Critical behaviour in the QCD Anderson transition
aa r X i v : . [ h e p - l a t ] D ec Critical behaviour in the QCD Anderson transition
Matteo Giordano ∗ † Institute for Nuclear Research of the Hungarian Academy of Sciences,Bem tér 18/c, H-4026 Debrecen, HungaryE-mail: [email protected]
Tamás G. Kovács † Institute for Nuclear Research of the Hungarian Academy of Sciences,Bem tér 18/c, H-4026 Debrecen, HungaryE-mail: [email protected]
Ferenc Pittler † Institute for Nuclear Research of the Hungarian Academy of Sciences,Bem tér 18/c, H-4026 Debrecen, HungaryE-mail: [email protected]
We study the Anderson-type localisation-delocalisation transition found previously in the QCDDirac spectrum at high temperature. Using high statistics QCD simulations with N f = + ∗ Speaker. † Supported by the Hungarian Academy of Sciences under “Lendület” grant No. LP2011-011. TGK and FP ac-knowledge partial support by the EU Grant (FP7/2007 -2013)/ERC No. 208740. We also thank the Budapest-Wuppertalgroup for allowing us to use their code to generate the gauge configurations. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ritical behaviour in the QCD Anderson transition
Matteo Giordano W c E WE c ( W )extended localised T c λ T λ c ( T )extended localised Figure 1:
Schematic phase diagram of the 3D Anderson model (left) and QCD (right) in the disor-der/eigenvalue plane.
1. Introduction
It is well known that the properties of the low-lying modes of the Dirac operator are intimatelyrelated to the behaviour of QCD under chiral symmetry transformations, as clearly exemplified bythe Banks-Casher relation [1]. In particular, it has been realised in recent years that their localisa-tion properties change completely across the chiral transition/crossover. While below the criticaltemperature T c all the eigenmodes are delocalised, it has been shown [2, 3, 4, 5] that above T c the low-lying ones, up to some critical point l c , become localised; modes above l c remain delo-calised. Initially the evidence for this was mainly obtained in the quenched approximation and/orfor the SU ( ) gauge group, but recently this scenario has been demonstrated in full QCD [5], bystudying the spectrum of the staggered Dirac operator in numerical simulations of lattice QCDwith N f = + T c , is a well known phenomenon in condensed matter physics, and it repre-sents the main feature of the celebrated Anderson model [8] in three dimensions. The Andersonmodel aims at a description of electrons in a “dirty” conductor, by mimicking the effect of impuri-ties through random interactions. In its lattice version, the model is obtained by adding a randomon-site potential to the usual tight-binding Hamiltonian, H = (cid:229) n e n | n ih n | + (cid:229) n (cid:229) m = | n + ˆ m ih n | + | n ih n + ˆ m | , (1.1)where | n i denotes a state localised on the lattice site n , and e n are random variables drawn fromsome distribution, whose width W measures the amount of disorder, i.e., of impurities in the system.The phase diagram of this model is sketched in Fig. 1. While for W = E c separating localised and delocalised modes is called “mobility edge”,and its value depends on the amount of disorder, E c = E c ( W ) . As W increases, E c moves towardsthe center of the band, and above a critical disorder W c all the modes become localised. From thephysical point of view, this signals a transition of the system from metal to insulator.2 ritical behaviour in the QCD Anderson transition Matteo Giordano
In Fig. 1 we also sketch a schematic phase diagram for QCD. Here the role of disorder isplayed by the temperature, while the energy is replaced by the eigenvalue of the Dirac operator.Localised modes are present in the low end of the spectrum above T c , up to the “mobility edge” l c ( T ) . Around the critical temperature l c vanishes [5], and below T c all the modes are extended.In both models, localised modes appear where the spectral density is small. One then expectsthat they are not easily mixed by the fluctuations of the random interaction, which in turn suggeststhat the corresponding eigenvalues are statistically independent, obeying Poisson statistics. On theother hand, eigenmodes remain extended in the region of large spectral density also in the presenceof disorder, and so one expects them to be basically freely mixed by fluctuations. The correspond-ing eigenvalues are then expected to obey the Wigner-Dyson statistics of Random Matrix Theory(RMT). This connection between localisation of eigenmodes and eigenvalue statistics provides aconvenient way to detect the localisation/delocalisation transition and study its critical properties.The transition from Poisson to RMT behaviour in the local spectral statistics is most simplystudied by means of the so-called unfolded level spacing distribution (ULSD). Unfolding con-sists essentially in a local rescaling of the eigenvalues to have unit spectral density throughout thespectrum. The ULSD gives the probability distribution of the difference between two consecu-tive eigenvalues of the Dirac operator normalised by the local average level spacing. The ULSDis known analytically for both kinds of behaviour: in the case of Poisson statistics it is a simpleexponential, while in the case of RMT statistics it is very precisely approximated by the so-called“Wigner surmise” for the appropriate symmetry class, which for QCD is the unitary class, P Poisson ( s ) = e − s , P RMT ( s ) = p s e − p s . (1.2)Rather than using the full distribution to characterise the local spectral statistics, it is more practicalto consider a single parameter of the ULSD. Any such quantity, having different values for Poissonand RMT statistics, can be used to detect the Poisson/RMT transition. In our study, we used theintegrated ULSD and the second moment of the ULSD, I l = Z s ds P l ( s ) , s ≃ . , h s i l = Z ¥ ds P l ( s ) s , (1.3)defined locally in the spectrum. The choice of s was made in order to maximise the differencebetween the Poisson and RMT predictions, namely I Poisson ≃ .
398 and I RMT ≃ . h s i Poisson = h s i RMT = p /
2. Numerical results
The results presented here are based on simulations of lattice QCD using a Symanzik-improvedgauge action and 2 + N t = b = .
75, correspondingto lattice spacing a = .
125 fm and physical temperature T =
394 MeV ≃ . T c . For differentchoices of spatial size L = , , , , , , ,
56 in lattice units, we collected large statisticsfor eigenvalues and eigenvectors of the staggered Dirac operator in the relevant spectral range - seeRef. [7] for more details. Here and in the following the eigenvalues l are expressed in lattice units.Unfolding was done by ordering all the eigenvalues obtained on all the configurations (for a given3 ritical behaviour in the QCD Anderson transition Matteo Giordano I λ λ L Poisson RMT 2432404856 h s i λ λ L Poisson RMT 2432404856
Figure 2:
Integrated ULSD (left) and second moment of the ULSD (right), computed locally along thespectrum, for several lattice sizes. Here
D l = · − . volume) according to their magnitude, and replacing them by their rank order divided by the totalnumber of configurations. We then computed locally the integrated ULSD and the second momentof the ULSD, by dividing the spectrum in small bins of size D l , computing the observables in eachbin, and assigning the resulting value to the average value of l in each bin. We used several valuesfor D l , ranging from 1 · − to 6 · − .In Fig. 2 we show the integrated ULSD I l and the second moment of the ULSD h s i l , forseveral values of the spatial volume. A transition from Poisson to RMT is clearly visible, andmoreover it gets sharper and sharper as the volume of the lattice is increased. This suggests thatthe transition becomes a true phase transition in the thermodynamic limit.
3. Finite size scaling
To check if the Poisson/RMT transition in the spectral statistics (i.e., the localisation/deloca-lisation transition) is a genuine, Anderson-type phase transition, we have performed a finite sizescaling analysis, along the lines of Refs. [11, 12, 13]. The Anderson transition is a second-orderphase transition, with the characteristic length of the system x ¥ diverging at the critical point l c like x ¥ ( l ) ∼ | l − l c | − n . To determine the critical exponent n and the critical point l c , one picksa dimensionless quantity Q ( l , L ) , measuring some local statistical properties of the spectrum, andhaving different thermodynamic limits on the two sides of the transition (and possibly at the criticalpoint), i.e., lim L → ¥ Q ( l , L ) = Q Poisson l < l c (localised) , Q c l = l c (critical) , Q RMT l > l c (delocalised) . (3.1)As the notation suggests, Q ( l , L ) is computed on a lattice of linear size L . For large enough volume,and close to the critical point, finite size scaling suggests that the dependence of Q on L is of theform Q ( l , L ) = f ( L / x ¥ ( l )) . As Q ( l , L ) is analytic in l for any finite L , we must have Q ( l , L ) = F ( L / n ( l − l c )) , (3.2)with F analytic. Here we have assumed that corrections to one-parameter scaling can be neglected.4 ritical behaviour in the QCD Anderson transition Matteo Giordano n max νδννL min = 36 ∆ λ · = 1 . w · = 2 . ν ∆ λw · = 2 . L min = 36 ν w ∆ λ · = 1 . L min = 36 Figure 3:
Dependence of the fitted value of n and corresponding relative error as a function of the numberof terms n max , in the case of L min = D l · = . w · = . n on the bin size D l for the smallest fitting range (center) and on the width w of the fitting rangefor the smallest bin size (right). Here L min = If one determines l c and n correctly, the numerical data for Q ( l , L ) obtained for differentlattice sizes should collapse on a single curve, when plotted against the scaling variable L / n ( l − l c ) . We then proceeded as follows: expanding the scaling function F in powers of l − l c , one gets Q ( l , L ) = ¥ (cid:229) n = F n L n / n ( l − l c ) n . (3.3)By truncating the series to some n max and performing a fit to the numerical data, using severalvolumes at a time, one can then determine n and l c , together with the first few coefficients F n . Forour purposes, the best quantity turned out to be the integrated ULSD I l . Our fitter was based onthe MINUIT library [9]. Statistical errors were determined by means of a jackknife analysis. Tocheck for finite size effects, we repeated the fit using only data from lattices of size L ≥ L min forincreasing L min .Systematic effects due to the truncation of the series for the scaling function, Eq. (3.3), arecontrolled by including more and more terms in the series, and checking how the results change.In order to circumvent the numerical instability of polynomial fits of large order, we resorted tothe technique of constrained fits [10]. The basic idea of constrained fits is to use the availableinformation to constrain the values of the fitting parameters. In our case, they are needed only toavoid that the polynomial coefficients of higher order take unphysical values. One then checks theconvergence of the resulting parameters and of the corresponding errors as the number of termsis increased. After convergence, the resulting errors include both statistical effects and systematiceffects due to truncation [10].To set the constraints, we shift and rescale F as follows, ˜ F ( x ) = ( F ( x ) − F RMT ) / ( F Poisson − F RMT ) , so that ˜ F interpolates between 1 (localised/Poisson region) and 0 (delocalised/RMT region).The data indicate that ˜ F changes rapidly, monotonically and almost linearly between 1 and 0 over arange d x . Any reasonable definition of d x has then to satisfy 1 + ˜ F d x ≃
0. Moreover, d x providesa reasonable estimate of the radius of convergence r of the series. Furthermore, it is known that ( ˜ F n + / ˜ F n ) r → n → ¥ , and so we expect ˜ F n r n ∼ n ). One then finds that˜ F n / ( − ˜ F ) n is expected to be of order 1. This constraint was imposed rather loosely, by asking˜ F n / ( − ˜ F ) n to be distributed according to a Gaussian of zero mean and width s =
10 for n ≥
4. Wedid not impose any constraint on the coefficients F n with n <
4, as well as on n and l c . The resultsof the constrained fits converge rather rapidly as n max is increased, see Fig. 3. We went as far as n max =
9, and we used the corresponding results for the following analyses.5 ritical behaviour in the QCD Anderson transition
Matteo Giordano
SUO ν L min I λ h s i λ RMT Poisson a = 0 .
125 fm T = 394 MeV L Figure 4:
Dependence of the fitted value of n , averaged over 2 . ≤ w · ≤ . . ≤ D l · ≤ . L min . The values of n obtained in the three symmetry classes of the 3D Anderson model (symplectic, n S = . ( ) [16], unitary n U = . ( ) [14] and orthogonal n O = . ( ) [15]) are shown for comparisontogether with their errors (left). Plot of I l against h s i l for several lattice sizes (right). The effects of the choice of bin size and fitting range were checked by varying the bin size
D l and the width w of the fitting range, which was centered approximately at the critical point.The results show a slight tendency of n to decrease as D l is decreased, but it is rather stable for
D l · .
3. There is also a slight tendency of n to increase as w is decreased, becoming ratherstable for w · .
3. See Fig. 3. To quote a single value for n , we averaged the central valuesobtained for 1 ≤ D l · ≤ . ≤ w · ≤
3. As the error is also rather stable within theseranges, we quote its average as the final error on n for each choice of L min . We have checked thatother prescriptions (e.g., extrapolating to vanishing w and/or D l , or changing – within reasonablebounds – the ranges of w and D l over which the final average is performed) give consistent resultswithin the errors.Concerning finite size effects, the fitted value of n increases with L min , stabilising around L min =
36, see Fig. 4. This signals that our smallest volumes are still too small for one-parameterscaling to work, and that finite size corrections are still important there. On the other hand, as thedifference between the values obtained with L min =
36 and L min =
40 is much smaller than thestatistical error, one-parameter scaling works fine for our largest volumes.The value for the critical point l c ≃ .
336 was obtained through the same procedure describedabove. As a function of L min , the fitted value of l c shows no systematic dependence, and differentchoices of L min give consistent values within the errors.Our result for the critical exponent n = . ( ) is compatible with the result obtained for thethree-dimensional unitary Anderson model n U = . ( ) [14]. This strongly suggests that the tran-sition found in the spectrum of the Dirac operator above T c is a true Anderson-type phase transition,belonging to the same universality class of the three-dimensional unitary Anderson model.
4. Shape analysis
From the point of view of random matrix models, Fig. 2 shows that the local spectral statis-tics along the spectrum are described by one-parameter families of models, with spectral statisticsinterpolating between Poisson and Wigner-Dyson along some path in the space of probability dis-tributions. To check if the appropriate one-parameter family depends on the size of the lattice,6 ritical behaviour in the QCD Anderson transition
Matteo Giordano one can simply plot a couple of parameters of the ULSD against each other (thus projecting thepath onto a two-dimensional plane in the space of probability distributions): if points are seen tocollapse on a single curve, irrespectively of L , then the intermediate ULSDs lie on a universal pathin the space of probability distributions [17].In Fig. 4 we show I l and h s i l plotted against each other for several volumes, and we seethat they indeed lie on a single curve. As L is increased, points corresponding to a given valueof l flow towards the Poisson or RMT “fixed points”, while flowing away from an unstable fixedpoint corresponding to l c , where a different universality class for the spectral statistics is expected.Similar plots made by changing T and/or a are compatible with a similar universality of the path,but statistical errors are still too large to reach a definitive conclusion.The transition from Poisson to Wigner-Dyson behaviour in finite volume is therefore expectedto be described by a universal one-parameter family of random matrix models [18]. Comparingwith analogous results for the Anderson model, it turns out that the spectral statistics at the criticalpoint in the two models are compatible [18]. References [1] T. Banks and A. Casher, Nucl. Phys. B
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