th e d im en sio n less cosm ological c o n s ta n t a n d A — a fu n c tio n of th e p a r a m e te r a , th e ra tio of th e s p a tia l a n d tim e edge len g th s (for a d e ta ile d discu ssio n we refer to [35] a n d to th e m o st re cen t review [36] a n d fo r th e o rig in al lite ra tu re to [3 7 , 38]).
T h e a m p litu d e is defined for K 4 > K 4rit a n d th e lim it K 4 ^ K 4rit c o rre sp o n d s to a (d is
c re te ) in fin ite v olum e lim it. In th is lim it, th e p ro p e rtie s of th e m o del d e p e n d on values o f th e tw o re m a in in g co u p lin g c o n s ta n ts . T h e m o d el w as ex ten siv e ly s tu d ie d in th e case, w h ere th e s p a tia l to p o lo g y w as a ssu m ed to b e sp h eric al ( S 3) [39- 45]. T h e m o d el could n o t b e solved a n a ly tic a lly a n d th e in fo rm a tio n a b o u t its p ro p e rtie s w as o b ta in e d u sin g M o n te C arlo sim u la tio n s. I t w as fo u n d t h a t th e m o d el h as a su rp risin g ly rich p h a se s tru c tu re , w ith fo u r d ifferent phases. T h e m o st in te re stin g am o n g th e fo u r p h ases is p h a se C, w h ere th e m o d el d y n a m ic a lly develops a sem iclassical b a c k g ro u n d g e o m e try w hich in som e re sp e c t is like (E u c lid e an ) de S itte r geo m etry , i.e. like th e g e o m e try o f S 4. B o th th e sem iclassical vol
u m e d is trib u tio n a n d flu c tu a tio n s a ro u n d th is d is trib u tio n c a n be in te rp re te d in te rm s o f a m in isu p e rsp a c e m o d el [46- 49]. F or in cre asin g K 0 p h a se C is b o u n d e d by a first-o rd e r p h ase tra n s itio n to p h a se A, w h ere th e tim e c o rre la tio n b etw e en th e co n secu tiv e slices is a b se n t.
F o r sm aller A p h a se C h as a p h a se tra n s itio n to a so-called b ifu rc a tio n p h ase , w h ere one ob serves th e a p p e a ra n c e of local c o n d e n sa tio n s of g e o m e try a ro u n d som e v ertices o f th e tr ia n g u la tio n [50- 53]. T h e p h a se tra n s itio n is in th is case of seco nd o r h ig h er o rd e r. F o r still low er A th e b ifu rc a tio n p h a se is linked w ith th e fo u rth ph ase, th e so-called B p h ase, w h ere o ne observes a sp o n ta n e o u s co m p a c tific a tio n of v olum e in th e tim e d ire c tio n , such t h a t effectively all v olum e con d en ses in on e tim e slice. T h e p h a se tra n s itio n b etw e en th e b ifu rc a tio n p h a se a n d th e B p h a se is also of second o r h ig h er o rd e r [44]. T h e b e h a v io r o f th e m odel n e a r co n tin u o u s p h a se tra n s itio n s is c ru cial if one w a n ts to define a ph ysical larg e-v o lu m e lim it (a carefu l discu ssio n of th is c a n b e fo u n d in [54]). In th is re sp e c t p h a se C s ta n d s o u t, th e re a so n b ein g t h a t o n ly in th is p h a se th e larg e scale s tr u c tu r e o f th e average g e o m e try is “o b se rv e d ” (v ia th e M o n te C arlo sim u la tio n s) to b e fo u r-d im e n sio n a l, iso tro p ic a n d hom ogeneous, a n d one c a n define a n in fra re d sem iclassical lim it w ith a c o rre c t scaling o f th e p h y sical v olum e [4 2 , 46]. V ia p h a se C we th u s w a n t a re n o rm a liz a tio n g ro u p flow in th e b a re co u p lin g c o n s ta n t space to w a rd s an U V fixed p o in t (th e a s y m p to tic safety fixed p o in t), w hile keeping p hysical ob serv ab les fixed. T h e n a tu r a l e n d p o in t o f such a flow w ould b e a p o in t in th e p h a se d ia g ra m w h ere several p h ases m eet. In th e e a rly stu d ie s it w as s p e c u la te d t h a t th e re could be a q u a d ru p le p o in t, w h ere all fo u r p h ase s m eet. U n fo rtu n a te ly th e n u m eric al a lg o rith m used w as n o t efficient in th is m o st ph ysically in te re stin g
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
F i g u r e 1. T h e phase stru c tu re of C D T for a fixed n u m b er of tim e slices T = 4 an d average lattice volum e N4 1 = 160k. B lue color rep resen ts th e b ifurfaction phase, black color th e crum pled phase, green color th e C phase an d orange color th e A phase.
ra n g e in th e co u p lin g c o n s ta n t space. As a con seq u en ce it w as n o t p o ssib le to an a ly z e th e m o d el in th is range.
T h e p re se n t a rtic le discusses a new fo rm u la tio n of th e m o d e l, w h ere th e s p a tia l to p o lo g y is assu m ed to be t h a t of a th re e -to ru s ( T 3) [55- 57], r a th e r th a n t h a t o f a th re e -s p h e re , w hich w as th e to p o lo g y used in all th e fo rm e r stu d ie s. I t w as fo u n d t h a t th e fo u r p h ase s in th is case a re th e sam e as in th e sp h eric al m od el, w ith th e p o s itio n of p h a se b o u n d a rie s sh ifted a l i ttle.2 T h e a d d itio n a l, im p o rta n t b o n u s in th is new fo rm u la tio n com es from th e fa ct t h a t th e p h y sically in te re stin g regio n in th e b a re co u p lin g c o n s ta n t space m e n tio n e d above becom es n u m eric ally accessible w ith th e s ta n d a r d a lg o rith m used in th e ea rlie r stu d ies. W e cou ld th e n observe t h a t th e sp e c u la tiv e q u a d ru p le p o in t, m a y b e n o t su rprisin gly, s e p a ra te s in to tw o trip le p o in ts, co n n e c te d by a p h a se tra n s itio n line b etw e en p h a se C a n d th e B ph ase , a n d n o t s e p a ra te d by th e b ifu rc a tio n p h a se (see figure 1) . A n im p o rta n t p o in t is t h a t we now have access to th e s e trip le p o in ts d ire c tly fro m p h a se C a n d it is th u s possible to have a re n o rm a liz a tio n g ro u p flow from th e in fra re d to th e p o te n tia l U V fixed p o in t e n tire ly in th e “p h y sical” C phase.
T h e p h ases of th e m od el w ere id en tified for a sy ste m w ith N41 = 160k, a n a ly z in g th e s tr u c tu r e o f g e o m e try a t th e g rid o f p o in ts in th e co u p lin g c o n s ta n t p lan e show n in figure 1, th e d ifferent p h ase s re p re se n te d by d o ts w ith d ifferen t colors. In th e p re se n te d p h a se d ia g ra m th e p recise p o sitio n of p h a se tra n s itio n s w as n o t d e te rm in e d . T h is req u ires a carefu l s tu d y of th e in fin ite v o lum e lim it a n d scalin g of th e p o sitio n o f p h a se tra n s itio n
2This may be a finite-size effect. The diagram was determined by analyzing systems with only one volume.
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
lines w ith th e la ttic e volum e. T h e m o st in te re stin g regio n is th e one s e p a ra tin g p h a se C a n d B w h ere we m ay observe tw o trip le p o in ts. T h e p re se n t p a p e r is th e first ste p in th e a n a ly sis of th is m o st p h y sically in te re stin g region. W e will p erfo rm a d e ta ile d an a ly sis of th e b e h a v io r of th e m odel a t K0 = 4.0 in th e n e ig h b o rh o o d of th e p h a se tra n s itio n line. W e w ill t r y to d e te rm in e th e o rd e r of th e p h a se tra n s itio n a t th is p o in t. W e will show t h a t th e tra n s itio n seem s to b e a first o rd e r tra n s itio n . T h e re s u lts p re se n te d in th is a rtic le show t h a t th e m o st in te re stin g region in th e b a re p a r a m e te r sp ace c a n successfully b e an a ly zed u sin g th e s ta n d a r d M o n te C arlo a lg o rith m u sed in th e e a rlie r sim u la tio n s.
2 T h e p h a se s tr u c tu r e o f C D T
A s m e n tio n e d , th e p h a se d ia g ra m of th e C D T m o del w ith a to ro id a l s p a tia l to p o lo g y p e r
m its us to in v e stig a te th e p ro p e rtie s o f th e m od el in a n im p o rta n t ra n g e o f th e b a re co u p lin g c o n s ta n ts , p re v io u sly inaccessible to n u m eric al m e a su re m e n ts. F o r sy stem s w ith a sp h eric al s p a tia l to p o lo g y a d e ta ile d an a ly sis o f th e p h a se d ia g ra m w as p e rfo rm e d follow ing tw o lines in th e b a re co u p lin g c o n s ta n t space. T h e se w ere th e v e rtic a l line w ith v ary in g A a t K0 = 2.2 a n d th e h o riz o n ta l line a t A = 0.6. In th e first case it w as p o ssible to an a ly z e th e p h ase tra n s itio n b etw e en C a n d b ifu rc a tio n p h ases a n d b etw e en th e b ifu rc a tio n a n d B p h ases. In th e second case a tra n s itio n b etw e en th e C a n d A p h ases w as s tu d ie d (see [58] for re c e n t re
su lts). T h e belief com ing from th e an a ly sis of th e sp h eric al case w as t h a t if we d ec rea se th e value of A for a fixed value of K0 we n ecessarily m ove from C p h a se to th e b ifu rc a tio n p h ase a n d only, for still lower A , to th e B ph ase. H ow ever, ch a n g in g to to ro id a l s p a tia l to p o lo g y we d iscovered t h a t th is is n o t th e case, p ro b a b ly also in th e sp h eric al topolog y. T h e re exists a ra n g e of b a re co u p lin g c o n s ta n ts w h ere C a n d B p h ases are d ire c tly n eig h b o rin g . T h is h a p p e n s close to th e A = 0 line in th e ra n g e o f K0 b etw een , ap p ro x im a te ly , 3.5 a n d 4.5.
O ne m ay ex p e c t th e ex isten c e of tw o trip le p o in ts (in ste a d of th e p re v io u sly co n je c tu re d q u a d ru p le p o in t): one trip le p o in t w h ere C, A a n d B p h ase s m eet, an d a seco nd trip le p o in t w h ere C, b ifu rc a tio n a n d B p h ase s m eet. F in d in g th e p recise lo c a tio n of th e trip le p o in ts m ay b e n u m eric ally m o re difficult th a n a n a ly z in g th e gen eric tra n s itio n b etw e en p h a se C a n d B. As a first s te p in th e d e ta ile d an a ly sis we have chosen to d e te rm in e th e p o sitio n an d th e o rd e r of th e p h a se tra n s itio n b etw e en C a n d B p h ase s alo ng a v e rtic a l line a t K0 = 4.0.
T h is is a p p ro x im a te ly in th e m id d le b etw e en th e p o s itio n of th e tw o trip le p o in ts. Since th e c h a ra c te ris tic b e h a v io r in th e tw o p h ases c o rre sp o n d s to d ifferent sy m m etries o f th e co n fig u ratio n s (we have tr a n s la tio n a l s y m m e try in tim e in th e C p h a se a n d a sp o n ta n e o u s b re a k in g o f th is s y m m e try in th e B p h ase ) we e x p e c t a re la tiv e ly large h y stere sis w h en we cross th e p h a se b o u n d a ry . W e w a n t to find m e th o d s w hich m ak e th e h y stere sis effect as sm all as possible. W e also e x p e c t re la tiv e ly larg e fin ite size effects. A n im p o rta n t p o in t in th e an aly sis w ill be to check how th e h y stere sis beh av es w h en th e sy ste m size goes to infinity.
T h e an a ly sis p re se n te d in th e p a p e r is b ase d o n a s tu d y o f sy stem s w ith a fixed tim e p e rio d T = 4 a n d d ifferen t (a lm o st) fixed volum es N 41. In th e e a rlie r stu d ie s, it w as show n t h a t re d u cin g th e p e rio d T does n o t p ro d u c e significan t fin ite-size effects [58]. O n th e o th e r h a n d , in p a r tic u la r in th e C p h ase, th e av erage v olum e p e r tim e slice for a fixed to ta l v olum e g ets re la tiv e ly large, w h ich is v ery im p o rta n t. In th e M o n te C arlo sim u la tio n s we
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
enforce th e la ttic e v olum e N 41 to flu c tu a te a ro u n d a chosen valu e N 41, so t h a t th e m easu red (N 41) = N 41. T h is is realized by a d d in g to th e R egge a c tio n ( 1.3) a volu m e-fixing te rm
Sr ^ Sr + e(N 41 — N 41)2- (2 -1)
In th e th e rm a liz a tio n p rocess it is esse n tial to fin e -tu n e th e value o f K4 in such a w ay t h a t one g e ts s ta b ility of th e sy stem volum e. T h is is realized by le ttin g th e v alu e of K 4 d y n a m ic a lly ch a n g e by sm all step s, u n til th e re q u ire d s ta b le s itu a tio n is realized. If a value o f K 4 is to o high, we o bserv e t h a t sy stem vo lu m e stab iliz es below th e ta r g e t v alue N 41.
Sim ilarly, if we ta k e it to o sm all, th e v o lu m e w ill b e to o large. O n ly fo r K 4 K4Crit(N41) flu c tu a tio n s of v olum e are ce n te re d a ro u n d N 41 w ith th e w id th co n tro lle d by e. D u rin g th e th e rm a liz a tio n p a r t of th e M o n te C arlo sim u la tio n s th e a lg o rith m trie s to find th e o p tim a l value of K 4 for a given fixed set o f p a ra m e te rs K 0, A a n d N 41. T h e w hole p rocess of m e a su re m e n ts is o rg a n iz ed in th e follow ing way:
• W e s ta r t a sequence of th e rm a liz a tio n ru n s a t a set of A values in th e n e ig h b o rh o o d of th e e x p e c te d p o sitio n o f th e p h a se tra n s itio n . T h e in itia l c o n fig u ra tio n o f th e sy stem is ta k e n to b e th e sm all h y p e r-c u b ic co n fig u ra tio n discu ssed in referen ce [55]. W e choose th e ta r g e t v olum e N 41 a n d let th e sy ste m size grow to w a rd s N 41 a n d a d a p t th e K4 value from th e guessed in itia l value. T h e in itia l K4 c a n b e chosen e ith e r a little below o r a little above th e guessed c ritic a l value.
• W e find t h a t on th e g rid of A values we c a n d e te rm in e ran g es co rre sp o n d in g to th e a p p e a ra n c e of tw o d ifferen t phases, w ith a re la tiv e ly s u d d e n ju m p b etw e en th e phases.
In g en e ral th e ju m p is o b served b etw e en tw o n eig h b o rin g values o n th e g rid of A . T h e co rre sp o n d in g values o f K4 are m a rk e d ly d ifferent in th e tw o p hases. T y p ica lly th e value is sm aller for th e C p h a se t h a n for th e B ph ase. W e c a n d e te rm in e th e p h a se of th e sy stem by th e m e a su re d values o f th e o rd e r p a ra m e te rs (see la te r for d efin itio n s), w hich are very d ifferen t in th e d ifferen t p hases.
• T h e value of A w h ere th e p h a se tra n s itio n is o b serv ed d e p e n d s on th e in itia l value of K 4 used in th e th e rm a liz a tio n process. As a con sequ ence, we ob serv e in g en eral tw o values AfOW(N41) a n d Ah1gh (N 41). B o th values a re d e te rm in e d w ith th e ac c u ra c y d e p e n d in g o n th e g rid o f A .
• W e re p e a t th e an a ly sis o n a finer g rid , w hich covers th e ra n g e w h ere we observed p h a se tra n s itio n s . W e fo u n d th e m o st effective p ro c e d u re is to r e s ta r t th e M o n te C arlo e v o lu tio n from th e sam e sm all in itia l c o n fig u ra tio n as before, b u t usin g as th e in itia l values of K4 th e ones d e te rm in e d for th e C o r th e B p h a se from e a rlie r ru n s in th e n e ig h b o rh o o d o f th e tra n s itio n s , co rre sp o n d in g to A£11W(N41) o r Ahigh (N 41) respectively.
• A finer g rid p e rm its to d e te rm in e th e tw o p o sitio n s of th e p h a se tra n s itio n w ith b e t te r accuracy. T h e d ifferen t p o sitio n of ju m p s b etw e en th e tw o p h ase s (low or h ig h ) c a n b e in te rp re te d as th e h y stere sis effect in a p rocess w h ere we slowly in crease th e value of th e A p a r a m e te r o r slow ly d ec rea se its v alue. W e observe th a t the size
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
A
F i g u r e 2. T he p lo t illu stra te s th e hysteresis m easured d u rin g sim ulations for th e ta rg e t volum e NN4i = 160k. T he green an d blue d o ts correspond to th e location of th e phase C side of th e p h ase-tran sitio n , while th e red an d black d o ts correspond to th e location of th e p hase B side of th e p h ase-tra n sitio n . T he sam e colors will be used in th e n e x t plots, w here we com pare resu lts for different volumes.
o f the h ystere sis fo r a p a rtic u la r choice o f IV41 does n o t decrease w ith in reasonable th e rm a liza tio n tim e s. B y ta kin g a fin e r g rid in A we can on ly d e te rm in e the en d p o in ts o f a h ystere sis curve w ith a better accuracy. W e illu s tra te th e s itu a tio n in figure 2 . T h e lines show n w ere o b ta in e d from th e m e a su re d values of A an d K4 for N41 = 160k.
• In th e ra n g e of A values b etw e en AfOW(N41) a n d Ah1jgth(N 41), d e p e n d in g on th e in itia l value of K4 a sy stem en d s e ith e r in th e B o r C p h ase . T h is c a n b e in te rp re te d as a ra n g e o f p a ra m e te rs , w h ere th e tw o p h ases m ay co e x ist. T h e d is trib u tio n of th e values of th e o rd e r p a ra m e te rs (to be d efin ed below ), c h a ra c te ris tic for th e tw o phases, is very n arro w . As a co nsequ ence, a tu n n e llin g b etw e en th e tw o p h ases is nev er o b serv ed a fte r we have re ach ed a “s ta b le ” en sem b le of co n fig u ratio n s in th e th e rm a liz a tio n stage.
T h e th e rm a liz a tio n p a th chosen abo v e m ean s in p ra c tic e , t h a t in th e b eg in n in g , th e sy ste m grow s in a re la tiv e ly ra n d o m w ay from th e in itia lly sm all c o n fig u ra tio n to th e d esired ta r g e t v olum e IV41 a n d th e g e o m e try evolves to a s ta b le ra n g e in th e c o n fig u ra tio n space.
T h e first s te p c a n b e in te rp re te d as a s te p in th e d ire c tio n ty p ic a l for th e p h a se A, w here c o rre la tio n s b etw e en th e s p a tia l co n fig u ratio n s in th e co n se cu tiv e tim e slices are sm all o r a b s e n t. O n ly a fte rw a rd s we reach th e d o m a in s co rre sp o n d in g to th e tw o p h ases we stud y.
As a con sequence, we ex p e c t t h a t th e d e sc rib e d m e th o d will b e v ery well su ite d to th e fu tu re an a ly sis of th e trip le p o in t involving th e A ph ase.
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
F i g u r e 3. T he pseudo-critical value K 4rlt(N4 1) as a function of A crlt(N4 1). T he d a ta points m easured for increasing la ttic e volum e N4 1 are going from left to rig h t. C en ter of th e black ellipse corresponds to th e e stim a te d p o sitio n of (A crlt(TO), K 4rlt(TO)) and its radii correspond to th e e stim a te d uncertain tie s. Colors of th e fits follow th e convention used in figure 2 .
T h e b e h a v io r of th e p se u d o -c ritic a l values K 4rit(N 41) is v ery s im ila r to t h a t of A crit(N 41). T h is c a n b e seen in figure 3 , w h ere we show th e values of K 4rit(N 41) p lo tte d as a fu n c tio n of A crit(N 41). O n b o th sides o f th e h y stere sis th e d e p e n d e n c e is a p p ro x im a te ly lin ear, w hich m ean s t h a t values o f b o th p se u d o -c ritic a l p a ra m e te rs ( K 4rit a n d A crit) scale in th e sam e w ay w ith th e la ttic e v olu m e N 41. E x tra p o la tin g th e lines to a p o in t w h ere th e y cross p e rm its to d e te rm in e values fo r K 4rit a n d A crit in th e lim it N41 ^ to . T h e fit gives K 4rit(TO) = 1.095 ± 0.001 a n d A crit(TO) = 0.022 ± 0.002. T h e e rro rs o n th is a n d o th e r p lo ts a re th e e s tim a te d s ta tis tic a l erro rs a n d in clu d e th e g rid sp acin g for A .
A lth o u g h th e size of th e h y stere sis sh rin k s w ith v olu m e N 41, th e p lo ts in d ic a te t h a t th e sh rin k in g p rocess is re la tiv e ly slow a n d th u s in o rd e r to g et rid o f th e h y stere sis one sh o u ld use e x tre m e ly large la ttic e volum es, n o t tr a c ta b le nu m erically . T h e d e p e n d e n c e of A crit o n th e la ttic e volum e, ra n g in g b etw e en N41 = 40k an d N41 = 1600k is p re se n te d in figure 4 . As it w as ex p la in e d above, th e p lo t c o n ta in s fo u r sets of d a t a co rre sp o n d in g to th e fo u r d ifferent p o in ts d e sc rib in g th e h y stere sis (see figure 2) . T h e d a t a p o in ts c a n be fitte d w ith th e curve
A crit(N41) = A crit(TO) - A ■ N ~1 / 7. (2.2)
T h e b e st fit for th e co m b in ed sets of d a t a (w ith fixed A crit(TO) = 0.022 d e te rm in e d above) w as o b ta in e d for 7 = 1.64 ± 0.18. A n a lte rn a tiv e fit w ith 7 = 1 (a n d th e sam e valu e of A crit(TO)) is ex c lu d e d as c a n be seen in figure 4 (th e d a sh e d line). T h e value 7 = 1 w ould b e a s tro n g evid en ce for a first o rd e r tra n s itio n . T h e fits w ere b ase d o n d a t a m easu red for v olum es ra n g in g from N41 = 40k to N41 = 720k. T h e la rg e st v olu m e N41 = 1 6 0 0 k w as u sed o n ly for checking c o n sisten cy w ith th e e x tra p o la tio n s T h e an a lo g o u s p lo t p re se n tin g th e fo u r sets of th e p se u d o -c ritic a l K4rit(N 41) values for th e sam e ra n g e of volum es is show n in
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
F i g u r e 4. T he pseudo-critical value A crlt as a function of NN4 1. T h e solid lines are (one p a ram eter) fits of form ula (2.2) w ith fixed com m on values of 7 = 1.64 an d A crlt( w ) = 0.022. Colors of th e fits follow th e convention used in figure 2 . T h e d ash ed line shows a com m on fit of all d a ta p o in ts to th e scaling function (2.2) w ith enforced value of 7 = 1 and A crlt( w ) = 0.0 2 2.
figure 5. T h e e x p e rim e n ta l p o in ts a re a g a in well fitte d by th e fo rm u la
K4Crit(N 4i) = K 4ritM - B ■ N - 1 h , (2.3)
w h ere th e m e a su re d value of 7 = 1.62 ± 0.25 agrees well w ith th e re su lt o b ta in e d for A crit.
T h e fits a re re p re se n te d by curves w ith d ifferen t colors, w hich a g a in follow th e co n v e n tio n used in figure 2 . O n th e scale used in th is p lo t th e g re en a n d b lu e cu rv es p ra c tic a lly o verlap.
3 O rder p a r a m e te rs
To id en tify th e p h ases of C D T w ith to ro id a l s p a tia l to p o lo g y we follow m e th o d s used in th e p re v io u s stu d ie s. T h ese are b ase d o n th e an a ly sis of o rd e r p a ra m e te rs w hich h av e a d ifferen t b e h a v io r in th e differen t ph ases. W e use o rd e r p a ra m e te rs w hich c h a ra c te riz e b o th g lobal a n d local p ro p e rtie s of th e sim plicial m anifolds. T h e g lo bal o rd e r p a ra m e te rs w ere called O 1 a n d O 2, w here
O i = N T ' ° 2 = (3A)
In each p h a se th e d is trib u tio n s of N 0 a n d N 32 a re v ery n arro w , a n d p ra c tic a lly G au ssia n . P h a se s B a n d C are c h a ra c te riz e d by v ery d ifferen t averag e values for th e tw o d is trib u tio n s . T h e d e p e n d e n c e of th e o rd e r p a ra m e te rs O 1 a n d O 2 o n N 41 a t th e e n d p o in ts o f th e h y steresis is p re se n te d in figure 6 . T h e colors follow th e co n v e n tio n u sed in figure 2 .
T h e d a t a p re se n te d o n th e p lo ts co rre sp o n d for each N 41 to th e fo u r v alu es of th e A crit(N 41) p o in ts, follow ing a g a in th e n o ta tio n of figure 2 . I t is seen t h a t a lth o u g h b o th p se u d o -c ritic a l values K 4rit(N 41) a n d A crit(N 41) b eco m e v ery close for in cre asin g N 41, this is n o t the case fo r the order pa ra m eters, w hich in fa c t behave in a w ay sim ila r to th a t
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
F i g u r e 5. T he pseudo-critical value K 4rlt as a function of NN4 1. T he solid lines are (one p ara m eter) fits of form ula (2.3) w ith fixed com m on values of 7 = 1.62 an d K 4rlt( w ) = 1.095. Colors of th e fits follow th e convention used in figure 2.
ch ara cterizin g the fir s t order tra n sitio n . I t m ean s t h a t a tra n s itio n b etw e en th e B an d C p h ases becom es v ery ra p id . O n th e o th e r h a n d , d u e to th e o b serv ed h y steresis, th e m e th o d u se d in th is an aly sis chooses a p o s itio n of m e a su re d values for th e o rd e r p a r a m e te rs slig h tly aw ay from th e tru e tra n s itio n p o in t (lo c a te d in sid e th e h y stere sis region) a n d th u s in fa c t we w ere n o t ab le to p erfo rm s ta b le sim u la tio n s e x a c tly a t K 4rit(N 41) a n d A crit(N 41) co rre sp o n d in g to such a tra n s itio n p o in t.3
A sim ilar b e h a v io r is o b serv ed for th e set o f local o rd e r p a ra m e te rs O3 a n d O4 defined by
O3 = ^ ( n t + 1 - n t ) 2, O4 = m a x op . (3.2)
H ere nt is th e n u m b e r of t e tr a h e d r a s h a re d by {4,1} a n d {1, 4} fo ur-sim plices w ith bases a t tim e t a n d nt =
1
N 41(t) =2
N41
. m a x op is th e m a x im a l o rd e r o f a v e rte x in a tria n g u la tio n . T h e ty p ic a l b e h a v io r o f th e se tw o o rd e r p a ra m e te rs is e x p e c te d to be d ifferen t in p h ases B a n d C. P h a s e B is c h a ra c te riz e d by h av in g a m acro sco p ic fra c tio n of th e fo u r-v o lu m e c o n c e n tra te d a t a single s p a tia l slice co rre sp o n d in g to som e tim e t (in th e sense t h a t a lm o st all {4,1} a n d {1,4} fou r-sim plices h ave fo u r v ertic es a t th is s p a tia l slice).T h is is ac co m p an ied by th e a p p e a ra n c e of tw o sin g u la r v ertic es lo c a te d a t tim e s t ± 1 an d s h a re d by a m acro sco p ic n u m b e r of fou r-sim p lices in a tria n g u la tio n . As a co nseq uen ce, in p h a se B —3 a n d -O4 sh o u ld b e of o rd e r one. In p h a se C th e re is n o such d e g e n era cy an d for larg e N
41
b o th —3 a n d — 4 sh o u ld a p p ro a c h zero. T h e b e h a v io r o f th e se tw o o rd e r p a ra m e te rs is p re se n te d in figure 7 .3We are currently working on the numerical algorithm which would enable tunneling between both sides of the hysteresis region in a single Monte Carlo run and thus enable to define a more precise position of the transition point.
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
F i g u r e 6 . T he o rd e r-p a ram e te rs O i and O2 as a function of N 41 a t th e en d p o in ts of th e hysteresis.
T he colors correspond to th e convention used in figure 2 .
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
F i g u r e 7. T he o rd er-p aram eters ° 3/n41 an d ° 4/n41 as a function of NN4 1 a t th e e n d p o in ts of th e hysteresis. T h e colors correspond to th e convention used in figure 2 .
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
4 C o n c lu sio n and d isc u ssio n
In th e p re se n t a rtic le we m a d e a d e ta ile d s tu d y of th e p h a se tra n s itio n o b serv ed b etw een th e p h a se C a n d th e p h a se B a t th e v alu e of th e dim en sio n less g ra v ita tio n a l co u p lin g c o n s ta n t K0 = 4.0. T h e tra n s itio n a p p e a rs to b e lo c a te d close to A = 0. T h e id e n tific a tio n o f th is region, a n d th e p o ssib ility t h a t o ne c a n m ove all th e w ay to th e trip le p o in ts of th e p h a se d ia g ra m , sta y in g e n tire ly insid e th e “p h y sical” C ph ase, is a goo d new s for th e re n o rm a liz a tio n g ro u p p ro g ra m s ta r te d in [54] (a n d te m p o ra rily p u t on hold by th e discovery of th e b ifu rc a tio n p h ase ). T h e re n o rm a liz a tio n g ro u p an aly sis is p ro b a b ly th e cle a n e st w ay to co n n e c t C D T la ttic e g ra v ity a p p ro a c h to a s y m p to tic safety. T h e an aly sis o f th e re lev an t co u p lin g c o n s ta n t regio n w as m a d e p o ssib le by sw itch in g from sp h eric al s p a tia l to p o lo g y to to ro id a l s p a tia l topology. In th is first s tu d y of th e in te re stin g regio n we p o sitio n e d ourselves in th e m id d le of th e B -C p h a se tra n s itio n line, b etw e en th e tw o trip le e n d p o in ts a n d from th e an a ly sis of th e M o n te C arlo d a t a we co n c lu d e t h a t th e tra n s itio n is m o st likely of first o rd e r. Since e n d p o in ts of p h a se tra n s itio n lines o fte n a re o f h ig h er o rd e r, th e trip le p o in ts m ig h t well b e of second o rd e r a n d on e o f th e m cou ld th e n serve as a U V fixed p o in t for a q u a n tu m th e o ry of g ravity. W e a re ac tiv e ly p u rsu in g th is line o f research.
L e t us en d by som e re m a rk s a b o u t o u r q u a n tu m g ra v ity m od el, view ed as a s ta tis tic a l sy ste m o f fo u r-d im e n sio n a l g eo m etries. D e sp ite th e a lm o st triv ia l a c tio n ( 1.3) , th e m odel h as a n am az in g ly rich p h a se s tru c tu re , w ith fo u r d ifferent p h ase s, each c h a ra c te riz e d by v ery d ifferent d o m in a tin g g eo m etries. In a d d itio n , som e o f th e p h a se tra n s itio n s hav e q u ite u n u s u a l c h a ra c te ristic s . T h e tra n s itio n b etw e en p h a se B a n d th e b ifu rc a tio n p h a se is a second o rd e r tra n s itio n [44], b u t superficially, for a fin ite volum e, it looked like a first o rd e r tra n s itio n . H ow ever, an a ly z in g th e b e h a v io r as a fu n c tio n of th e in cre asin g la ttic e volum e th e first o rd e r n a tu r e fa d ed away. M oving to w a rd s la rg e r v alues o f K 0, i.e. to w a rd s th e reg ion we h ave b ee n in v e stig a tin g in th is a rtic le , th e tra n s itio n b ec a m e m o re a n d m o re like a first o rd e r tra n s itio n . W ith th e sp h eric al s p a tia l to p o lo g y used in [44] on e co uld n o t get to th e region in v e stig a te d in th e p re s e n t artic le , b u t it is n a tu r a l to c o n je c tu re t h a t passin g th e trip le p o in t m oving from th e b ifu rc a tio n -B line to th e C-B line, th e tra n s itio n ch ang es from second o rd e r to first o rd e r. H ow ever, th is first o rd e r tra n s itio n is still so m ew h a t u n u su a l. F irstly , it h as k ep t th e c h a ra c te ris tic s o f th e second o rd e r b ifu rc a tio n -B tra n s itio n t h a t th e fin ite size b e h a v io r of th e p se u d o -c ritic a l p o in ts, g iven by eqs. ( 2 .2) a n d (2 .3 ) have n o n -triv ia l e x p o n e n ts 7. Secondly, th e h y stere sis g a p goes to zero w ith in cre asin g volum e, w hich is a n o n -s ta n d a rd b e h a v io r in th e case o f a first o rd e r tra n s itio n . H ow ever, th e ju m p s o f th e o rd e r p a ra m e te rs seem v olum e in d e p e n d e n t a n d t h a t is th e m a in re a so n t h a t we classify th e tra n s itio n as b ein g a first o rd e r tra n s itio n . T h e large fin ite size effects we observ e m ig h t be re la te d to th e global chan g es o f d o m in a n t co n fig u ratio n s w hich ta k e place betw e en p h a se C a n d p h a se B , an d th e se g lo bal re a rra n g e m e n ts m ig h t, for fin ite volum es, h ave a d ifferen t “p h a se -sp a c e ” in th e case o f sp h eric al a n d to ro id a l top olog ies. T h a t m ig h t e x p la in w h y o u r M o n te C arlo a lg o rith m c a n access th e B -C tra n s itio n on ly in th e case of to ro id a l topology. T h e s ta tis tic a l th e o ry o f g eo m etrie s is a fa sc in a tin g a re a w h ich is alm o st u n ex p lo re d for sp a c e tim e d im en sio n s la rg e r th a n tw o.
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
A c k n o w le d g m e n ts
D N w ould like to th a n k R e n a te R oll th e fru itfu l d iscussio ns a n d h o s p ita lity d u rin g his sta y a t R a d b o u d U n iv e rsity in N ijm eg en. J G S acknow ledges s u p p o rt from th e g ra n t U M O -2 0 1 6 /2 3 /S T 2 /0 0 2 8 9 from th e N a tio n a l Science ce n tre , P o la n d . J A acknow ledges s u p p o rt from th e D a n is h R e se a rc h C o un cil g ra n t Q u a n tu m G eo m etry, g ra n t 7014-00066B . A G a n d D N acknow ledges s u p p o rt by th e N a tio n a l S cience C e n tre , P o la n d , u n d e r g ra n t no. 2 0 1 5 /1 7 /D /S T 2 /0 3 4 7 9 .
O p e n A c c e s s . T h is a rtic le is d is trib u te d u n d e r th e te rm s o f th e C re a tiv e C o m m o ns A ttr ib u tio n L icense ( C C -B Y 4.0) , w hich p e rm its an y use, d is trib u tio n a n d re p ro d u c tio n in a n y m ed iu m , p ro v id ed th e o rig in al a u th o r(s ) a n d so urce are c re d ite d .
R e fer e n c es
[1] S. W einberg, Ultraviolet divergences in quantum theories o f gravitation, in General relativity:
E in ste in centenary survey, S.W . H aw king and W . Israel eds., C am bridge U niversity Press, C am bridge, U .K . (1979), pg. 790 [i nSPIRE].
[2] M. R eu ter, N onperturbative evolution equation fo r quantum gravity, P hys. Rev. D 5 7 (1998) 971 [h e p -th /9 6 0 5 0 3 0 ] [i nSPIRE].
[3] A. Codello, R. P ercacci and C. R ahm ede, Investigating the ultraviolet properties o f gravity w ith a W ilsonian renorm alization group equation, A nnals Phys. 3 2 4 (2009) 414
[a r X iv :0 8 0 5 .2 9 0 9 ] [i nSPIRE].
[4] M. R e u te r an d F. Saueressig, F unctional renorm alization group equations, asym ptotic safety and quantum E in ste in gravity, in G eom etric and topological m ethods fo r quantum field theory, C am bridge U niversity P re ss, C am bridge, U .K . (2010), pg. 288 [a r X iv :0 7 0 8 .1 3 1 7 ] [i nSPIRE].
[5] M. N iederm aier a n d M. R eu ter, The asym ptotic safety scenario in quantum gravity, Living Rev. Rel. 9 (2006) 5 [i nSPIRE].
[6] D .F. L itim , F ixed points o f quantum gravity, Phys. Rev. Lett. 9 2 (2004) 201301 [h e p -th /0 3 1 2 1 1 4 ] [i nSPIRE].
[7] F. D avid, P la n a r diagrams, tw o-dim ensional lattice gravity and surface models, Nucl. Phys.
B 2 5 7 (1985) 45 [i nSPIRE].
[8] A. Billoire an d F. D avid, M icrocanonical sim ulations o f random ly triangulated planar random surfaces, Phys. Lett. B 1 6 8 (1986) 279 [i nSPIRE].
[9] J. A m bjprn, B. D urh u u s an d J. Frohlich, Diseases o f triangulated random surface models and possible cures, Nucl. Phys. B 2 5 7 (1985) 433 [i nSPIRE].
[10] J. A m bjprn, B. D urhuus, J. Frohlich an d P. O rland, The appearance o f critical dim ensions in regulated string theories, Nucl. Phys. B 2 7 0 (1986) 457 [i nSPIRE].
[11] V.A. K azakov, A.A. M igdal an d I.K . K ostov, Critical properties o f random ly triangulated planar random surfaces, Phys. Lett. B 1 5 7 (1985) 295 [i nSPIRE].
[12] D.V. B oulatov, V.A. K azakov, I.K . K ostov an d A.A. M igdal, A nalytical and num erical study o f the m odel o f dynam ically triangulated random surfaces, Nucl. P hys. B 2 7 5 (1986) 641 [i nSPIRE].
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
[13] V.A. K azakov, The appearance o f m a tte r fields fro m quantum flu ctu a tio n s o f 2D gravity, Mod. Phys. Lett. A 4 (1989) 2125 [i nSPIRE].
[14] J. A m b jp rn an d Yu. M. M akeenko, Properties o f loop equations fo r the H erm itea n m a trix model and fo r tw o-dim ensional quantum gravity, Mod. Phys. Lett. A 5 (1990) 1753 [i nSPIRE].
[15] J. A m b j0rn, J. Jurkiew icz a n d Yu. M. M akeenko, M ultiloop correlators fo r tw o-dim ensional quantum gravity, P hys. Lett. B 2 5 1 (1990) 517 [i nSPIRE].
[16] J. A m b j0rn, L. Chekhov, C .F. K ristjan sen a n d Yu. M akeenko, M a trix m odel calculations beyond the spherical lim it, Nucl. P hys. B 4 0 4 (1993) 127 [Erratum ibid. B 4 4 9 (1995) 681]
[h e p -th /9 3 0 2 0 1 4 ] [i nSPIRE].
[17] V .G . K nizhnik, A.M . Polyakov an d A.B. Zam olodchikov, Fractal structure o f 2D quantum gravity, Mod. Phys. Lett. A 3 (1988) 819 [i nSPIRE].
[18] F. D avid, C onform al field theories coupled to 2D gravity in the conform al gauge, Mod. Phys.
Lett. A 3 (1988) 1651 [i nSPIRE].
[19] J. D istler a n d H. K aw ai, C onform al field theory and 2D quantum gravity, Nucl. Phys. B 321 (1989) 509 [i nSPIRE].
[20] J. A m b j0rn an d S. V arsted, Three-dim ensional sim plicial quantum gravity, Nucl. P hys. B 3 7 3 (1992) 557 [i nSPIRE].
[21] J. A m b j0rn an d S. V arsted, E ntropy estim ate in three-dim ensional sim plicial quantum gravity, Phys. Lett. B 2 6 6 (1991) 285 [i nSPIRE].
[22] J. A m b j0rn, D.V. B oulatov, A. Krzyw icki an d S. V arsted, The vacuum in three-dim ensional sim plicial quantum gravity, Phys. Lett. B 2 7 6 (1992) 432 [i nSPIRE].
[23] M .E. A gishtein a n d A.A. M igdal, T hree-dim ensional quantum gravity as dynam ical
triangulation, Mod. Phys. Lett. A 6 (1991) 1863 [Erratum ibid. A 6 (1991) 2555] [i nSPIRE].
[24] D.V. B oulatov an d A. K rzywicki, O n the phase diagram o f three-dim ensional sim plicial quantum gravity, Mod. P hys. Lett. A 6 (1991) 3005 [i nSPIRE].
[25] J. A m b j0rn an d J. Jurkiew icz, F our-dim ensional sim plicial quantum gravity, Phys. Lett. B 2 7 8 (1992) 42 [i nSPIRE].
[26] J. A m b j0rn an d J. Jurkiew icz, Scaling in fo u r-d im e n sio n a l quantum gravity, Nucl. Phys. B 4 5 1 (1995) 643 [h e p - th /9 5 0 3 0 0 6 ] [i nSPIRE].
[27] M .E. A gishtein an d A.A. M igdal, S im u la tio n s o ffo u r-d im e n sio n a l sim plicial quantum gravity, Mod. Phys. Lett. A 7 (1992) 1039 [i nSPIRE].
[28] P. B ialas, Z. B u rd a, A. K rzyw icki a n d B. P etersson, Focusing on the fixed p o in t o f 4D sim plicial gravity, Nucl. Phys. B 4 7 2 (1996) 293 [h e p - la t/9 6 0 1 0 2 4 ] [i nSPIRE].
[29] S. C atte ra ll, R. R enken and J.B . K ogut, Singular structure in 4D sim plicial gravity, Phys.
Lett. B 4 1 6 (1998) 274 [h e p - la t/9 7 0 9 0 0 7 ] [i nSPIRE].
[30] J. A m b j0rn, L. G laser, A. G orlich an d J. Jurkiew icz, E uclidian 4d quantum gravity w ith a non-trivial measure term , JH E P 10 (2013) 100 [a r X iv :1 3 0 7 .2 2 7 0 ] [i nSPIRE].
[31] J. L aiho an d D. C oum be, Evidence fo r a sym ptotic safety fro m lattice quantum gravity, Phys.
Rev. Lett. 1 0 7 (2011) 161301 [a r X iv :1 1 0 4 .5 5 0 5 ] [i nSPIRE].
[32] D. C oum be a n d J. Laiho, Exploring Euclidean dynam ical triangulations w ith a non-trivial measure term , JH E P 0 4 (2015) 028 [a rX iv :1 4 0 1 .3 2 9 9 ] [i nSPIRE].
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
[33] J. Laiho, S. B assler, D. C oum be, D. D u an d J .T . N eelakanta, Lattice quantum gravity and asym ptotic sa fety, P hys. Rev. D 9 6 (2017) 064015 [a r X iv :1 6 0 4 .0 2 7 4 5 ] [i nSPIRE].
[34] T. Regge, General relativity w ithout coordinates, N uovo Cim. 19 (1961) 558 [i nSPIRE].
[35] J. A m b j0rn, A. G orlich, J. Jurkiew icz an d R. Loll, Nonperturbative quantum gravity, Phys.
Rept. 5 1 9 (2012) 127 [a r X iv :1 2 0 3 .3 5 9 1 ] [i nSPIRE].
[36] R. Loll, Q uantum gravity fro m causal dynam ical triangulations: a review, a rX iv :1 9 0 5 .0 8 6 6 9 [i nSPIRE].
[37] J. A m b j0rn, J. Jurkiew icz a n d R. Loll, D ynam ically triangulating L orentzian quantum gravity, Nucl. Phys. B 6 1 0 (2001) 347 [h e p -th /0 1 0 5 2 6 7 ] [i nSPIRE].
[38] J. A m bj0rn, J. Jurkiew icz an d R. Loll, A nonperturbative L orentzian pa th integral fo r gravity, Phys. Rev. Lett. 8 5 (2000) 924 [h e p - th /0 0 0 2 0 5 0 ] [i nSPIRE].
[39] J. A m b j0rn, J. Jurkiew icz an d R. Loll, R econstructing the universe, P hys. Rev. D 72 (2005) 064014 [h e p - th /0 5 0 5 1 5 4 ] [i nSPIRE].
[40] J. A m b j0rn, J. Jurkiew icz a n d R. Loll, Em ergence o f a 4D world fro m causal quantum gravity, Phys. Rev. Lett. 9 3 (2004) 131301 [h e p - th /0 4 0 4 1 5 6 ] [i nSPIRE].
[41] J. A m b j0rn, J. Jurkiew icz an d R. Loll, Spectral d im en sio n o f the universe, Phys. Rev. Lett.
9 5 (2005) 171301 [h e p -th /0 5 0 5 1 1 3 ] [i nSPIRE].
[42] J. A m b j0rn, A. G orlich, J. Jurkiew icz an d R. Loll, The nonperturbative quantum de S itte r universe, P hys. Rev. D 78 (2008) 063544 [a r X iv :0 8 0 7 .4 4 8 1 ] [i nSPIRE].
[43] J. A m b j0rn, A. G orlich, J. Jurkiew icz an d R. Loll, P lanckian birth o f the quantum de S itte r universe, P hys. Rev. Lett. 1 0 0 (2008) 091304 [a r X iv :0 7 1 2 .2 4 8 5 ] [i nSPIRE].
[44] J. A m b j0rn, S. Jo rd an , J. Jurkiew icz an d R. Loll, Second- and first-o rd er phase transitions in CD T, Phys. Rev. D 8 5 (2012) 124044 [a r X iv :1 2 0 5 .1 2 2 9 ] [i nSPIRE].
[45] J. A m b j0rn, S. Jo rd an , J. Jurkiew icz an d R. Loll, A second-order phase tra n sitio n in CD T, Phys. Rev. Lett. 1 0 7 (2011) 211303 [a r X iv :1 1 0 8 .3 9 3 2 ] [i nSPIRE].
[46] J. A m b j0rn, J. Jurkiew icz a n d R. Loll, Sem iclassical universe fro m fir s t principles, Phys.
Lett. B 6 0 7 (2005) 205 [h e p -th /0 4 1 1 1 5 2 ] [i nSPIRE].
[47] J. A m b j0rn, A. G orlich, J. Jurkiew icz, R. Loll, J. G izb ert-S tu d n ick i an d T. Trzesniewski, The sem iclassical lim it o f causal dynam ical triangulations, Nucl. Phys. B 8 4 9 (2011) 144 [a r X iv :1 1 0 2 .3 9 2 9 ] [i nSPIRE].
[48] J. A m b j0rn, J. G izbert-S tudnicki, A. G orlich an d J. Jurkiew icz, The transfer m a trix in fo u r-d im e n sio n a l C D T, JH E P 0 9 (2012) 017 [a r X iv :1 2 0 5 .3 7 9 1 ] [i nSPIRE].
[49] J. A m bj0rn, J. G izbert-S tudnicki, A. G orlich and J. Jurkiew icz, The effective action in 4-dim C D T. The transfer m a trix approach, JH E P 0 6 (2014) 034 [a r X iv :1 4 0 3 .5 9 4 0 ] [i nSPIRE].
[50] J. A m b j0rn, D.N. C oum be, J. G izb ert-S tu d n ick i an d J. Jurkiew icz, Signature change o f the m etric in C D T quantum gravity?, JH E P 0 8 (2015) 033 [a r X iv :1 5 0 3 .0 8 5 8 0 ] [i nSPIRE].
[51] D.N. C oum be, J. G izb ert-S tu d n ick i and J. Jurkiew icz, Exploring the new phase tra n sitio n o f CD T, JH E P 0 2 (2016) 144 [a r X iv :1 5 1 0 .0 8 6 7 2 ] [i nSPIRE].
[52] J. A m b j0rn, J. G izbert-S tudnicki, A. G orlich, J. Jurkiew icz, N. K litg a a rd an d R. Loll, C haracteristics o f the new phase in C D T, Eur. Phys. J. C 77 (2017) 152
[a r X iv :1 6 1 0 .0 5 2 4 5 ] [i nSPIRE].
J H E P 0 7 ( 2 0 1 9 ) 1 6 6
[53] J. A m b j0rn, D. C oum be, J. G izbert-S tudnicki, A. G orlich an d J. Jurkiew icz, N ew
higher-order transition in causal dynam ical triangulations, P hys. Rev. D 9 5 (2017) 124029 [a r X iv :1 7 0 4 .0 4 3 7 3 ] [i nSPIRE].
[54] J. A m b j0rn, A. G orlich, J. Jurkiew icz, A. K reienbuehl an d R. Loll, R en o rm a liza tio n group flow in CD T, Class. Q uant. Grav. 31 (2014) 165003 [a r X iv :1 4 0 5 .4 5 8 5 ] [i nSPIRE].
[55] J. A m b j0rn, Z. Drogosz, J. G izbert-S tudnicki, A. G orlich, J. Jurkiew icz an d D. N em eth, Im pact o f topology in causal dynam ical triangulations quantum gravity, P hys. Rev. D 94 (2016) 044010 [a rX iv :1 6 0 4 .0 8 7 8 6 ] [i nSPIRE].
[56] J. A m b j0rn, J. G izbert-S tudnicki, A. G orlich, K. G rosvenor an d J. Jurkiew icz, F our-dim ensional C D T w ith toroidal topology, Nucl. Phys. B 9 2 2 (2017) 226
[a r X iv :1 7 0 5 .0 7 6 5 3 ] [i nSPIRE].
[57] J. A m b j0rn, J. G izbert-S tudnicki, A. G orlich, J. Jurkiew icz and D. N em eth, The phase structure o f causal dynam ical triangulations w ith toroidal spatial topology, JH E P 0 6 (2018) 111 [a r X iv :1 8 0 2 .1 0 4 3 4 ] [i nSPIRE].
[58] J. A m b j0rn, D. C oum be, J. G izbert-S tudnicki, A. G orlich an d J. Jurkiew icz, Critical phen o m en a in causal dynam ical triangulations, a rX iv :1 9 0 4 .0 5 7 5 5 [i nSPIRE].
