Euclidean Dynamical Triangulation revisited: is the phase transition really first order?
CCERN-PH-TH/2013-261
Euclidean Dynamical Triangulation revisited:is the phase transition really first order?
Tobias Rindlisbacher ∗ Institute for Theoretical Physics, ETH Zürich, CH - 8093 Zürich, SwitzerlandE-mail: [email protected]
Philippe de Forcrand
Institute for Theoretical Physics, ETH Zürich, CH - 8093 Zürich, Switzerland andCERN, Physics Department, TH Unit, CH-1211 Genève 23, SwitzerlandE-mail: [email protected]
The transition between the two phases of 4D Euclidean Dynamical Triangulation [1] was longbelieved to be of second order until in 1996 first order behavior was found for sufficiently largesystems [3, 4]. However, one may wonder if this finding was affected by the numerical methodsused: to control volume fluctuations, in both studies [3, 4] an artificial harmonic potential wasadded to the action; in [4] measurements were taken after a fixed number of accepted instead of attempted moves which introduces an additional error. Finally the simulations suffer from strongcritical slowing down which may have been underestimated.In the present work, we address the above weaknesses: we allow the volume to fluctuate freelywithin a fixed interval; we take measurements after a fixed number of attempted moves; and weovercome critical slowing down by using an optimized parallel tempering algorithm [6]. Withthese improved methods, on systems of size up to N =
64k 4-simplices, we confirm that thephase transition is first order. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov DT revisited: is the phase transition really first order?
Tobias Rindlisbacher
1. Introduction
The transition between the two phases of 4D Euclidean Dynamical Triangulation (EDT) [1]was first believed to be of 2 nd order until 1996 where in [3] for the first time 1 st order behaviorwas reported for a system consisting of N =
32k 4-simplices. Shortly afterwards this finding wasverified in [4] and extended to larger systems with N = accepted (instead of attempted) moves,which introduces a systematic error.2. The use of an artificial harmonic potential to control volume fluctuations also introduces asystematic error.3. Autocorrelation and thermalization time could easily have been underestimated.Therefore we wanted to crosscheck these old results with our own, hopefully correct code whichsatisfies detailed balance, uses a potential well instead of a harmonic potential to control volumefluctuations, and makes use of parallel tempering to cope with critical slowing down. In 4-dimensional Euclidean Dynamical Triangulation (EDT) [1] the formal path integral for
Euclidean (local SO ( ) instead of SO ( , ) symmetry) gravity , Z = (cid:90) D (cid:2) g µν (cid:3) e − S EH [ g µν ] , (1.1)with S EH = − π G (cid:82) d x √ g ( R − Λ ) being the Einstein-Hilbert action , is regularized by approxi-mating the configuration space (space of all diffeomorphism inequivalent 4-metrics) with the spaceof simplicial piecewise linear (PL) manifolds consisting of equilateral 4-simplices with fixed edgelength a (such manifolds are also called abstract triangulations ). Under such a discretization, theEinstein-Hilbert action turns into the Einstein-Regge action which for equilateral 4-simplices anda space-time of topology S takes the simple form S ER = − κ N + κ N , where the N i label thenumber of i -simplices in the PL manifold, κ = V G and κ = ( / ) V + Λ V π G , with V n = a n √ n + n ! √ n being the volume of a n -simplex and G , Λ the gravitational and cosmological coupling respectively.The partition function (1.1) can now be written as Z ( κ , κ ) = (cid:88) T C T e κ N ( T ) − κ N ( T ) = (cid:88) N Z ( κ , N ) e − κ N , (1.2)where the sum after the first equality sign runs over all abstract triangulations T of S and C T is foreach T a corresponding symmetry factor . The second equality sign defines the canonical partitionfunction Z ( κ , N ) . The partition function (1.2) is suitable for use in a Markov chain Monte Carlosimulation with Metropolis updates consisting of the so-called Pachner moves [1]. C T is assumed to be ∼ DT revisited: is the phase transition really first order?
Tobias Rindlisbacher
The grand canonical partition function (1.2) is finite only if κ > κ cr ( κ ) . We therefore have acritical line for convergence in the ( κ , κ ) -plane, given by κ cr ( κ ) . To obtain the thermodynamiclimit ( N → ∞ ) we have to ensure that κ N → ∞ −→ κ cr ( κ ) . For quasi-canonical simulations aroundsome fixed volume ¯ N , it follows from (1.2) that we can define a pseudo-critical κ pcr ( κ , ¯ N ) = ∂ ln ( Z ( κ , N )) ∂ N (cid:12)(cid:12) N = ¯ N , which corresponds to the value of κ for which the N -distribution is flat around¯ N . Κ Κ (cid:45) . Κ critical point pseudo critical points N (cid:61) N (cid:61) N (cid:61) N (cid:61) N (cid:61) N (cid:61) N (cid:61) N (cid:61) c r u m p l ed e l o n g a t e d Figure 1:
Phase diagram for 4D EDT. The Figureshows κ pcr ( κ , N ) as a function of κ for different N together with the corresponding pseudo-criticalpoints (cid:0) κ pcr ( N ) , κ pcr (cid:0) κ pcr ( N ) , N (cid:1)(cid:1) . The dottedred line separates the crumpled from the elongated phase; in the limit N → ∞ this line ends at the crit-ical point : ( κ cr , κ cr ) . To improve readability, the y-axis shows ( κ − . κ ) instead of κ itself. N (cid:144) N p r obab . den s . Κ (cid:61) Κ (cid:61) Κ (cid:61) Figure 2: N -distribution for systems of size N =
32k (blue), 48k (red) and 64k (green): the solid linesare double-Gaussian fits to the data. It can be seenthat the double peak structure becomes more pro-nounced with increasing system size and there is nosign that the peaks will merge again in the thermody-namic limit. This is characteristic of a 1 st order tran-sition. For constant N , we can define (see Fig. 1) a line κ pcr ( κ , N ) as a function of κ , along whichtwo phases are separated by a pseudo-critical point at κ = κ pcr ( N ) . For κ < κ pcr ( N ) we are inthe crumpled phase where a typical configuration is highly collapsed in the sense that the distancebetween any two 4-simplices is very short, leading to a large (infinite) Hausdorff dimension. For κ > κ pcr ( N ) we are in the elongated phase with Hausdorff dimension ∼
2, where a typicalconfiguration consists of a so-called baby-universe tree : the total volume is subdivided into smallerparts, the baby-universes , which are pairwise connected by only a small neck. This structure ishierarchical in a treelike manner: consider the largest baby-universe as “mother” with outgrowingsmaller “babies” which in turn give birth to their own “babies”, and so on (see Fig. 3).The true critical point in the thermodynamic limit is obtained as ( κ cr , κ cr ) = lim N → ∞ (cid:0) κ pcr ( N ) , κ pcr (cid:0) κ pcr ( N ) , N (cid:1)(cid:1) . (1.3)
2. Simulation Methods
We are interested in the canonical system, but as there is no set of ergodic moves known for3
DT revisited: is the phase transition really first order?
Tobias Rindlisbacher
Figure 3:
Representative configurations in the crumpled (left, κ = .
24) and elongated (right, κ = . N ≈ the space of triangulations of fixed volume, we can only run quasi-canonical Markov chain MonteCarlo simulations with local updates consisting of the five Pachner moves.
Calling T k the current triangulation in our Markov chain, we obtain T k + as follows:1. randomly choose a move type n ∈ { , . . . , }
2. randomly choose one of the N T k and call it D
3. randomly choose one of the (cid:0) n + (cid:1) n -simplices contained in D and call it S
4. perform a Metropolis test with acceptance probability p n ( T k , S ) : • accept: T k + is obtained from T k by applying the n -move at S • reject: T k + = T k The acceptance probability at step 4 is given by [5] p n ( T , S ) = (cid:40) p n ( N ( T )) if n -move possible at S ∈ T else , (2.1)where p n ( N ) = min (cid:110) , N N + ∆ N ( n ) e κ ∆ N ( n ) − κ ∆ N ( n ) (cid:111) is the so-called reduced transition probabil-ity , ∆ N i ( n ) labels the change of N i under a n -move, and a n -move is considered as possible at S if S is contained in ( − n ) and the application of the move does not violate the manifoldconstraint which requires that the move will not produce any simplex which already exists in thetriangulation. I.e. grand canonical simulations where the volume is constrained to fluctuate around some desired volume ¯ N . If a n -simplex is contained in ( − n ) ( n + ) ( d − n ) -simplex (orthogonal/dual to the initial n -simplex). This is precisely the action of a Pachner n -move. DT revisited: is the phase transition really first order?
Tobias Rindlisbacher
In previous work [1, 3, 4], the volume was controlled by adding a harmonic potential, U = δ ( N − ¯ N ) to the Einstein-Regge action. This of course introduces a systematic error for allmoves which change N . We therefore decided to rather use an infinite potential well of somereasonable width w ≈ σ ( N ) / .
5, where 2 . = max n (cid:110) ∆ N ( n ) ∆ N ( n ) (cid:111) and σ ( N ) is the square root of the N -susceptibility.With such a potential well we can not use the saddle point expansion method from [5] to tune κ to its pseudo-critical value κ pcr ( κ , ¯ N ) . Instead we made use of a method mentioned in [2]: as the N -histogram has to be flat around ¯ N if κ = κ pcr ( κ , ¯ N ) , we have that¯ p geo ( ¯ N ) p pcr ( ¯ N ) = ¯ p geo ( ¯ N + ∆ N ( )) p pcr ( ¯ N + ∆ N ( )) , (2.2)where p pcrn ( N ) is the reduced transition probability p n ( N ) from Sec. 2.1 with κ = κ pcr ( κ , ¯ N ) and ¯ p geon ( N ) is the average probability to select within a configuration of size N a n -simplexwhere a n -move can be applied (we call these geometric probabilities ). One can then solve for κ pcr ( κ , ¯ N ) which leads to κ pcr ( κ , ¯ N ) = ∆ N ( ) (cid:20) ln (cid:18) ¯ p geo ( ¯ N ) ¯ p geo ( ¯ N + ∆ N ( )) (cid:19) − ln (cid:18) + ∆ N ( ) ¯ N (cid:19)(cid:21) + ∆ N ( ) ∆ N ( ) κ . (2.3)As the 4-move is always possible, only ¯ p geo has to be measured. Critical slowing down is much worse for 1 st order transitions than for 2 nd order. This is becausetransitions between the two phases are exponentially suppressed with increasing system size. Toovercome this problem, we use an optimized parallel tempering algorithm as described in [6]: fora fixed average volume ¯ N , we simulate in parallel 48 systems (called replicas ) along the pseudo-critical line κ pcr ( κ , ¯ N ) , such that they connect a region with fast relaxation in the crumpled phasewith a region with fast relaxation in the elongated phase and thereby pass through the critical regionaround κ pcr ( ¯ N ) . At regular intervals, a swap of configurations between neighboring ensembles isattempted. This motion of configurations through coupling space permits a faster evolution. Westart with equally spaced κ values and apply after some runtime the optimization procedure of [6]which gives us a new set of couplings for which the configuration exchange between replicas ismore frequent. A study of the efficiency of this procedure can be found in [6].
3. Data Analysis and Results
Due to the use of a potential well instead of a harmonic potential to control the system volumeand due to the tuning of κ to its pseudo-critical value, we have significant volume fluctuations inthe data which also affect for example the N distribution. To take this into account, we project thedata in the ( N , N ) -plane along the "correlation direction" before evaluating any observables, i.e.instead of N we use ¯ N = N − (cid:104) ( N − (cid:104) N (cid:105) ) ( N − (cid:104) N (cid:105) ) (cid:105)(cid:104) N − (cid:104) N (cid:105)(cid:105) ( N − (cid:104) N (cid:105) ) (3.1)5 DT revisited: is the phase transition really first order?
Tobias Rindlisbacher to evaluate observables depending on N . We checked that this leads to the same results as whenevaluating the observables only on data subsets corresponding to single, fixed N values. From nowon, when talking about the N -distribution, we mean the corrected, fixed N -version (3.1).After that, we use multi-histogram reweighting [7] with respect to κ . The errors are determinedwith the Jack-Knife method with 20 sets. In multi-histogram reweighting, these sets consist of thesimultaneous data of all the simulations at different κ values, therefore cross-correlations shouldautomatically be taken into account. N Distribution
As stated in [4], the N -distribution starts to be double-peaked for systems consisting of morethan ∼ st order transition. B A more quantitative method to determine the order of a phase transition is to study finite-sizescaling of the 4 th order Binder cumulant (Kurtosis) of the N distribution, B [ N ] = (cid:104) ( N −(cid:104) N (cid:105) ) (cid:105)(cid:104) ( N −(cid:104) N (cid:105) ) (cid:105) .According to [8], this quantity should for large N scale like B pcr [ N ] ( N ) ≈ B cr [ N ] + c N − ω , (3.2)where B cr [ N ] is the critical, infinite volume value of the Binder cumulant. For a 2 nd order transitionone should get 1 < B cr [ N ] < ω = / d H ν , where ν is the critical exponent of the correlationlength ξ N ≈ | κ cr − κ | − ν and d H the Hausdorff dimension , whereas for a 1 st order transition weshould obtain B cr [ N ] = ω = B pcr [ N ] ( N ) assuming 1 st and 2 nd order scaling ansaetze (see Figs. 4and 5). As is typical for a weak 1 st order transition, the 2 nd order fit seems to work fine, but theobtained values B cr [ N ] = − . ± . ν = / d H ω = . ± .
74 do not make much sense.Now, fixing ω = st order ansatz, it is not possible to obtain B cr [ N ] = B cr [ N ] ≈ .
7. As the Binder cumulant involves a 4 th order moment, it requires rather large statis-tics which we probably have not accumulated yet for the largest system. A fit to the data of the nextsmaller pair of systems, i.e. those consisting of 48k and 32k 4-simplices, leads to B cr [ N ] ≈ .
4. Conclusion
Our study confirms the qualitative findings of [3, 4]: for κ ≈ κ pcr ( N ) we find for N ≥
32k a clear double peak structure in the N distribution, which becomes more pronounced with The parallel tempering optimization procedure mentioned above also leads to a good distribution of simulationpoints for the reweighting. DT revisited: is the phase transition really first order?
Tobias Rindlisbacher (cid:165) (cid:165) N B c r (cid:64) N (cid:68) Figure 4:
Binder cumulant B pcr [ N ] as a function of1 / N . The red lines correspond to fits of the form(3.2) with ω = st order transition) to the data ofthe largest and second largest pair of systems. Thevalue B cr [ N ] ≈ . B cr [ N ] ≈ .
10 5020 30151.82.02.22.42.62.8 L pcr B c r (cid:64) N (cid:68) Figure 5:
Binder cumulant B pcr [ N ] as a functionof average linear system size L pcr ( N ) , assuming a2 nd order transition, together with a fit of the form(3.2) where N ω = ( L pcr ) / ν . We also included higherorder corrections. It can be seen that the fit seems towork fine, but the obtained values B cr [ N ] = − . ± . ν = . ± .
74 do not make much sense. increasing system size (and there is no sign that the two peaks will eventually merge again in thethermodynamic limit). This is characteristic of a weak 1 st order transition. But the standard 1 st orderansatz for finite-size scaling of B pcr [ N ] did not work so far; presumably our systems are still toosmall and we have to include higher order scaling corrections. References [1] J. Ambjorn, J. Jurkiewicz,
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