P u b l i s h e d f o r SISSA b y S p r i n g e r R e c e i v e d: December 2, 2015
A c c e p t e d: February 10, 2016
P u b l i s h e d: February 22, 2016
Exploring the new phase transition of C D T
D .N . Coum be,a J . Gizbert-Studnicki6 and J . Jurk iew icz6
a The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
bFaculty of Physics, Astronom y and Applied Computer Science, Jagiellonian University, ul. prof. Stanislawa Ł ojasiewicza 11, Krakow, P L 30-348, Poland
E-m ail: D a n i el.Coumbe@nbi.ku.dk, jakub.gizbert-studnicki@uj.edu. pl,
jerzy.jurkiewicz@uj.edu.pl
Ab s t r a c t: This work focuses on th e newly discovered b ifurcation phase tra n sitio n of C D T q u a n tu m gravity. We define various order param eters and investigate which is m ost suitable to stu d y th is tra n sitio n in num erical sim ulations. By analyzing th e behaviour of th e order param eters we present evidence th a t th e tra n sitio n sep aratin g th e bifurcation phase and th e physical phase of C D T is likely a second or higher-order tra n sitio n , a result th a t m ay have im p o rta n t im plications for th e continuum lim it of CD T.
Ke y w o r d s: M odels of Q u an tu m Gravity, L attice M odels of G ravity
ArXi v ePr i n t: 1510.08672
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C o n te n ts
1 I n t r o d u c t i o n 1
2 D e f i n in g a n o r d e r p a r a m e t e r t o s t u d y t h e b i f u r c a t i o n t r a n s i t i o n 3
2.1 Overview 3
2.2 T h erm alization and error estim ates 4
2.3 R esults 6
3 D i s c u s s i o n a n d o u t l o o k 8
1 In tr o d u c tio n
A ssum ing only key aspects of q u a n tu m m echanics and general relativity, and including few ad d ition al ingredients, causal dynam ical trian g u latio n s (C D T s) define a p articu larly simple ap proach to q u a n tu m gravity. T he sim plicity of co n stru ctio n and plenitude of results has m ade C D T a serious contender for a n o n p e rtu rb ativ e th eo ry of q u a n tu m gravity. T here now exists strong evidence th a t C D T has a classical lim it th a t closely resem bles general relativ ity on large distance scales [1], while on short distance scales it has produced some exciting hints ab o u t th e n a tu re of spacetim e a t th e P lan ck scale, including evidence th a t th e num ber of spacetim e dim ensions m ay dynam ically reduce [2], a result th a t has also been rep o rted in num erous o th er approaches to q u an tu m gravity [3- 6].
C D T gives an app rox im ate description of continuous spacetim e via th e connectivity of an ensem ble of locally flat n-dim ensional simplices. In order to reproduce general relativ ity in th e classical lim it it seems th e in tro d u ctio n of a causality condition is a necessary requirem ent [7], such th a t th e lattic e can be foliated into spacelike hypersurfaces of fixed topology. By only including geom etries in th e p a th integral m easure th a t adm it such a foliation, th e unphysical features observed in dynam ical trian g u latio n s w ith o u t a causal
ity condition (see [8- 10] and m ore recently [11, 12]) a p p e ar to be suppressed, yielding a semi-classical geom etry th a t closely resem bles general relativity.
C D T discretises th e continuous p a th integral into a p a rtitio n function [13]
Z e = E C - e~ SEH, (1.1)
T T
and transform s th e E instein -H ilb ert action into th e discretised E instein-R egge action S E gg = — (k 0 + 6A ) No + k4 (N 4,1 + N 3,2) + A (2N 4,1 + N 3,2) • (1.2) E q u a tio n ( 1.1) is defined by th e sum over all possible trian g u latio n s T , w here CT is a sym m etry factor dividing o ut th e num ber of equivalent ways of labelling vertices in T .
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K 0
F ig u r e 1. The phase diagram of 4-dimensional CDT. Filled points denote actual measurem ents while dashed lines represent extrapolations.
C D T defines two types of 4-dim ensional trian g u latio n s, th e (4 ,1 ) and (3,2) simplices (see ref. [7] for m ore d etails). T he num ber of (4,1) simplices in eq. ( 1.2) is given by N 4,i, th e num ber of (3,2) simplices is denoted by N 3,2 and th e num ber of vertices in a tria n g u la tio n is given by N 0. E q u a tio n ( 1.2) is a function of th re e b are coupling constants: ko, A and k4. ko is inversely p ro p o rtio n al to N ew ton’s co n stan t, A defines an asym m etry p a ra m ete r quantifying th e ratio of th e length of space-like and tim e-like links on th e lattice and k4 is related to th e cosmological co n stan t, and is typically tu n ed in num erical sim ulations to a (pseudo-)critical value. Fixing k4 in this way allows one to tak e an infinite-volum e lim it, leaving a tw o-dim ensional p a ra m ete r space spanned by th e bare couplings k o and A.
T he p a ra m ete r space of C D T has now been m ap ped o ut in some detail, as shown in figure 1, and consists of four distin ct phases. P hases A and B are generally regarded as lattic e artifacts containing unphysical geom etric properties [7]. P h ase C, however, closely resem bles 4-dim ensional de S itter space on large distance scales [1]. T he possibility of tak in g a continuum lim it w ithin phase C seemed a real possibility following th e discovery of a second-order phase tra n sitio n dividing phases B and C. However, th e discovery of a fo u rth so-called bifurcation phase (D) existing betw een phases B and C makes it difficult or im possible to approach th is second-order tra n sitio n from w ithin th e physically in te rest
ing phase C. T his m otivates th e need to investigate th e location and order of th e (C-D) b ifurcation phase tra n sitio n , since if th e tra n sitio n was second-order it would re-establish th e possibility of tak in g a continuum lim it in CD T.
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2 D e fin in g an ord er p a r a m e te r to s tu d y th e b ifu r ca tio n tr a n sitio n
2 .1 O v e r v i e w
In order to locate and stu d y th e critical behaviour of th e tra n sitio n dividing th e bifurcation and de S itte r phases we seek an o rder p a ra m ete r (O P) th a t is approxim ately zero in one phase and non-zero in th e other. Hence, by tak in g th e n th -o rd e r derivative of an ap p ro p ri
ately defined order p a ra m ete r one should in principle be able to determ ine th e order of th e tra n sitio n . For exam ple, in th e infinite volum e lim it a first order tra n sitio n is characterised by a discontinuity in th e first order derivative a t th e tra n sitio n point, w hereas a continu
ous function should be observed for higher-order tran sitio n s. In num erical sim ulations one usually considers th e susceptibility x defined via th e variance of th e order p a ra m ete r O P,
Xo p = (O P 2) - (O P )2. (2.1)
One th e n searches th e p a ra m ete r space for peaks in th e susceptibility, whose presence would indicate th e existence of a (pseudo-)critical point. By m easuring how th e position of such points changes w ith increasing volum e one can in principle determ in e th e location of th e tra n sitio n in th e infinite volum e lim it via ex trap o latio n . C ritical exponents can also be determ ined using th e sam e m ethod, th ereb y helping to determ ine th e order of th e tran sitio n .
It is im p o rta n t to carefully define a suitable order p aram eter. A good order p a ra m ete r should c a p tu re th e tru e n a tu re of th e tra n sitio n and provide a strong signal/noise ratio.
We now investigate various order p aram eters to find one th a t gives th e strongest signal of th e bifurcation tra n sitio n , and therefore is th e m ost suitable to m easure its precise location and order. O rder p aram eters analysed in this article can be divided into two m ajo r groups. T he first group com prises order p aram eters which c a p tu re only global features of C D T trian g u latio n s. Such global order p aram eters have already been proposed in refs. [14, 15], w here th ey were used to locate and analyse th e previously discovered A-C and B-D tra n s itio n s .1 Exam ples of such global O P s include: N o, N 1, N 2 and N 4 which denote th e to ta l num ber of vertices, links, triangles and 4-simplices in a trian g u latio n , respectively. We have analysed all of th e above O Ps, finding sim ilar qu alitativ e behaviour.
In th e following sections we will focus on a p a rticu la r com bination, nam ely
OPo = conj(A ) = 2N4,i + N3,2 - 6No. (2.2) In order to analyse th e b ifurcation tra n sitio n we perform ed a series of m easurem ents of th is O P for a range of bare coupling co n stan ts th a t begin in phase D and end in phase C. We stu d y a p a rticu la r p a th w ithin th e phase diagram for which we fix k0 = 2.2 and vary A . Therefore O P o given by eq. (2.2) , which is conjugate to A in th e bare C D T action ( 1.2) , seems to be a p articu larly good choice. T he sam e order p a ra m ete r was also used in refs. [14, 15] to analyse th e form er B -D 1 tra n sitio n in a sim ilar way.
T he second group of o rder p aram eters focuses on m icroscopic geom etric properties of th e underlying C D T trian g u latio n s. It was shown in ref. [16] th a t th e d istrib u tio n of volum e in th e bifurcation phase is m arkedly different th a n in phase C, w ith spatial
1As we now know th a t phase D exists, th e form er “B -C ” tra n sitio n now becom es th e B-D transition.
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volum e con centrated in clusters connected by vertices of very large coordination num ber (th e num ber of 4-simplices sharing a given vertex). T his change of th e geom etric stru c tu re can be exploited to signal th e phase tra n sitio n . Inside th e bifurcation phase b o th th e average scalar cu rv atu re R (t) = 2 n — C (where C = 6 arccos(1/3) — 2n > 0) and th e m axim al coordination num ber of a vertex O (v (t)) differ significantly betw een sp atial slices of odd and even tim e t, w hereas th ere is no such difference in phase C. One can quantify th is difference by defining th e order param eters [16]
O P \ = |-R(to) — -R(to + 1) | (2.3)
and
O P2 = m a x [O (v (t0)] — m a x [O (v (t0 + 1)] , (2.4) w here th e (integer) tim e to is chosen to be th e closest to th e centre of volum e of a triangu - la tio n .2 A detailed analysis of all th re e order p aram eters is presented in section 2.3.
2 .2 T h e r m a l i z a t i o n a n d e r r o r e s t i m a t e s
W hen perform ing M onte C arlo sim ulations it is im p o rta n t to ensure th e lattice is th er- m alized before beginning to tak e m easurem ents. E nsuring therm alizatio n is p articu larly im p o rta n t to th is work as we aim to explore phase tra n sitio n lines th a t are typically as
sociated w ith very long auto-co rrelation lengths. For each of our m easurem ent series we perform ed th erm alizatio n checks by dividing th e d a ta series into two sets and statistically com paring them . An exam ple of such a check is presented in figure 2 w here we plot th e O P 2 order p a ra m ete r defined in eq. (2.4) as a function of M onte Carlo tim e (pro po rtio nal to th e n um ber of a tte m p te d moves). We check w h ether a given configuration range is therm alized by sp littin g th e d a ta set in two and com paring th e average and sta n d a rd deviation of each set. A com parison betw een th e two d a ta sets gives good sta tistic a l agreem ent, as shown in figure 2 . We find th e longest au to co rrelation lengths closest to th e phase tran sitio n , and th a t th e au to co rrelatio n tim e increases w ith to ta l volume. A t th e tra n sitio n point, th e order p a ra m ete r tu n n els betw een two m etastab le values, w ith th e frequency of tra n sitio n decreasing for larger to ta l volumes (see figure 2 (righ t)). T he sta tistic a l agreem ent be
tw een subsets of d a ta for th e larger volum e ensem ble is slightly worse th a n for th e sm aller ensem ble because for th e sam e physical sim ulation period we observe fewer m etastab le tran sitio n s, m eaning local variations have had less tim e to average o u t.3
W hen perform ing M onte C arlo sim ulations it is also im p o rta n t to accurately estim ate sources of sta tistic a l errors. S tatistical errors in th is work are calculated using a single
elim ination (binned) jackknife procedure, after blocking th e d a ta to account for au to co rre
lation errors. W hen au to co rrelatio n errors are im p o rta n t th e sta tistic a l error increases w ith increasing block size, and w hen auto co rrelatio n errors are insignificant th e error is largely independent of block size. For th is reason we calculate th e associated error for various block sizes, selecting th e block size for which th e sta tistic a l error is m axim ised. An exam ple of
2In our approach th e discrete centre of volume t 0 is defined up to one tim e slice, therefore to calculate O P i and O P 2 we first choose t 0 and m easure 3 values of O P for t 0 —1, t 0 and t 0+ 1 and th e n choose th e highest one.
3For th e 160k ensemble we only observe two m etastab le tran sitio n s over th e entire sim ulation period of alm ost nine m onths, so th e average tra n sitio n period is around th ree m onths.
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^4,1 = 80k n41 = 160k
o p2 o p2
F ig u r e 2. An example therm alization check based on the O P 2 order param eter. The order param eter is plotted as a function of sim ulation tim e (proportional to the num ber of attem p ted Monte Carlo moves) for our point closest to th e phase transition and for lattice volumes of N4ji = 80, 000 (left) and N 4ji = 160,000 (right), respectively. The d a ta is divided into two subsets (blue and red), whose statistical properties are compared. The m ean value is denoted by a solid line and the dashed lines indicate 1 stan d ard deviation error bounds.
F ig u re 3 . The statistical error of the susceptibility x o p2 calculated for the point closest to the phase transition and for lattice volumes of N 4ji = 80,000 (left) and N 4ji = 160,000 (right), respectively.
The d a ta set is divided into blocks of identical size and then a single-elimination (binned) jackknife procedure is used to determ ine the statistical error. The size of the error depends on the num ber of blocks. We take the largest value (red dashed line) as our final error estim ate.
such a procedure is presented in figure 3 w here we plot th e error in th e m easurem ent of th e susceptibility Xo p2 a t th e point closest to th e phase tra n sitio n . T he erro r is estim ated by a jackknife procedure for each block size and is p lo tted as a function of th e num ber of blocks.
T he erro r typically increases w ith th e num ber of blocks, eventually stabilising around a c o n sta n t, as shown for th e sm aller volum e ensem ble presented in figure 3 (left). In some cases th e largest error is observed for a small num ber of blocks, which app ears to be th e case for th e larger volum e ensem ble close to th e phase tra n sitio n point (see figure 3 (rig h t)).
As already discussed, for th is em pirical d a ta we observe only two m etastab le tran sitio n s in th e o rder p a ra m ete r over th e entire sim ulation p erio d , th is likely m eans th e jackknife procedure is overestim ating th e error. We ad o p t a cautious a ttitu d e and tak e th e highest value as our erro r estim ate.
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F ig u re 4 . The m ean value (OP) as a function of A for three different order param eters OPo (left), O P i (centre) and O P 2 (right) and for two different lattice volumes N 4j1 = 80,000 (top) and N4 1 = 160, 000 (bottom ). O P 1 and O P 2 b o th clearly change around A = 0.27 — 0.325 and A = 0.325—0.375 for N 4 1 = 80, 000 and N 4 1 = 160, 000, respectively, suggesting a phase transition.
However, there is no clear signal of a transition when using OPo.
2 .3 R e s u l t s
We now present th e results of our order p a ra m ete r studies. We focus on th re e order param eters defined in section 2 . Figure 4 shows th e m ean value of th e o rder p aram eters (O P) p lo tted as a function of A for fixed ko = 2.2 and for two different lattic e volumes N 4,1 = 80,000 and N 4,1 = 160,000. One clearly sees th a t all order param eters ten d to zero (or a co n stant) for large A (inside phase C) and increase in value for sm aller A (inside phase D ). A clear change in behaviour of O P 1 and O P 2 can be seen around A = 0.27 — 0.325 and A = 0.325 — 0.375 for system s w ith 80, 000 and 160, 000 simplices of ty p e (4,1), respectively, w hereas th ere is no clear signal of th e tra n sitio n using th e p a ra m ete r O P 0.
In figure 5 we plot th e susceptibility %OP of each order p a ra m ete r defined in eq. (2.1) . A clear signal of th e phase tra n sitio n is observed only for th e O P 2, w here one can see a peak of susceptibility at th e (pseudo-)critical points A crit(80k) = 0.30 ± 0.01 and A crit(160k) = 0.35±0.01. Interestingly, if one plots th e ratio %OP/( O P ) one can also observe th e tra n sitio n peaks using O P 1 (see figure 6) .
T he above results indicate th a t for th e bifurcation tra n sitio n th e d etails of th e geom etry play an im p o rta n t role, and therefore order p aram eters based solely on global properties of th e tria n g u la tio n do not cap tu re these details. T h e central difference betw een phase C and phase D is related to th e form ation of periodic clusters of volum e aro un d singular vertices, which form a kind of tu b e stru c tu re (see ref. [16] for d etails). Such a stru c tu re does not exist in phase C, b u t is a generic p ro p erty of phase D. Therefore, in o rder to observe th e phase tra n sitio n it is im p o rta n t to analyse th e m icroscopic sim plicial geom etry. Even order p aram eters such as O P1 only c a p tu re general features of th e geom etry (i.e. th e difference
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F ig u r e 5. The susceptibility x o p as a function of A m easured for three different order param eters:
OPo (left), O P i (centre) and O P 2 (right), and for two different lattice volumes N4,1 = 80, 000 (top) and N4 1 = 160,000 (bottom ). The (pseudo-)critical A value at which the bifurcation transition occurs appears to be at A crlt = 0.30 ± 0.01 for N4 1 = 80, 000 and at A crlt = 0.35 ± 0.01 for N 4 1 = 160, 000, as determ ined using O P 2.
Na i = 8 0 k
A
X (O P i)
A
X(O Po) / <OP0> X (O Pi )/< O Pi> x(0P2 )/<o p2>
x(o p2 )/<OP2>
F ig u r e 6. The ratio x o p /(O P ) as a function of A measured for three different order param eters OPo (left), O P i (centre) and O P2 (right) for two different lattice volumes N4,i = 80,000 (top) and N4,i = 160, 000 (bottom ). The (pseudo-)critical A value at which the bifurcation transition occurs appears to be at A crlt = 0.30± 0.01 for N4,i = 80, 000 and at A crlt = 0.35± 0.01 for N4,i = 160,000, as determ ined via the order param eters O P i and O P2.
Na 1 = 8 0 k N 4 1 = 8 0 k
A
A
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in average curv atu re for different tim e slices), and are therefore not capable of cap tu rin g th e m icroscopic d etails of th e phase tra n sitio n . T his sim ple observation explains why th e existence of th e b ifurcation phase went unnoticed during previous phase tra n sitio n studies.
3 D is c u ss io n an d o u tlo o k
S ta rtin g from a point in th e p a ra m ete r space w ith good semi-classical features, th e hope is th a t one can establish a continuum lim it by approaching a second order tra n sitio n , thereb y defining a sm ooth in terp o latio n betw een th e low and high energy regim es of C D T . T he infinite correlation length associated w ith such a tra n sitio n should allow one to shrink th e lattic e spacing to zero while keeping observables fixed in physical units. Such a continuous tra n sitio n has been shown to exist in th e C D T p a ra m ete r space [14, 15] and was originally th o u g h t to divide th e semi-classical phase C from phase B. However, recent results [16, 17] show th a t a new bifurcation phase (D) exists betw een phases C and B, which m ay prevent th e possibility of tak in g a continuum lim it from w ithin phase C. A nalysing th e new tra n sitio n betw een phases C and D is therefore very im p o rta n t, since a second order tra n sitio n would re-establish th e possibility of defining a continuum lim it. To stu d y this tra n sitio n one m ust define an order p a ra m ete r which signals th e tra n sitio n . In th is article we have analysed two groups of order p aram eters, related to general and detailed features of th e C D T sim plicial geom etries, respectively. We have shown th a t th e p aram eters from th e first (general) group, which were used in previous phase tra n sitio n studies, do not work well w ith th e new phase tra n sitio n . However, th e second (detailed) group of order p aram eters give a clear tra n sitio n signal. A m ong th e num erous order p aram eters tested , th e strongest tra n sitio n signal was given by O P 2, as defined by eq. (2.4) .
T he order of th e new b ifurcation tra n sitio n rem ains an open question, altho ug h at least we now have an order p a ra m ete r capable of determ ining it. It seems th a t th e order p a ra m ete r m easured a t th e (pseudo-)critical point jum p s betw een two different values (see figure 2) and th a t th e frequency of such ju m p s decreases w ith increasing volume. This result m ay suggest th a t th e tra n sitio n is first order. This is illu strated in figure 7 w here we plot a histogram of th e O P 2 (norm alised by th e lattic e volume) m easured for two different volum es N 4,i = 80,000 (blue) and N 4,4 = 160,000 (red), respectively. By fittin g a double G aussian function to th e m easured d a ta we observe two clearly sep arated p eaks.4 T he peak sep aratio n is slightly sm aller for a larger to ta l volume. A sim ilar situ a tio n was previously observed a t th e ‘o ld ’ B-C (now called th e B-D) phase tra n sitio n (which is very likely second order) [15], w here th e peak sep aratio n reduced w ith increasing volume.
M easuring th e behaviour of th e order p a ra m ete r for a num ber of different lattice vol
um es will enable us to calculate critical exponents and to analyse th e o rder of th e phase tra n sitio n in detail. T his work is still in progress, however prelim inary results are prom is
ing. In figure 8 we plot th e position of (pseudo-)critical points A crit as a function of lattice volum e N 4,1. Using this em pirical d a ta we fit th e function
A crlt(N4,i) = A critM - a ■ N 4,i-1/v (3.1)
4As we are able to establish th e phase tran sitio n point only w ith finite precision th e height of th e two peaks is different. T he peaks would be th e sam e height a t th e (pseudo-)critical point.
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N41= 8 0 k , A =0.30 N41= 1 6 0 k , A =0.35
F ig u re 7 . A histogram of the O P 2/N 4ji order param eter m easured at the phase transition point for two different lattice volumes N4ji = 80, 000 (blue) and N4ji = 160, 000 (red). We fit the histogram d a ta to a double Gaussian function (solid line). The position of the two peaks is m arked by dashed lines. The peak separation appears to shrink slightly with increasing lattice volume.
F ig u re 8 . Prelim inary results of C-D phase transition dependence on lattice volume. The (pseudo- )critical points A crlt were estim ated for fixed k0 = 2.2 and various to tal volumes N 4ji by looking at peaks in susceptibility x OP2 as described in section 2. The solid red line corresponds to a fit of eq. (3.1) to the m easured d a ta (v = 2.6), while the dashed blue line uses the same fit bu t w ith a critical exponent of v = 1.
and estim ate th e critical exponent v = 2.6 ± 0.6 (solid red line in figure 8) . T his value of v suggests a continuous tra n sitio n . For com parison we also m ade a fit using a fixed value of v = 1 th a t would correspond to a first order tra n sitio n (dashed blue line in figure 8), which cann o t be com pletely excluded b u t appears m uch less likely. We are curren tly collecting d a ta a t th e C -D tra n sitio n for addition al lattic e volumes as well as increasing sta tistic s of previous m easurem ents. U nfortunately, th is process is co m p u tatio n ally very tim e consum ing and a com prehensive stu d y of th e bifurcation tra n sitio n o rder will be presented in a sep arate article.
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A c k n o w le d g m e n ts
T he au th o rs wish to acknowledge th e su p p o rt of th e g ran t D E C -2 0 1 2 /0 6 /A /S T 2 /0 0 3 8 9 from th e N atio n al Science C entre Poland.
O p e n A c c e s s . This article is d istrib u ted under th e term s of th e C reative Com m ons A ttrib u tio n License (CC -B Y 4.0) , which perm its any use, d istrib u tio n and reprodu ction in any m edium , provided th e original au th o r(s) and source are credited.
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