Higher-order level spacings in random matrix theory based on Wigner's conjecture
HHigher-order level spacings in random matrix theory based on Wigner’s conjecture
Wen-Jia Rao ∗ School of Science, Hangzhou Dianzi University, Hangzhou 310027, China.
The distribution of higher order level spacings, i.e. the distribution of { s ( n ) i = E i + n − E i } with n ≥ isderived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poissonensemble. It is found s ( n ) in Gaussian ensembles follows a generalized Wigner-Dyson distribution with rescaledparameter α = νC n +1 + n − , while that in Poisson ensemble follows a generalized semi-Poisson distributionwith index n . Numerical evidences are provided through simulations of random spin systems as well as non-trivial zeros of Riemann zeta function. The higher order generalizations of gap ratios are also discussed. I. INTRODUCTION
Random matrix theory (RMT) was introduced half a cen-tury ago when dealing with complex nuclei , and since thenhas found various applications in fields ranging from quan-tum chaos to isolated many-body systems . This roots inthe fact that RMT describes universal properties of randommatrix that depend only on its symmetry while independentof microscopic details. Specifically, the system with time re-versal invariance is represented by matrix that belongs to theGaussian orthogonal ensemble (GOE); the system with spinrotational invariance while breaks time reversal symmetry be-longs to the Gaussian unitary ensemble (GUE); while Gaus-sian symplectic ensemble (GSE) represents systems with timereversal symmetry but breaks spin rotational symmetry.Among various statistical quantities, the most widelyused one is the distribution of nearest level spacings { s i = E i +1 − E i } , i.e. the gaps between adjacent energy lev-els, which measures the strength of level repulsion. The exactexpression for the P ( s ) can be derived analytically for ran-dom matrix with large dimension, which is cumbersome .Instead, for most practical purposes it’s sufficient to employthe so-called Wigner surmise that deals with × matrix(this will be reviewed in Sec. II), the out-coming result for P ( s ) has a neat expression that contains a polynomial partaccounting for level repulsion and an Gaussian decaying part(see Eq. (6)).Different models may and usually do have different den-sity of states (DOS), hence to compare the universal behaviorof level spacings, an unfolding procedure is required to erasethe model dependent information of DOS. To overcome thisobstacle, Oganesyan and Huse proposed a new quantity tostudy the level statistics, i.e. the ratio between adjacent gaps { r i = s i +1 /s i } , whose distribution P ( r ) is later analyticallyderived by Atas et al. . The gap ratio is independent of localDOS and requires no unfolding procedure (provided the DOSdoes not vary in the scale of the spacings involve), hence hasfound various applications, especially in the context of many-body localization (MBL) .Both the nearest level spacing and gap ratio account for theshort range level correlations. However, long range correla-tions are also important, especially when studying the MBLtransition phenomena. Indeed, there’re several effective mod-els describing the level distribution at the MBL transition re-gion. For example, the Rosenzweig-Porter model , meanfield plasma model , short-range plasma models (SRPM) and its generalization – so-called weighed SRPM , Gaussian β ensemble and the generalized β − h model . All of thesemodels more or less describe the short-range level correlationsin the MBL transition region well, and their difference canonly be revealed when long-range correlations are concerned.For a comparison of these models in describing MBL transi-tion point, see Ref. [22].Commonly, the long-range correlations in a random matrixcan be described by the number variance Σ or the Dyson-Mehta ∆ statistics , however, both of them are very sensi-tive to the concrete unfolding strategy and have already beena source of misleading signatures . Instead, it’s more directand numerically easier to study the higher order level spac-ings and gap ratios. There’re existing works that generalizethe level spacing and gap ratios to higher order, as well astheir applications in studying MBL transitions . How-ever, most of these works are numerical or phenomenological,and an analytical derivation for the distribution of level spac-ing/gap ratio is still lacking. Given the importance of higher-order level correlations, it’s desirable to have an analytical for-mula for them, it is then the purpose of this work to fill in thisgap.In this work, by using a Wigner-like surmise, we succeededin obtaining an analytical expression for the distribution ofhigher order spacing (cid:110) s ( n ) i = E i + n − E i (cid:111) in all the Gaus-sian ensembles of RMT, as well as the Poisson ensemble. Theresults show the distribution of s ( n ) i in the former class fol-lows a generalized Wigner-Dyson distribution with rescaledparameter; while in Poisson ensemble it follows a general-ized semi-Poisson distribution with index n . Interestingly, therescaling behavior of higher-order level spacing is identical tothat of the high-order gap ratio found numerically in Ref. [28],for which we will provide a heuristic explanation.This paper is organized as follows. In Sec. II we review theWigner surmise for deriving the distribution of nearest levelspacings, and present numerical data to validate this surmise.In Sec. III A we present the analytical derivation for higherorder level spacings using a Wigner-like surmise, and numer-ical fittings are given in Sec. III B. In Sec. IV we discuss thegeneralization of gap ratios to higher order. Conclusion anddiscussion come in Sec. V. a r X i v : . [ c ond - m a t . d i s - nn ] A ug II. NEAREST LEVEL SPACINGS
We begin with the discussion about nearest level spac-ings, our starting point probability distribution of energy lev-els P ( { E i } ) in three Gaussian ensembles, whose expressioncan be found in any textbook on RMT (e.g. Ref. [5]), P ( { E i } ) ∝ (cid:89) i FIG. 1. (a) The density of states (DOS) ρ ( E ) of random field Heisen-berg model at L = 12 and h = 1 in orthogonal case, the DOS is moreuniform in the middle part, we therefore choose the middle half lev-els to do level statistics. (b) Distribution of nearest level spacings P ( E i +1 − E i ) , we see a GOE/GUE distribution for h = 1 in theorthogonal/unitary model, while a Poisson distribution is found for h = 5 in orthogonal model, the result for h = 5 in unitary model isnot displayed since it coincides with that in the orthogonal model. We choose a L = 12 system to present a numerical sim-ulation, and prepare samples at h = 1 and h = 5 forboth the orthogonal and unitary model. In Fig. 1(a) we plotthe density of states (DOS) for the h = 1 case in orthogonalmodel. We can see DOS is much more uniform in the middlepart of the spectrum, which is also the case for h = 5 and uni-tary model. Therefore we choose the middle half of energylevels to do the spacing counting, and the results are shownin Fig. 1(b). We observe a clear GOE/GUE distribution for h = 1 in orthogonal/unitary model and a Poisson distributionfor h = 5 in orthogonal model as expected, the fitting resultfor h = 5 in unitary model is not shown since it almost coin-cides with that in orthogonal model. It is noted the fitting forPoisson distribution has minor deviations around the region s ∼ , this is due to finite size effect since there will alwaysremain exponentially-decaying but finite correlation betweenlevels in a finite system. As we will demonstrate in subsequentsection, the fitting for higher order level spacings will be bet-ter since the overlap between levels decays exponentially withtheir distance in MBL phase.A technique issue is, when counting the level spacings, wechoose to take the middle half levels of the spectrum, whilewe can also employ a unfolding procedure using a spline in-terpolation that incorporates all energy levels , and the fittingresults are almost the same . III. HIGHER ORDER LEVEL SPACINGS Now we proceed to consider the distribution of higher or-der level spacings (cid:110) s ( n ) i = E i + n − E i (cid:111) , using a Wigner-like surmise. We first give the analytical derivation, then providenumerical evidence from simulation of spin model in Eq. (7)as well as the non-trivial zeros of Riemann zeta function. A. Analytical Derivation Introduce P n ( s ) = P (cid:0) s ( n ) = s (cid:1) ≡ P ( | E i + n − E i | = s ) ,to apply the Wigner surmise, we are now considering ( n + 1) × ( n + 1) matrices, the distribution P n ( s ) then goesto P n ( s ) ∝ (cid:90) ∞−∞ (cid:89) i GUE GSE Poisson TABLE I. The order of the polynomial term in P n ( s ) for the threeGaussian ensembles as well as Poisson ensemble, the decaying termis Gaussian type for the former class and exponential decay for thelatter. one spacing remains in the spectrum. Finally, we want toemphasize that the levels are well-correlated in the Gaussianensembles, hence the derivation of P n ( s ) for Poisson ensem-ble in Eq. (19) do not hold, otherwise the result will deviatedramatically .For convenience we list the order of the polynomial part in P n ( s ) for the three Gaussian ensembles as well as Poissonensemble up to n = 8 in Table I, note that the exponentialparts in the former class are Gaussian type and that for Poissonensemble is a exponential decay. B. Numerical Simulation To show how well the distributions in Eq. (16) and Eq. (20)work for matrix with large dimension, we now perform nu-merical simulations for the random spin model in Eq. (7),where we also pick the middle half levels to do statistics. Wehave tested the formula up to n = 5 , and in Fig. 2 we displaythe fitting results for n = 2 and n = 3 .As expected, the fittings are quite accurate for both GOEand GUE as well as Poisson ensemble. In fact, the fittings forhigher order spacings in the Poisson ensemble are better thanthat for nearest spacing in Fig. 1(b). This is because in MBLphase the overlap between levels decays exponentially withtheir distance, hence the fitting for higher order level spacingsis less affected by finite size effect.For another example we consider the non-trivial zeros ofthe Riemann zeta function ζ ( z ) = ∞ (cid:88) n =1 n z , (21)it was established that statistical properties of non-trivial Rie-mann zeros { γ i } are well described by the GUE distribution .Therefore, we expect the gaps (cid:110) s ( n ) i = γ i + n − γ i (cid:111) followsthe same distribution as those in GUE. The numerical resultsfor n = 1 , , are presented in Fig. 3, as can be seen, thefittings are perfect. IV. HIGHER ORDER GAP RATIOS As mentioned in Sec. I, besides the level spacings, anotherquantity is also widely used in the study of random matri-ces, namely the ratio between adjacent gaps { r i = s i +1 /s i } , s (2) = E i + 2 E i P ( s ( ) ) (a) = 4(GSE)= 7semi-PoissonOrthogonal, h = 1Unitary, h = 1Orthogonal, h = 5 s (3) = E i + 3 E i P ( s ( ) ) (b) = 8= 14 n = 3Orthogonal, h = 1Unitary, h = 1Orthogonal, h = 5 FIG. 2. Distribution of next-nearest level spacings P ( s (2) ) in (a) andnext-next-nearest level spacings P ( s (3) ) in (b), where α and n arethe index in Eq. (16) and Eq. (20) respectively. s P ( s ) = 2(GUE)= 7= 14 i + 1 ii + 2 ii + 3 i FIG. 3. The distribution of n -th order spacings of the non-trivialzeros { γ i } of Riemann zeta function, where α is the index in gener-alized Wigner-Dyson distribution in Eq. (16). The data comes from levels starting from the th zero, taken from Ref. [38]. which is independent of local DOS. The distribution of nearestgap ratios P ( ν, r ) is given in Ref. [8], whose result is P ( ν, r ) = 1 Z ν (cid:0) r + r (cid:1) ν (1 + r + r ) ν/ (22)where ν = 1 , , for GOE,GUE,GSE, and Z ν is the normal-ization factor determined by requiring (cid:82) ∞ P ( ν, r ) dr = 1 .This gap ratio can also be generalized to higher order,but in different ways, i.e. the “overlapping” and “non-overlapping” way. In the former case we are dealing with (cid:101) r ( n ) i = E i + n − E i E i + n − − E i − = s i + n + s i + n − + ... + s i +1 s i + n − + s i + n − + ... + s i ,(23)which is named “overlaping” ratio since there is shared spac-ings between the numerator and denominator. While the “non-overlapping” ratio is defined as r ( n ) i = E i +2 n − E i + n E i + n − E i = s i +2 n + s i +2 n − + ... + s i + n +1 s i + n + s i + n − + ... + s i .(24)Both these two generalizations reduce to the nearest gap ratiowhen n = 1 , but they are quite different when studying theirdistributions using Wigner surmise: for overlapping ratio (cid:101) r ( n ) i ,the smallest matrix dimension is ( n + 2) × ( n + 2) ; while itis (1 + 2 n ) × (1 + 2 n ) for non-overlapping ratio. Naively, wecan expect the distribution for (cid:101) r ( n ) is more involved due tothe overlapping spacings. Indeed, the n = 2 case for P (cid:0)(cid:101) r ( n ) (cid:1) has been worked out in Ref. [26] and the result is very com-plicated. Instead, for the non-overlapping ratio, Ref. [28] pro-vides compelling numerical evidence for its distribution to fol-low P (cid:16) ν, r ( n ) (cid:17) = P ( ν (cid:48) , r ) , (25) ν (cid:48) = n ( n + 1)2 ν + n − . (26)Surprisingly, the rescaling relation Eq. (26) coincides withthat for higher order level spacing in Eq. (17). We have alsoconfirmed this formula by numerical simulations in our spinmodel Eq. (7), and the results for n = 2 in GOE ( ν = 1 )case is presented in Fig. 4, where we also draw the distribu-tion of overlapping ratio (cid:101) r (2) for comparison. As can be seen,they differ dramatically, and the fitting for non-overlappingratio is quite accurate. This result strongly suggest the non-overlapping ratio is more universal than the overlapping ratio,and its distribution P (cid:0) r ( n ) (cid:1) is homogeneously related withthat for the n − th order level spacing, at least in the sense ofWigner surmise, for which we provide a heuristic explanationas follows.For a given energy spectrum { E i } from a Gaussian ensem-ble with index ν , we can make up a new spectrum (cid:110) E (cid:48) i (cid:111) bypicking one level from every n levels in { E i } , then the n -th or-der level spacing s ( n ) in { E i } becomes the nearest level spac-ing in (cid:110) E (cid:48) i (cid:111) , and the n -th order non-overlapping ratio in { E i } becomes the nearest gap ratio in (cid:110) E (cid:48) i (cid:111) . Since we have analyt-ically proven the rescaling relation in Eq. (17), we conjecturethe probability density for (cid:110) E (cid:48) i (cid:111) (to leading order) bear thesame form as { E i } in Eq. (1) with the rescaled parameter α in Eq. (17). Therefore, the higher order non-overlapping gapratios also follow the same rescaling as expressed in Eq. (25)and Eq. (26). For this point of view, numerical evidences areprovided in a recent work of the author . r (2) P ( r ( ) ) = 4OverlappingRationon-OverlappingRatio FIG. 4. The distribution of second-order gap ratio in the orthogo-nal model, where red and blue dots correspond to overlapping andnon-overlapping ratios respectively, the latter fits perfectly with theformula in Eq. (25) with ν (cid:48) = 4 . Note the data is taken from thewhole energy spectrum without unfolding. V. CONCLUSION AND DISCUSSION We have analytically studied the distribution of higher orderlevel spacings (cid:110) s ( n ) i = E i + n − E i (cid:111) which describes the levelcorrelations on long range. It is shown s ( n ) in the Gaussianensemble with index ν follows a generalized Wigner-Dysondistribution with index α = νC n +1 + n − , where ν = 1 , , for GOE,GUE,GSE respectively. This results in a large num-ber of coincident relations for distributions of level spacingsof different orders in different ensembles. While s ( n ) in Pois-son ensemble follows a generalized semi-Poisson distributionwith index n . Our derivation is rigorous based on a Wigner-like surmise, and the results have been confirmed by numeri-cal simulations from random spin system and non-trivial zerosof Riemann zeta function.We also discussed the higher order generalization of gap ratios, which come in two different ways – the “overlapping”and “ non-overlapping” way – and point out their differencein studying their distributions using Wigner-like surmise. No-tably, the distribution for the non-overlapping gap ratio hasbeen studied numerically in Ref. [28], in which the authorsfind a scaling relation Eq. (26) that is identical to the one wefind analytically for higher order level spacings. This stronglyindicates the distribution of higher order spacing and non-overlapping gap ratio is correlated in a homogeneous way, forwhich we provided a heuristic explanation.It’s noted the higher-order level spacings have played animportant role in the study of the spacing distribution in aspectrum with missing levels , where the second order levelspacing distribution in GOE is derived by a method differentfrom this work. Our derivations for P (cid:0) s ( n ) (cid:1) in Guassian en-sembles are purely mathematical that work for arbitrary posi-tive values of ν , although the ν = 1 , , for GOE,GUE,GSEare of most physical interest. Therefore, it is possible for ourresults to find applications in models that goes beyond thethree standard Gaussian ensembles. For example, the ν = 3 behavior for level spacing has been found in a 2D lattice withnon-Hermitian disorder .It is also interesting to note the distribution of next-nearestlevel spacing in Poisson class is semi-Poisson P ( s ) ∝ s exp ( − s ) , which is suggested to be the distribution fornearest level spacing at the thermal-MBL transition point inorthogonal model . This indicates – to leading order – theuniversality property of this transition point is more affectedby the MBL phase than the thermal phase, a fact already no-ticed by previous studies . This observation thus moti-vates a natural question: how will the thermal phase affectthe universality of the MBL transition point? To answer thisquestion, a comparison between the GOE-Poisson and GUE-Poisson transition points is suggested, which is left for a futurestudy.Last but not least, in this paper the distribution of higherorder level spacing is derived only in ( n + 1) × ( n + 1) ma-trix, its exact value in large matrix as well as the differencebetween them can in principle be estimated using the methodin Ref. [8], this is also left for a future study. ACKNOWLEDGEMENTS The author acknowledges the helpful discussions with XinWan and Rubah Kausar. This work is supported by the Na-tional Natural Science Foundation of China through GrantNo.11904069 and No.11847005. ∗ [email protected] C. E. Porter, Statistical Theories of Spectra: Fluctuations (Aca-demic Press, New York), 1965. T. A. Brody et al., Rev. Mod. , 385 (1981). T. Guhr, A. Muller-Groeling, H. A. Weidenmuller, Phys. Rep. , 189 (1998). M. L. Mehta, Random Matrix Theory, Springer, New York (1990). F. Haake, Quantum Signatures of Chaos (Springer 2001). E. P. Wigner, in Conference on Neutron Physics by Timeof-Flight(Oak Ridge National Laboratory Report No. 2309, 1957) p. 59. V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111 (2007). Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys. Rev.Lett. , 084101 (2013). V. Oganesyan, A. Pal, D. A. Huse, Phys. Rev. B , 115104(2009). A. Pal, D. A. Huse, Phys. Rev. B , 174411 (2010). S. Iyer, V. Oganesyan, G. Refael, D. A. Huse, Phys. Rev. B ,134202 (2013). X. Li, S. Ganeshan, J. H. Pixley, and S. Das Sarma, Phy. Rev. Lett. , 186601 (2015). Y. Bar Lev, G. Cohen, and D. R. Reichmman, Phys. Rev. Lett. , 100601 (2015). K. Agarwal, S. Gopalakrishnan, M. Knap, M. Mueller, and E.Demler, Phys. Rev. Lett. David J. Luitz, Nicolas Laflorencie, and Fabien Alet, Phys. Rev.B , 081103(R) (2015). Y. Avishai, J. Richert, and R. Berkovits, Phys. Rev. B , 052416(2002). N. Regnault and R. Nandkishore, Phys. Rev. B , 104203 (2016). S. D. Geraedts, R. Nandkishore, and N. Regnault, Phys. Rev. B , 174202 (2016). P. Shukla, New Journal of Physics , 021004 (2016). M. Serbyn and J. E. Moore, Phys. Rev. B , 041424(R) (2016). E. B. Bogomolny, U. Gerland and C. Schmit, Eur. Phys. J. B ,121 (2001). P. Sierant and J. Zakrzewski, Phys. Rev. B , 104205 (2019). W. Buijsman, V. Cheianov and V. Gritsev, Phys. Rev. Lett. ,180601 (2019). P. Sierant and J. Zakrzewski, Phys. Rev. B , 104201 (2020). J. M. G. Gomez, R. A. Molina, A. Relano, and J. Retamosa, Phys.Rev. E , 036209 (2002). Y. Y. Atas, E. Bogomolny, O. Giraud, P. Vivo, and E. Vivo, J.Phys. A: Math. Theor. , 355204 (2013). S. H. Tekur, S. Kumar and M. S. Santhanam, Phys. Rev. E, ,062212 (2018). S. H. Tekur, U. T. Bhosale, and M. S. Santhanam, Phys. Rev. B , 104305 (2018). P. Rao, M. Vyas, and N. D. Chavda, arXiv:1912.05664v1. A. Y. Abul-Magd and M. H. Simbel, Phys. Rev. E , 5371 (1999). M. M. Duras and K. Sokalski, Phys. Rev. E , 3142 (1996). R. Kausar, W.-J. Rao, and X. Wan, J. Phys.: Condens. Matter ,415605 (2020). W.-J. Rao, J. Phys.:Condens. Matter , 395902 (2018). M. L. Mehta and F. J. Dyson, Journal of Mathematical Physics, (1963). E. B. Bogomolny, U. Gerland and C. Schmit, Phys. Rev. E ,R1315(R) 1999. Definition of the Riemann ζ ( z ) function given in Eq. (21) is validonly for Re ( z ) > . To overcome this problem, see, e.g., H. M.Edwards, ”Riemann’s Zeta Function”, Chap.1.4. H. L. Montgomery, Proc. Symp. Pure Math. , 181 (1973); E. B.Bogomolny and J. P. Keating, Nonlinearity , 1115 (1995); ibidNonlinearity 9, 911 (1995); Z. Rudnick and P. Sarnak, Duke Math.J. , 269 (1996); J. P. Keating and N. C. Snaith, Comm. Math.Phys. , 57 (2000). ∼ odlyzko/zeta tables/index.html. W.-J. Rao and M. N. Chen, arXiv:2006.07774. O. Bohigas and M. P. Pato, Phys. Lett. B , 171-176 (2004). A. F. Tzortzakakis, K. G. Makris, and E. N. Economou, Phys. Rev.B101