JHEP07(2020)193 Springer July 27, 2020 June 10, 2020 June 18, 2020 : : : February 28, 2020 : Revised = 4. We show that = 32 show that only Published Accepted q N Received -body interactions, but also q = 32 and N ) from one realization of the ensemble Published for SISSA by https://doi.org/10.1007/JHEP07(2020)193 N -model, which is derived through a Fierz σ = 4 or higher even q , . 3 -model results in an ill-behaved expansion of the resolvent, and it gives 1912.11923 σ The Authors. c Matrix Models, Random Systems, Field Theories in Lower Dimensions
We present a detailed quantitative analysis of spectral correlations in the , [email protected] = 2 SYK model albeit with a much smaller Thouless energy while the correct q Stony Brook, New York 11794,E-mail: U.S.A. [email protected] Department of Physics and Astronomy, Stony Brook University, Open Access Article funded by SCOAP spacings for the numberform variance factor. or exponentially short times inKeywords: the case of the spectral ArXiv ePrint: result in this case shouldthe have been number Poisson variance statistics. andthe In our quadratic spectral numerical deviation form studies we ofsmall factor analyze times the for in number the variance spectralwe form for factor. find large After quantitative energies eliminating agreement the appears with long-wavelength fluctuations, as RMT a up peak to for an exponentially large number of level transformation, and evaluate the oneloop and order. two-point spectral The correlationcorrections functions whose wide to lowest-order correlator two- term isexpansion corresponds of given to the scale by fluctuations. theuniversal RMT sum However, fluctuations the of not loop only thefor for universal the RMT result and the lowest eight polynomials arefluctuations. needed to The eliminate covariance the matrixanalytically nonuniversal of from part the low-order of coefficients double-trace thethe of spectral moments. this first We expansion evaluate six can thespectral moments be covariance correlations and matrix obtained in of find terms that of it a agrees nonlinear with the numerics. We also analyze the Matrix Theory (RMT) behavior are(of due the to order a small of number theto of number long-wavelength the of fluctuations Majorana next fermions one.orthogonal polynomials, These the modes mainwe contribution can give being a be due simple parameterized to analytical effectively scale estimate. in fluctuations terms Our for numerical of which results Q-Hermite for Abstract: Sachdev-Ye-Kitaev (SYK) model. We find that the deviations from universal Random Spectral fluctuations in the Sachdev-Ye-Kitaev model Yiyang Jia and Jacobus J.M. Verbaarschot JHEP07(2020)193 18 33 19 53 24 = 4) 47 , q 21 = 32 23 N 44 24 13 33 3 7 – i – 52 26 17 37 43 39 40 ) by 3-cross-linked examples 30 45 6.12 44 10 24 -model 8 )–( σ 4 6.9 1 -model σ 5.2.1 Higher order corrections E.1 Values for low-order double-trace contractions ( C.1 One-point function C.2 Two-point function 8.2 Long-wavelength fluctuations 7.1 Number variance 7.2 The effect7.3 of ensemble averaging Spectral form factor 8.1 The calculation of high-order moments 6.1 General expression6.2 for double-trace Wick Some contractions useful6.3 properties of double-trace Low-order covariances moments of Q-Hermite expansion coefficients 5.1 One-point function 5.2 Loop expansion of the two-point function 5.3 Does the expansion in powers of the resolvent of the semicircle make sense? D Illustration of ( E Low-order double-trace moments B Some combinatorial identities C Replica limit of the GUE partition function 9 Discussion and conclusions A Derivation of the 8 Double-trace moments and the validity of random matrix correlations 7 Numerical analysis of spectral correlations 6 Analytical calculation of the fluctuations of the expansion coefficients 3 Spectral density 4 Collective fluctuations of eigenvalues 5 Nonlinear Contents 1 Introduction 2 The Sachdev-Ye-Kitaev (SYK) model JHEP07(2020)193 ] ] ]. 2 20 28 – as a – 18 26 N ]. However, 13 ]. The complex ] exactly as the 43 ] and decay of the 24 – ]. In this limit, the – 31 4 41 , ]. Lattices of coupled 22 3 , 40 12 [ 0 E ] or from a moment calculation − ]. This approach also showed that E 15 , √ 21 14 , 5 ] for the nuclear level density, which was 25 ], the Sachdev-Ye-Kitaev (SYK) model [ 1 formulation of the SYK model [ – 1 – ] was first introduced in nuclear physics [ ], eigenstate thermalization [ G 2 30 , ], and because of this an initial state has to thermalize ] and black hole microstates [ 29 mod 8)-dependent anti-unitary symmetries of the SYK 5 39 ]. Coupled SYK models have been used to get a deeper N – -model formulation [ σ 35 34 , ], which was shown by the calculation of Out-of-Time-Order ]. The chaotic-integrable phase transition has been studied ]. 7 33 32 8 , , ]. 6 3 , , 17 ] stating that if the corresponding classical system is chaotic, the 1 , 3 9 , 1 16 ] is to formulate the SYK model as a path integral which isolates 2 ]. This has been understood analytically in terms of a two-replica nonper- 12 – 10 The SYK model has become a paradigm of quantum many-body physics. It has been The so-called complex SYK model [ A different measure of chaos in quantum systems is the extent to which correlations of thermofield double state [ in a mass-deformed SYKunderstanding [ of wormholes [ SYK models describe phaseSYK transitions involving model non-Fermi has liquids [ a conserved charge (the total number of fermions) and its effects were phenomenologically successful Bethe formulaactually [ not realized inmodel the is early not expected nuclear to physics belike literature. important, models, and specifically The it tensor has randomness been models, of shown have for the a several SYK similar non-random SYK- melonicused mean to field understand behavior thermalization [ [ prefactor of the action,by making mean it field possible theory. to Thismodel, analysis evaluate revealed namely the one that Green’s of its functions thefor ground most of non-Fermi striking state the liquids properties entropy model and of is the blackthe nonzero SYK hole and level physics extensive, density alike making [ of it the a SYK model model increases as exp to reflect the four-body naturein of the the nuclear nuclear interaction physics (knownand as literature) a the as two-body random interaction well matrixwas as made behavior the in exponential of [ increase nuclear of level the correlations. level density The great advance that model [ turbative saddle point ofthere the are so also called deviations from Σ which random follow matrix from theory a at nonlinear manyof level the spacings SYK or model small [ times Schmidt conjecture [ eigenvalue correlations of the quantumSYK system model are this given can by beIt random investigated was by matrix the theory. found exact In that diagonalizationanti-unitary the of level symmetries the as statistics SYK the are Hamiltonian. ( given by the random matrix theory with the same which is only possible ifmaximally its chaotic dynamics [ are chaotic.Correlators In (OTOC) fact [ the SYK model turned out to be eigenvalues are given by random matrix theory. This goes back to the Bohigas-Giannoni- Starting with thehas seminal attracted talk a by greatdimensional Kitaev deal gravity. of [ The attention low-energyaction in limit which recent of can years, also the in beSYK SYK particular obtained model model as from is is dual Jackiw-Teitelboim a gravity to given model [ a by black for the hole two- [ Schwarzian 1 Introduction JHEP07(2020)193 n 2. / 2 N/ = 4 becomes important q ]. However, these are not the 17 – which for 15 q N / 2 n ]. It has also been used to construct a model ∼ – 2 – 44 independent random variables, the relative error ], it is natural to expand the deviations in terms q N 54 . Such fluctuations in the average level density result , 2 / 1 23 , ]. The disadvantage of these measures is that they include q N ]. Recently, it was shown that the main long-range spectral 12 formulation [ 51 ]. / G 50 53 , – ]. The duality been the SYK model and Jackiw-Teitelboim gravity 52 46 45 ) to the spectral form factor, which becomes the dominant contribution τ ( δ . It was already realized many years ago that these scale fluctuations can , which is not large in comparison to the total number of levels of 2 1 2 − N q N ∼ n The goal of this paper is to determine quantitatively the agreement of level correlations The average spectral density is not universal, and to analyze the universality of spectral There are different observables to study level correlations. The best known one is the τ < an analytical calculation of the covariance matrix of the first six moments. the Q-Hermite polynomials [ of Q-Hermite polynomials.als, It suggesting turns a out separation thatdensity of we and scales only the between short-wavelength long-wavelength need fluctuations fluctuationsThe a of of long-wavelength small the the fluctuations number universal spectral are of RMT determined spectral polynomi- by correlations. low-order moments, and we present for each configuration [ fluctuations in theonly SYK long-range model fluctuations: are they aresmoothened of just spectral this the density first nature term forsuch [ in each long-range a fluctuations. realization. “multipole” expansion Since In of the the this spectral paper density we is systematically close to study the weight function of when This effect gives aproportional constant to contribution to thefor pair correlation function,be and eliminated a by function rescaling the eigenvalues according to the width of the spectral density correlations within one specific realizationfluctuations of from the one SYK realization modelthe to and SYK the model level is correlations next. determined due by The to only in latter a are observable expected is to ofin be order a large. 1 contribution to Since the number variance of numbers and no further unfoldingdo is so). needed The (although number in varianceeach practice and with it the their is spectral own often form advantages factor convenient and to are disadvantages. complementary observables, in the SYK model with random matrix theory. To do this we distinguish between level correlators, one has toplaces. eliminate First, this one non-universalsecond, has one part to has subtract which to the appears unfolddensity the disconnected that in spectrum is part two by uniform. of a different Note the smooth that transformation two-point the resulting correlator, number in and variance a is spectral already defined in terms of level of two consecutive spacings [ both two-point and higher-pointvariance, correlations. which is In the thiseigenvalues variance paper on of we average, the will and number the focuspair of spectral on correlation eigenvalues form the in function. factor number a Note whichthe interval is that that pair the the correlation contains Fourier number function transform variance (by of is definition) the an and integral the transform spectral of form both factor. for quantum batteries [ has been further explored inmodel random at matrix low theories energies with [ the spectral density ofspacing the distribution SYK of neighboring levels or the ratio of the maximum and the minimum recently analyzed in the Σ JHEP07(2020)193 . C and (2.2) (2.1) ], we in the rather 57 D N N = 4 these q -model for- σ . The replica B we give a simple ]. The covariance -body Hamiltonian 4 q 15 ]. N, ] (for values of 60 , . Concluding remarks are ≤ 11 q 8 . 59 , E < i ] of the SYK model make use . In section 37 , 2 ··· 15 15 , – < 14 = 2, with energies given by sums of 2 . We calculate the one-point function 13 , 5 α q < i Γ -model for the spectral determinant from a α 1 i σ ]. We evaluate the wide correlator to two- J ≤ α 56 , X 1 – 3 – = 55 , , } H q integer elements: 15 , q ] and the numerical results of [ 14 we derive the interacting Majorana fermions with a , . . . , i 15 , where we give explicit results for the first six double-trace A 2 N 6 , i 1 i { := α ] and the spectral determinant [ . In this section we also discuss the effects of ensemble averaging and the 13 . In appendix 7 = 4. It is clear though, that for 9 q ] while solutions that couple the two replicas, but are otherwise diagonal, after introducing the SYK model in section 58 3 for 2 being a multi-index set of N α formulation [ Our strategy to eliminate the collective fluctuations of the spectral density is discussed We also analyze the deviations from random matrix theory in terms of the replica = 2 are qualitatively the same albeit with a Thouless energy that scales as G given by with Examples for the calculationexplicit of results double-trace for moments low-order are moments worked are out given in in appendix appendix 2 The Sachdev-Ye-Kitaev (SYK)The model SYK model is a model of and the spectral formconvergence factor. to the The random structure matrixmade result of in is section high-order discussed double-trace in moments section Fierz and transformation. the Several combinatorial formulaslimit are of given the in appendix one-point and two-point functions of the GUE are calculated in appendix resummation of the expansionmatrix in is semicicular obtained resolvents in obtained section moments. in [ Numerical resultssented in for section the numberfluctuations variance relative and to the spectral ensemble form average, the factor latter are contribute to pre- the number variance argument to determine themulation leading of correction the to SYK the modeland the number is two-point variance. discussed function in to The section the two-loop order. expansion of We also the obtainto resolvent a the in very terms actual efficient expansion of width for powers of of the the spectrum resolvent of of the a SYK semicircle model. rescaled This expansion is obtained by a expect that at thefunctions mean field [ level onlyappear replica in diagonal the solutions calculation contribute of to the one-point two-point function [ in section q than single-particle energies, spectralthe correlations mechanism are that given nullifiesΣ the by contribution Poisson from statistics. theof zero the modes Presently, replica is trick. not clear. Although the Both replica the trick may result in an incorrect answer [ limit of a spectralloop determinant order. [ Thepart random of matrix the propagator, contribution whereasresults to the are this deviations in correlator are agreement due with isuniversality class to [ given of massive by the modes. Gaussian the For Unitary massless Ensemble (GUE)). However, the expressions for JHEP07(2020)193 2 v (3.5) (3.1) (3.2) (3.3) (3.4) (2.3) (2.4) (2.5) ] by the , , 54 , k q 23 k 2 η ) ) − denotes ensemble − k η ). In this paper we η q N − , 3.5 ij by (1 h· · · i δ 2 (1 + q k x x = 2 k − . − } ) ) q 1 j s , 1) + +1 , γ 2 i j j v − =1 ∞ γ u u ( Y k { 2 − − q q =0 N , see equation ( x k X . ) (1 (1 1 N η , − 1 , 1)! =0 = =0 η q − − ∞ j ∞ j 2 i , q i − γ σ q 2 − Q N (1 Q σ E 4 q N s 1 1 4 ( q H ··· 2 − 2 = 1 = 4. The Majorana fermions are represented as 2 – 4 – = J − Tr = i ) h x q γ 2 1 0 u 2 β 1 = i E Γ r − γ N/ 2 α η v − 1) ) Γ − 1 2 2 β q ( ( Γ = 2 q 2 1 + α η ) = (1 2 is related to dimensionless energy √ 1) s Γ σ ( π , especially − u E TrΓ q Γ √ configurations. Furthermore, β ) = X x q N 1 ( := ( − α QH Γ ρ q ) N α J 2 i ≈ ) ]: N/ x can have are independently Gaussian distributed random variables with variance ( − 61 α α ) is defined by J SYK = 2 matrices which is an effective way to obtain a Hamiltonian that can be diagonalized x ρ ( η h γ u is one) [ 2 and the ground state energy is given by In physical units the energy and Γ σ where 3 Spectral density The average spectralweight density function of of the the SYK Q-Hermite model polynomials (in is units well where approximated the [ many-body variance and a ground state energyonly that consider scales even linearly values with of Dirac numerically. This choice results in aaverage many-body over variance all (the following bracket and the given by and hence JHEP07(2020)193 (3.6) (3.7) (3.8) (3.11) (3.12) (3.10) = 1 it η . = 4 we find that q 2 ) k k η η 2 . , ] ! E η ! (1 + 4 61 ) n ) (3.9) 2 0 = 32 and x x ( E x. ( nm N η 1 k δ − − . This results in the dimensionful . H η n ) = 1 i x . N k x ) = H . ( c k x h η 1 ( η ! =1 ∞ = 0 Y s k 1 H =1 . The expansion converges very well in η ∞ m i η 2 k X − =0 k 4 ) k X H n c 0 k h =0 ) s X − x 1 + norm of the difference between the numerical ( )
η x 2 n E/E 1 ) ( ( – 5 – L − =1 , η H x n is extensive in Y k so that all odd coefficients vanish after averaging. n ) ( − 0 x x 1 ( xH = 0 E QH ! = i ρ p η 2 QH η n c ) = = h ) x ) = 1 and i 1 2 ( dxρ x ) ( 1 + ( 2 x η η η +1 0 ( η − √ η n 2 − 2 1 1 Γ H √ H √ πσ SYK ). Note that − ρ Z h 2.5 ) = are normalized to unity. E ) is a generalization of the Gram-Charlier expansion — for ( QH 3.7 QH ρ ρ and 01 and it decreases for larger values of . was given in ( ! is the Q-factorial defined as 0 η 2 SYK n σ < ρ The expansion coefficients can be calculated in two ways. First, from the scalar product The Q-Hermite polynomials satisfy the recursion relation [ The average spectral density of the SYK model can be expanded in terms of the i| k c |h this case. of the numerically calculated averagenomials, spectral and density second, and from the minimizing Q-Hermite the orthogonal poly- The expansion ( becomes the Gram-Charliercoefficients expansion. are sufficiently It small. For is the only SYK model positive-definite with when the expansion where with The orthogonality relations are given by Both The average spectral density isSince even in the second andHermite spectral fourth density moments we of also have the SYK model coincide with those of the Q- orthogonal polynomials corresponding to this weight function, spectral density In this paper we will only use the dimensionless energy where JHEP07(2020)193 i k 1 c h (3.14) (3.13) = 32 and 1.0 4
) N E = (
q
correspond to QH,8 ρ 32,
)- k QH,0 = E 0.5
( ρ c N each configuration 10 η SYK ρ H E 0.0 , they will correspond to . k . We thus expect that the # 2 ) / 1 levels on average behaves as x 0.5 ( - n q η N k H / k 1 c . We expect that these fluctuations independent stochastic variables, the 1.0 − - =1 /k ∞ 0.010 0.005 0.000 0.005 0.010 q k X 0 N - - ). The fluctuations of q N E Δρ 3.7 2 1 + ¯ n " ) – 6 – 1.0 ∼ x ) ) is of order 1 ( E ) ( x n , but for large values of ( QH,8 (¯ QH ρ k ρ 4 2 ρ )- =
E ( Σ q 0.5
SYK ρ 32,
) = =
x N ( are determined by E 0.0 k SYK c ρ as well as of k 0.5 - c value of the difference not taking into account 2.5% of the total number 2 χ 1.0 - are stochastic variables whose averages give the expansion coefficients 0.0010 0.0005 0.0000 0.0005 0.0010 k - - c Δρ . Difference of the of the ensemble average of the spectral density of the SYK Hamiltonian Since the coefficients We can also expand the spectral density of the SYK Hamiltonian for = 4. What is plotted is the difference of the average spectral density and the eighth order relative error of the variance of the number of levels in an interval containing ¯ of the average spectralspectral density fluctuations in with equation wavelengthare scale ( non-universal for of smalluniversal values random of matrix correlations. Thetransition aim to of universal this random paper matrix is behavior to study takes at place. which scale this Now, the order Q-Hermite polynomial (red curve).minimization In of the left figure,of the eigenvalues coefficients at are each obtained end by an of accuracy the of spectrum. 1 part This in gives a 10000. fit thatin is terms a of factor Q-Hermite 10 polynomials better with products, but they stillfor the can average be spectral obtained densityq by of minimization. an ensemble of WeQ-Hermite 400 illustrate approximation. SYK this In Hamiltonians in the for the figure right scalar products figure and the the coefficients difference are (black curve) obtained is by close calculating to the contribution of the tenth result and the expansion upever, to an a expansion given in order.of orthogonal The the polynomials two should spectrum, does give not andspectrum the converge in a same well this results. much near region. better How- the In fit end that points is case, obtained the coefficients by cannot excluding be a obtained small by fraction taking scalar of the Figure 1 and the eighth order Q-Hermitecalculated approximation by (black minimization, curve). and Inin in the the left the right figure, right figure the figure is coefficients by a are calculating fit of scalar the products. contribution The of the red tenth curve order Q-Hermite polynomial. JHEP07(2020)193 ]. by l 17 (4.7) (4.3) (4.4) (4.5) (4.6) (4.2) , , with ξ . This 2 ]. Such , 15 i y ]: 2 and ) 1+ ˜ z 53 17 k M ( , , ρ ρ h , 52 ξ ) , 2 ξ 4 y ξ 17 ( x 1+ ) = – z O 1 + . Since we consider ( 15 ξ ρ + ρ i ) ξ ξ y ( x . 1 ρ 1+ 6 1 + ih for the SYK model [ ) ρ ≈ x i ξ ( ) 2 2 ρ k x , . x H ih 1+ i 1 2 n − − ξ Tr can be obtained from the normalized h 2 n H ) (4.1) ξ − 2 + ξ . q H = ], see section N ξ x Tr m ,n i 0 Tr σ ) m 1 + 64 h ξ ˜ y ( – N (1 + M ( H δ y 2 = 2 n ρ ξ 1+ 62 Tr x 2 , 2 ih := – 7 – h 1 ) ˜ 1, terms involving the first and second derivative n ρ 20 M x → 1 + ( ˜ := M n ρ − k ξ x | 2 , X y x 2 m,n 1+ ˜ i − h ˜ − 2 M M )
x y )) = 2 | ρ ( ) ξ 2 ρ can be calculated analytically for small values for ξ ) ) 2 ξ i x y l ( 2 c (1 + ρ (1+ x k h k
c ). As a special case we introduce the notation for (normalized and (1+ h x − =
2.5 − i 4 ) ) x . ( ξ
δ 2 x, y dxdy ξ ( k 2
(1+ X ρ 4 Z h denotes averaging over the square of the stochastic variable ≈ ≈ ≈ in terms of scale fluctuations is given by i 2 ) = 2 is the ensemble-averaged spectral density function so that is given by ( 2 2 is a stochastic variable with a zero average and a finite variance. The corresponding x , ˜ ξ ( M 2 ρ σ h ξ defined as ) can be ignored. The expectation value of ρ ˜ − x M 2 ( , ρ 2 m,n ˜ ˜ M On the other hand, this quantity can be calculated directly in the SYK model [ The where rescaled) single-trace moments where spectral correlations on a scale of (by Hilbert space dimension)M and rescaled (by variance) double-trace moment where results in a contribution to the two-point correlation function where spectral density fluctuates as As was noticed in earlieroverall work, rescaling the of main the contribution eigenvalues from tofluctuations one the can configuration number be to variances written the comes as next from [ The covariance matrix calculation double-trace moments usingcalculation methods of similar the to single-trace those moments [ first introduced for the 4 Collective fluctuations of eigenvalues in agreement with the explicit calculation of JHEP07(2020)193 ] 14 (5.1) (4.8) (4.9) (4.12) (4.11) (4.10) n 2 ˜ M -model for σ m 2 ˜ ]. However, we M 15 i n 2 ) ξ . U µ n 2 X = 2 universality class (see (1 + ˜ µ dydz M a i ih β ) 0 µ m m z 2 2 ( P ) ˜ ρ M ξ D σ 1 ih √ ’s are all the linearly independent ). ) − . µ + i y z ) ( . (1 + X y ρ q 1 N 6.13 ( − ih ρ . 2 h n ξ i − h Tr log 2 h q -model for the n dy N − mn 2 2 ˜ σ 2 µ 2 / 2 2 ) M a / 2 / n ξ 1 2 / m 2 tr ∆ ≈ +∆ 2 – 8 – 1 ∆ ˜ +∆ x − ˜ − M n = µ x − x M 2 T x m i 1 Z (1 + 0 µ ˜ 2 Z 2 − M 2 m ξ ˜ / 2 P 2 h M m 2 = / ) i 2 2 1 ¯ n q ξ ∆ 2 N +∆ ¯ ˜ n i − ξ x M − 2 e h x dimensions and they form a basis for the vector space of all ξ µ Z h − (1 + N n mn mn h = Da 2 4 ) = m, Z n ≈ ≈ = 2 2 (¯ ˜ n 2 = M 2 Σ ˜ is not positive definite, but the integrals can be made convergent by Z ] the following nonlinear M 2 µ -model m 15 2 a σ ˜ M − matrices. Because the Hamiltonian commutes with the Dirac chirality matrix n for a derivation), 2 2 A m, N/ 2 2 ˜ M -model reliably. The SYK model with Majorana fermions is simpler and Altland and × We can also calculate the contribution of scale fluctuations to arbitrary double-trace σ 2 N/ an appropriate rotation of theevaluate integration the contour integral in perturbatively thechoice in complex of plane a the [ loop integration expansionproducts contour of about is Dirac its matrices irrelevant. in saddle2 The point when the Bagrets obtained [ appendix The coefficient of Using standard random matrixthe techniques SYK it model. is For possible the complexbut to SYK because derive model of a this the coupling was nonlinear done betweenthe the already massive in and the massless early modes, eighties [ it was hard to analyze We will see below thatmoment in with the two cross SYK contractions, model, see this equation result ( is equal5 to the average double-trace Nonlinear This follows from the elementary calculation moments: with the average number of levels in the interval given by This results in the number variance We thus find JHEP07(2020)193 . 2 x µ X = 4 N/ (5.3) (5.4) (5.5) (5.2) = 2, = N p † | ≤ µ µ X α . In the an- ≤ | Γ } µ 4 γ X 1 = 1 and α Γ , iγ p µ 2. 3 , / γ X } 2 1 4 N/ Tr . , ≤ ) , iγ i } 2 3 α 0 i γ = 2 ). We use the normalization X , a 1 . As an example, for | } 1 − : < i U 22 µ D 4 µ A − j , iγ P 2 A , y X ) ≤ 1 − − | tr q { n 2 N q {z < i 0 ··· N , x, y a ) and 1 ( } i 2 < i N 11 i, 4 Z ), and only ranges from 0 1 , 2.3 − P ≤ | i 3 − µ d j µ dy , tr 1 , ] we use Hermitian generators | | | h ≤ } 2 -dimensional in case of the one-point func- = 2 3 , y 4 d , i 2 15 1 n dx } j γ 1 | γ i U α σ 3 2 2 2 2, see appendix i i will be denoted by 11 or 22, respectively. The 1) Γ γ 1 – 9 – = 1 + 1 γ γ D D 2 is a combinatorial factor due to the commutation U µ − y 1 1 i , γ N/ i i ( 2 2 | µ } . 1 i 1 1 , x X and correct for that by including the appropriate n n 0 T iγ iγ γ 1 γ q or = =0 U α . It is a Fierz coefficient of the Fierz transformation a 0 0 { { { { { n j X µ {z | Γ -model, see appendix µ x ··· → → µ σ U = = = = = µ | n n q is, as indicated by the prime, over the upper block of the 1 N i, } } } } } X = lim µ µ µ µ µ µ + ) = lim Tr = X X X X X matrix in replica space and x | { { { { { µ α n x, y T X ( 2 = ( would be a sum over the set 1 C z × − 0 µ = 1 : = 2 : = 3 : = 4 : = 0 : | | | | | n P µ µ µ µ µ = 1 and denote the upper block by q | | | | | N 2 µ (the length of the multi-index -dimensional replica space ( X | 1 D n is an 2 µ | -body operator defined in equation ( = µ q a µ are the projections onto the 11 block and 22 block for T the the cardinality of index set matrices: pp | P α γ as a derivative with respect to µ | y The pair correlation function is given by with even , the Hamiltonian splits into two blocks, and the partition function is for one of the two µ c where respectively. To obtain thisand expression we have written the derivatives with respect to with Γ with that arises in the derivation of the tion) is denoted byspace tr, is while denoted the by Tr. combined Theof trace coefficient the over the Clifford algebra and the replica The matrix block correspondingtrace to over the 2 and in this casealytical the calculation we sumcombinatorial over factor. all Note thatThe contrary matrix to [ blocks. Therefore, the sum over X (with only half theconvention generators that for we have γ JHEP07(2020)193 - ) σ is z µ 5.1 (5.8) (5.7) (5.6) a ) gives ). The } , x 5.1 n {z ··· x, | , ), but now model ( U µ . The model ( X σ 5.1 µ a C.2 0 µ = diag( z P D σ , √ ) reduces to the universal , as in case of the GUE, and + 2 z ) = 0 , 5.1 y i 0 D − a √ x Tr log matrix and ( / − tr Z / µ h 2 µ n D a a model ( tr U log D µ × √ 1 σ 1 / √ − X n µ d U dz µ 0 µ T σ 0 µ 1 n Tr ], these couplings give large corrections which . P σX P 0 0 1 is an σ 1 2 – 10 – D a 14 → lim − µ n 1 n + e a = 4. ) z − − 0 q a = ). The resolvent is given by = 2 case, which is integrable and should have Poisson − ) = } ( + Tr z q , x ( µ D σ ) symmetry of the replica space of the advanced and retarded a G n {z √ 1 ··· = 2 and µ − ), but now µ n, n q x, | T Da 5.1 = ( Z ). Because of that we know that the low energy effective Lagrangian z n ) = U( z ( × G ) n is the diagonal element of the replica matrix. We evaluate the integral by a loop matrix and 0 n a × The aim of this section is to show that the The generating function for the one-point function has the same form as in the case of n where expansion about the saddle point. The saddle point equation is given by function has the samean form as in the case of the two-point function, ( 5.1 One-point function To better understand thestructive convergence to properties work of out the the nonlinear one-point sigma function. model, The it generating is function in- for the one-point rise to long range eigenvalueare correlations consistent scaling as with 1 lowaddress, order the moments issue, which thatstatistics, can the for be corrections the terms calculated should independently.not nullify the able We universal to also term. solve this Alsothe problem, in spectral this but correlator case we for we do were find a significant quantitative difference between neither do we knowmodes. the effect As of was the alreadyultimately coupling realized should between in cancel the 1984 in Goldstonedate [ order modes it to and arrive has massive at notpaper the been is universal more shown GUE modest. that result. We such show However, by cancellations to a this indeed perturbative do calculation that happen. the The goal of this model of the Gaussian Unitaryspontaneouls Ensemble breaks the which U( is given insectors appendix to U( is based on this patternGUE. of However, spontaneous this argument symmetry does breaking not and determine is the given prefactor of by this the universal one term, of and the where the right-handderivative side as a is derivative with obtained respect after to a partial integration by expressing the the two-point function, ( resolvent is given by JHEP07(2020)193 (5.9) (5.19) (5.15) (5.16) (5.17) (5.18) (5.10) (5.11) (5.12) (5.13) (5.14) . k U µ k X µ U µ α X 0 µ µ α , 0 µ 2 X . k D ¯ X . ¯ Gσ Gσ 2 ! √ ¯ a − µ σ D U . µ 4 T µ , X T . − 1 1 k k µ = 0. The saddle point result for the = 0 . 2 − α D = 0 0 µ , z =1 =1 ¯ a U ∞ ∞ 1 µ 0 D √ α k k X X ¯ µ G p µ σ X √ kn 2 2 X + δ D aδ − σ 1 σ 1 ¯ a/ ααααα. a 2 √ lm = ¯ = Tr σ D ¯ + Gσ δ σ – 11 – tr = ¯ ) = Tr µ ) = √ − z + − α µν µ a U z µ 2 δ ( z a +
tr z σ X ¯ , we expand the logarithm as G 2 µ ¯ = 1 k + a G , vanishes in the replica limit. The first nonvanishing , is given by the diagram a i 0 i ¯ a i = 3 5 =3 ∞ ) ) k X mn ν µ ¯ G U U µ X µ α Tr X kl µ X D µ µ σ α h α √ α 0 µ 0 µ + P P z Tr( Tr( 0 0 a a Tr log( about its saddle point Tr Tr ] h − h µ 14 a 6= 0 this follows from the fact that Tr µ We now calculate the resolvent to two-loop order in this expansion. The one-loop contribution, contribution, while the vertices of the loop expansion are given by This results in the propagator We expand Using the saddle point equation for and is given by For resolvent is equal to The resolvent thus satisfies the saddle point equation is given by [ with where the trace Tr is over the gamma matrices. By inspection one realizes that a solution JHEP07(2020)193 s ). do . 5.6 U ρ U ν and (5.27) (5.25) (5.26) (5.23) (5.24) (5.20) (5.21) (5.22) X X U U ν ν p m X X , − − U and µ + n only depends N X 4 U . µ µ ! U ν Tr T X U µ U ρ X p m X U µ X µ U ν X p α 0 U ν X 1) U µ µ X − X ( U µ X commute or anti-commute σ X =0 m , ) Tr U p ν X D s gamma matrices in z 2 Tr ( 2 X √ factor of the replica limit ( . 1) 2 ¯ 1 G D lk ν − m ¯ Gσ ( 2 α
¯ /n , Gσ ρ and kl µ n T U ρ µ 1 Tr s α T n T ¯ Gσ U 3 X µ m T lk ν ν − η. − ! X α αααα T ν − 2 − 1 U µ kl µ T tr 1 m 2 N α X 2 − µ kk 0 1 = 2 + α ¯ α 0 – 12 – Gσ 2 ααα 1 ¯ 4 4 ν Gσ s kk 0 µ tr T m ν σ . M m α X ) T ¯ α T m Gσ 9 contains the gamma matrices that we obtain σ µ T µν P − ) tr 1 /z T µ U D µ U z ρ . Since the propagtor is replica symmetric, the sum over 1 − l T ( N =1 m (1 √ n X 2 X 1 ¯ P G − = k,l O ) gamma matrices in 2 2
. This results in 1 k ¯ 1 + ¯ D Gσ µ ) G 4 which is cancelled by the 1 2 ) µ Tr m expansion can be obtained by using various combinatorial iden- σ 2 ¯ 0 − T µ G 5 n 4 η ) it is immediately clear that the only nonvanishing contractions α T ¯ /z σ G (1 − 5 2 − )tr 7 . The normalized fourth moment is thus given by + 2 ¯ G z z (1 1 5.17 D η ( 2 0 B ¯ G (6 D 6 0 m N µνρ 1 X 12 σ 2 ν 0 * X ¯ + G 6 0 2 n 1 σ η D 5 X σ 7 4 µ 0 z 1 n ¯ σ X G − 0 m 1 = → X lim 4 n 1 D gives an overall factor The traces arenot only have nonzero in if common.common For gamma matrices this gives the combinatorial factor we find after taking the replica limit which permits two different contraction patterns. For the diagram Next we consider the contribution The coefficients of the 1 tities, see appendix The trace candepending be on evaluated the number by ofon observing gamma the matrices that they number have of in indices common, in while k Combining this with the factors From the progator ( are those with replica indices Writing out the indices and the traces over the replica space we obtain JHEP07(2020)193 ) y ). 5.3 5.8 and 2 (5.33) (5.30) (5.29) (5.31) (5.32) (5.28) 5.3 x only runs ¯ U ν Gσ 2 s X 2 s matrix. The m U 2 − µ . 2 7 − n 4 is to account X 2 defined in ( m 2 / U /z ν m + and µ ]. 1 . T × + X 1 1 m replaced by 11 U n µ T m m z = 0 T X is Hermitian. The dia- − Tr D is a 2 U 1 µ expansion of the resolvent √ U close to the support of the ν 2 / µ X D X z a /z 22 U µ U √ µ , we have obtained the normal- ¯ a ¯ X Gσ σ σ Z X 2 s 6 Tr 2 i m + T 2 T m ), and y T + . + 2 ¯ 5.2 ) − Gσ η 9 ν 1 22 αααα T ν ¯ a clashes with our notation of /z 2 vanishes anyway. We thus obtain the large expansion of the resolvent can be evaluated T tr + 3 6 2 (1 . The combinatorial factor 1 T η − /z ¯ O m Gσ ). Note that the sum over 1 B 1 – 13 – , ααα + 2 1 m ) tr 6 7 2 T m and 5.26 = 0 z T T α 1 = 5 + 6 . s D tr − ¯ 6 m , see equation ( Gσ 6 2 µ 1) 1 n √ σ M T / µ − 1 ( D T =0 11 m s X see appendix √ ¯ a − 1 s σ σ 1 =0 , m ], 2 N ) (1 − + X 0 9 m − 2 62 x [ /z ] refers to the single-trace chord diagrams with three chords all intersecting with m µν =0 N X 6 + 1 N (1 62 X T 2 m O 11 ¯ G 6 4 1 ¯ a in [ = 0 is the same as for the one-point function with 1 2 + σ 2 s 6 σ 7 m T ¯ 12 µ 2 G 6 ¯ a 7 η G 2 − σ 6 z σ 7 1 σ is a vector of length 2 1 ) is thus equal to ¯ 4 G − N 1 z 2 while the sum over odd m = D 1 5.26 2 N/ 1 × D If we continue this expansion we will recover the full 1 To summarize, starting from the generating function The notation 1 expansion of the correction for the 11-sector and the 22-sector, respectively, each other. It is a bit unfortunate that the notation 5.2 Loop expansionThe of saddle-point the equation two-point for function thebut two-point now function has thesolution same with form as equation ( in agreement with an explicit moment calculation of theof SYK the model SYK [ model.spectrum and This will expansion have is tofor not be more well resummed discussion to of behaved obtain this for nonperturbative issue. results, see section where we have used combinatorial identities to obtain theized coefficient sixth of moment 1 is equal to z The second diagram The sums for the leadingas order term the in contribution the 1 for the prime on theup sums to in equation ( gram ( where the phase factor is due to the fact that only JHEP07(2020)193 (5.38) (5.35) (5.36) (5.37) (5.39) (5.40) (5.34) + . , ), in this order. U µ 1 k − + X ! 4 k 5.11 U µ 21 µ )) ! q ! y α N X U . ( µ U µ µ 2 ··· 22 0 X 4. α + ¯ α X G 0 µ 0 / + = µ 2 U µ α α 3 2 µ σ α 0 µ µ . 0 X T X 22 T ) 0 − q µ 12 µ P µ X | x α X D µ tr ¯ ( µ Gσ N X | ¯ 11 0 , √ G ¯ a ))(1 D D D − ¯ 4 ) σ Gσ α x p ! y √ √ (
1 µ − 4 x 2 Tr n =0 T ( | 0 I x N 1 µ 1 1 k k ¯ D )
G ) X µ ¯ | T 2 G 6 y 2 0 y 1 n µ 2 ( ( =1 =1 1 4 σ ) we have two contributions. The first σ ∞ ∞ Tr D X 0 2 σ k k X X ¯ according to equation ( 2 G 1 4 ¯ α µ G − 1 22 0 + 0 ) T D n 22 11 α α 1 I x C.2 a P ) ( − − tr = Tr = 22 2 0 x ))(1 tr – 14 – ), as indicated by the primes, only runs over the 1 ( ) = Tr P ¯ 2 11 0 y µ G ¯ ( q U D α G µ and ¯ T tr N µν σ ¯ 0 G δ tr 5.39 √ X )
D 11 σ µ µ x * a = 1 a √ D ( X = ) 0 4 i ¯ 1 y 2 n G 1 y ¯ D ( 1 G 1 2 qp ν 4 µ 2 D * 1 σ n X α x ¯ µ G 0 6 pq µ ) T D α σ σ x α ) = ( h √ 11 − 2 , in equation ( P + 2 ¯ G + x, y (1 tr 3 ( 2 µ 0 z = 0 2 resulting in the combinatorial factor of 1 y σ 1 C D 3 3 µ ) are related to ¯ σ µ x 1 X y √ T D 4 0 ( 2 2 σ * ¯ 2 G 1 σ µ Tr log( n D 1 X µ < N/ D − = = = 1 n ) and x ( ¯ G We first calculate the one-loop contribution to the two-point correlator The propagator and vertices follow from the expansion of the logarithm in the action and that the sum over even values of where we have used that As is thecontribution case is for given by the GUE (see appendix The propagator is thus given by where The saddle point value of the resolvent is thus given by JHEP07(2020)193 + = 4. U ν (5.42) (5.43) (5.44) (5.45) (5.41) q X 12 ν α i U , but is not ν 2 )) X )) N i y 21 ν ( = 2 and α − ¯ G q ) y 22 0 we show the GUE . ( y α ( 2 G )) ¯ G Tr y 0 2 ( − 2 σ ) , + ν ¯ 3 G 2 i X − ) 2 ! y )) σ U µ + U µ y y − ( X − ))(1 1 X ( ··· 2 x x 2. In figure µ 21 µ ¯ G ( , it is puzzling how the correction ( G 6= 0) both for / + α α µ 2 2 2 ¯ 0 ··· ))(1 G 3 T µ U µ σ ))( ) µ D /D ) x ) T N/ µ + ( x 0 i y X 2 y − ( 2 X ( ( 3 1 µ π ¯ 2 12 µ ¯ 2 − G µ G 2 T = 2 ¯ α 2 ¯ 0 2 X G D G ¯ x Gσ ) ))(1 σ ) σ ( 4 11 0 √ x D µ x x − y α ( G ( ( ≈ − . 1 − X − 4 2 2 µ 2 2
4 ] x 4 ¯ − Tr T ¯ ¯ G / (1 – 15 – (1 0 G 2 G y ) 2 2 ) 6 D 1 2 2 2 2 4 = 4 result. We observe that in both cases we have y N ) )) σ Tr σ ( σ i D x µ )) 2 )) y q y 3 2 X y y 2 6 1 ) − ¯ + ( 1 ( 18 − G − σ y 22 0 D ¯ ) ¯ + 2 0 G x G x (1 x α x ) ( ) α 1 ( limit of the two-point correlator for the GUE (see ap- − 2 ( ( x x − tr G π 3 22 ( + 3 ( x [ )) ( ¯ ( 2 ¯ P ¯ h N G 11 0 1 y G G Dπ ( 2 2 q 2 2 ( − α tr N ¯ 1 1 π π sin σ G σ tr ) 4 2 D µ σ q x * − − T N = 2 result is much larger, by a factor of order − − √ ( ) 4 ¯ − y (1 G q 1 y = = 3 D ( 1 2 4 2 y 3 c 1 1 (1 σ n 1 x 3 i ¯ 0 D µ G 0 ) 6 x ) T y α σ 4 x ( µ 2 σ ( X 11 ρ − 3 ) P − 2 2 ¯ x G (1 ( = 2 σ tr 3 4 D 0 = 2 can nullify the GUE contribution in order to get Poisson statistics. The ρ y σ ). It is given by 1 h D 3 q 4 σ µ is the total number of eigenvalues, x = 0 and the sum of the correction terms 1 X √ D 4 = 0 term is the large 2 4 D C.2 µ σ * 2 1 σ D n µ 1 The sum of the two contributions is equal to The second one-loop contribution to the two-point function is given by D = = = 1 n The correction of the qualitatively different from that ofa the scale separation betweenSince the other GUE correction result terms andterms are the for of correction higher due order to in the 1 massive modes. where result This is exactly the asymptotic behavior of The corresponding spectral correlationreal function axis follows from the discontinuity across the The pendix after taking the replica limit. JHEP07(2020)193 1 we (5.46) (5.47) (5.48) (5.50) (5.49) = 0 or = 0 term | = 2 while | µ µ is the level | µ N | q 6= 0 terms. The µ πσ/D , 2 )) k values with (2 µ 1 , ), and ∆ = 1 + , k − 1 ( − iω . For ( q = 0 terms are strongly suppressed (0), we find N y | 6 ρ q . N . ) k µ 1 2 terms) are important for | → y N 2 1) = 4 (blue) is due to the − . Note that we only get the universal − 1 2 x q − − 4 2 ,N = q x 1 N . In the center of the spectrum this gives =1 , and in the figure its value is 142. , the 2 ) + N/ k X µ − µ k 2 ω T T 2 T = 0 = ( ( N ··· ( 1 π Re | 6 | ω σ ) are given by 2 2 µ 2 – 16 – + µ | k N π | ( GUE 2 terms for ∆ 2 = ρ ) = 2 | k N q/N = 2 (red) and = µ ( | | ) + X δR q µ ) = ω | 2 ω ( = 4. The details of this cancellation are not clear, in 2 | ( (0) 1 for D µ c 2 | q − 2 2 ], where it was argued that the corrections to the two-point 2 ¯ ρ ¯ ρ GUE 2 1 15 ρ − δR ∼ µ ) = = 0 or T ω | = 4 correction scales as ( 2 µ q | ρ 1. Therefore, we can approximate the correction to the GUE result by ]. Since 14 = 2 and µ is the degeneracy of massive modes with mass q T terms) and the massive modes ( . Perturbative calculation of the two-point correlator. The GUE result is the k N 2 ), while the correction N (1) from the correct SYK result. This result agrees with [ = 5.42 O | µ where spacing in the center of the band. The GUE result if weby rescale by the saddle point result forspectral the correlation spectral function density are which of differs the form In terms of the unfolded energy difference which is in agreement with the results in figure so that the correction to the two point correlator is given by SYK model [ with respect to the have that expanding the denominator tothe first result order in only way out| seems to betheir that contributions interaction cancel termsparticular for between because the the zero lowest modes order contribution ( of such terms was large in the complex Figure 2 in ( ratio of the JHEP07(2020)193 . , + N N U ρ A . U ρ X (5.52) (5.53) (5.54) (5.55) (5.51) X 12 ρ and the ω α U , ν 3 U ν U ν X X U µ X 2)!) 21 ν U µ X α q/ U ρ matrices for X U µ µ X γ ! T X U ν µ 2)!(( N 12 µ X α q/ , it cannot be naively P U µ U q 3 ρ X X − ) to , Tr 12 ρ 3 ) N α ( y U ν A.8 ( 3 + , 8 2)!) − = 0 terms include the Cooperon 1 X G ) | ! − and even 21 ν q/ x µ U µ q | ( α N ρ N q ! X G 22 0 N T µ 2 2 2)!(( α N σ α U q/ 0 µ 1 2 ρ q/ T 3 X 1) µ αααααααα. X − 12 µ − − ) + = 4. As was discussed at end of appendix 1 α tr ω ) ( D N β ¯ 11 0 Gσ α y ( a – 17 – ( √ 3 − 22 = 16( G − Tr GUE 2 P ) 0
U ρ ρ x tr ( µ q X = 1 or Tr N ν α X U G 1 8 ν T ) = β 2 11 0 22 0 2 ) to the 33 moment is given by X ω σ P α ( α q/ ν U µ 2 tr . The sum is only nonzero when the summation indices are 22 T tr ρ 1) X ρ 5.52 P − 11 0 U − ρ )) tr 1 α = 0 term after the Fierz transformation, gives the GUE result. y X 3( ) ) are much larger than the span of the spectrum, while ( D and tr | y U σ ν 2 k ( * µ ( ) √ ¯ ν ) | X G y G y ( 1 ) U µ ] is thus well-approximated by ( D − 4 -model was derived for arbitrary 4 x 1 X ¯ ( n σ 15 ¯ G G µ 0 ) 2, we thus get an overall combinatorial factor of 1/16, so that the total ) G , the ))(1 T Tr x α 2 q x ρ ( x ( σ ( 4 11 T N/ 4 µ 2 ν ¯ /N P ¯ G T ¯ G T 1 G 6 tr µ − − σ T ∼ 1 D 5 σ 0 (1 k √ D 1 T 5 6 µνρ X 2 * µνρ X σ Although the D n 4 1 1 × D = 4 case. D = 3 = 3 1 n either all even orindices all up odd. to Since thecontribution of sum the over diagram the ( many-body space is only over the even which can be proved bysame applying for the the Fierz sum transformation over ( There is another remarkable combinatorial identity (note that there is no prime) We find that go beyond scale fluctuations. We only calculate the three-loop diagram given by β 5.2.1 Higher orderIt corrections becomes increasingly hardof to the two-point calculate correlation higher function. order Such corrections terms to are the important for loop-expansion ensemble fluctuations applied to cases with Dysonthe index reason is thatTo the extract the(mod GOE 8) result, = 0 we beforecontributions the have Fierz characteristic exploited transformation. of the Then the the symmetry GOE of result. the Similar arguments can be made for the The correlator of [ which is exactly the result from the above perturbative calculation. Since JHEP07(2020)193 ], 14 (5.58) (5.62) (5.60) (5.61) (5.56) (5.57) ] . . 65 3 . , 2], while the resolvent , ) is strongly oscillating, 4 2 64 k 2)!) − p − + 2 5.59 q/ ) p η ) (5.59) . ! ) η − 2 . However, we can also expand ( z 2)!(( N / ( η p 1) (1 q/ QH . 2 2 +1 − 3 ) ˜ z k η k z ( M 2 ) gives the correct result for the 33 0 , − , and is a very nontrivial check of the ( k g p η 2 6 p p 1 2 k +1 η k N / in this case, k ( 5.42 2 1) 0 + − 2 1) k − 3 − g +1) a − k k − η 1 k − k ( ( a , p η k − p ( η q η − N p k + 1 – 18 – = 2 X k X − k =0 p ∞ 1 1) k X p = 2 z ) − − ) = 3 η µ ( z 1 T ) = ( η − k =0 z p G X ( | − ). Since the resolvent of the Q-Hermite spectral density is (1 µ N 1 | = QH 3.2 k p G a ) = η =0 | N ( X µ ) = | p z 2: QH 4 1 2 ( / η ˜ 2 4 0 M 1 g -model calculation. D − σ 2 z √ − 2 is given in equation ( z/ η = 1, one might think that a Borel resummation might lead to a convergent result [ -model results in an expansion of the resolvent of the SYK model in powers of sense? σ η The spectral density that can be derived from derived from ( ) = z ( 0 Naively, because the spectrum correspondingweight to function this resolvent of has the theexpansion. same Q-Hermite It support polynomials, turns as the out we that expect the that expansion this is surprisingly gives simple a [ much better but we were not ablethe to resolvent in work powers out of the sums for arbitrary and to make sense ofcase the result, the asymptotic series has to be resummed. Based on the with located on the interval where known analytically, we can get the coefficients g Each of the terms has aof cut the in the SYK complex model plane on has the a interval [ cut beyond that. In the Q-Hermite approximation, the cut is correctness of the 5.3 Does the expansion in powers ofThe the resolvent of the semicircle make moment after using the identity This agrees with the moment calculation in section which together with the last term of equation ( JHEP07(2020)193 ), by (6.2) (6.1) z ( +1 k η 2 0 g 6. Since the ≤ ) (5.63) ), and normalized z ( σ ), and three different η k, l 9 0 E g ( for ! (red curve). , ) SYK 2 i l ρ 4 ,n c 0 QH 6 k . The eighth order correction δρ ˜ c M h M 3 3 4 η ) 1% with the exact result, which . ) + . z − := ) 1 -δρ -δρ ( E x η ) (black curve). Second and third, the 1 + QH ( QH QH n 7 ( 0 = 4) to the spectral density. ˜ E η -ρ g − -ρ -ρ i M ( SYK k ) QH 6 ( H ρ SYK SYK SYK i ρ ρ ρ M QH 4 QH c 6 0 ρ E 7( δρ M , =0 i X i + − n ) x n H ( . QH 8 + – 19 – -1 SYK 4 6 Tr m M QH σ m ρ M = 3) and − N H k 2 ( ) = Tr -2 ) + ( 3 h SYK x 8 z ( ( 1, this expansion is convergent for the Q-Hermite spectral δρ ρ M :=
QH < | 0.004 0.002 0.000 G + -0.002 -0.004 m,n η | (blue curve), and 9th order, ˜ M 3 ) = z δρ and ( δρ(E) η ) + SYK − E ( G 1 √ QH ρ = = 32 the sixth order result agrees better than 0 | . The difference of the spectral density of the SYK model, η N 0 g | For We could also expand the exact resolvent of the SYK model in powers of numerics in this paper are doneto in one, dimensionless units, it is and convenient spectral to densities studymoments are the normalized rescaled (by the many-body variance which were already introduced in section spectral density in terms of the Q-Hermite density and the Q-Hermite polynomials, We will obtain explicit results for the expectation values only gives a slight improvement. 6 Analytical calculation ofIn the fluctuations this of section, the we expansion calculate coefficients the covariance matrix of the expansion coefficients of the with corresponding corrections is much better than the Q-Hermite approximation, see figure requiring that the momentsmoments up this to gives a given order are the same. To the eighth order in the expansion in powers of theto resolvent 7th of order, the semicircle scaled to the support of the Q-HermiteSince result density. Figure 3 approximations. First, the Q-Hermite approximation, JHEP07(2020)193 also (6.6) (6.7) (6.8) (6.3) (6.4) (6.5) mi = 0 for c ) must be mi m f H c ) coincide with γ ) y 6.8 ( ) = 0 realization by η j m ) we have H . An efficient way to ) . x H n ( x c n . ( γ k η i m H QH p . c , + H 2 ρ and γ ) nj p jn y ) 2 ) ( . 1 ) T 2 − 1 + are related to the double-trace / x = 7. In this paper we will not go is given by ) f , QH ( ( = 7, which means we might as well 1) T ) i j i η ρ i i i, j, m j η ) − c j f n c ( k c H x Tr i ( i ) = 0 for any (( k ) ( c k c m h + x h η ( QH m k H m,n m QH c mi , is ˜ ˜ ρ γ 1) f k M H M QH must also be triangular, we can consistently n = 7 and c x ρ − η ik y = 7 and = 4, this means equation ( i γ im ( a i,j η ik m ) p m mi X 2 q a 1 m f – 20 – − i p =0 k − = = 1 + X k X dxx f k i ( P up to j p = c Z dxdyx ) i i η j Tr c m,n X mi c = ) = h Z 1 ], implying Tr ( i f − x i c = nj ( = 32 and n 11 j h mi 2 f η i i c − (1 f j i c c N m H mi i ) are orthogonal with respect to h is the chirality Dirac matrix in even dimensions. However, ) can be obtained from Riordan-Touchard formula, introduced = f − c x η different Dirac matrices, we have Tr ( h σ c ( =0 ( mod 8 = 2 or 6 (GUE universality class), the two blocks have the same p γ η k i i,j X QH forms a triangular matrix of infinite size. The coefficients 2 i,j X +2 ): N + N H . For ˜ ) up to some finite values of N QH M m = = mi − ˜ f 6.5 5.60 2 M is odd. The covariance matrix m,n i , using that ˜ , where M + mi m < N/q f m,n due to computational complexity, but we do caution that Tr ( m ˜ 6 M , for the calculation of 6 ˜ is a product of M c m,n for the double-trace moments up to , which means γ ˜ An important caveat is in order: in the numerics we used an irreducible block of The covariances of the stochastic coefficients M Except in the case of 2 m,n ˜ use beyond confronted for higher moments. eigenvalues realization by realization [ instead of since realization if M the random Hamiltonians to calculatething the to two-point do correlations, for which probing is RMT the universalities. appropriate This means we really should be looking at where the moments earlier as equation ( truncate equation ( calculate the vanish when Since the inverse of the triangular matrix where Note that because the m < i moments by JHEP07(2020)193 ) belong c ) has four b ;( 4 , ) 4 2 ) ) and ( ˜ ) β M 2 b 2 β Γ a 1 Γ Γ β 2 3 β Γ a 2 Γ Γ β 1 4 a Γ a 1 Γ ) Γ β 2 c 1 ( a a )Tr(Γ 1 )Tr(Γ )Tr(Γ α ’s. 1 4 β a Γ Γ 2 Γ ; on the other hand ( Γ α 2 3 and a a Γ Γ 2 Γ , since all the chords connect only within variables which are the cardinality of the ’s 1 2 α 4 β a , Γ d Γ 4 Γ Γ 1 1 ˜ ) 1 α M a a a ( – 21 – Tr(Γ . It is important to keep in mind that notationally Tr(Γ Tr(Γ 2 ) is part of the disconnected piece in 2 4 ,β a ,β ,a . 1 single-trace links in some of the chord diagram literature. For example, 1 3 . In a self-evident manner they represent (we will not ,β :( 4 ,β ,a , 2 4 2 2 X , 4 m,n X X 4 ,α ,a ,a ˜ ˜ 1 ˜ 1 M 1 M M α a a cross links 4 4 4 because both contain chords that go from one backbone to the − − − 4 , ) 4 backbones b q ( ˜ q q N N N M N N N − − − ) 2 ) 2 ) 2 a b c ( ( ( ], with two additional rules for the ) has two cross links and two single-trace links. c we draw three chord diagrams with two backbones that represent some of the 62 . Three contractions in 4 ) belongs to the disconnected part of a Latin-letter subscripts on the cross-linked Greek-letter subscripts on the single-trace-linked Since the ensemble average is taken over a Gaussian distribution and thus reduced to • • The combinatorics for double-tracecussed moments in are [ much like the single-trace moments dis- Note ( single traces, and we call suchto chords the connected part of other, and we call such chords we used adopt Einstein’s summation convention unless otherwise stated) a sum over Wick contractions, wethere can are represent two the traces, double we traces willhorizontal as use lines two chord are different diagrams. called horizontal Since linesin to attach figure the chords on.contractions Such that contribute to 6.1 General expression forIn double-trace this Wick section contractions werescaled give double-trace a moments general expression for double-trace contractions contributing to Figure 4 cross links; ( JHEP07(2020)193 ) k , }} (6.9) 6.12 L (6.11) (6.10) (6.12) , . . . , i − )–( 1 G i 6.9 is the num- k i ,...,V 1 ...a 2 i a = 0. 1 (naturally, i i } ⊂ { a a s k d i d , , } ) d } { d ,...,j { 1 , . . . , a j 2 { i . , )∆( } } , a D d 1 i { single-trace links, as one of the subscripts. a ( , i L a ,...,L N M , ) -variables in 1 − ! } d G has a value of ( G β was primarily used to denote intersection graphs, subscripts, c V , }{ G G, G k 1) } ⊂ { a { r 1 − d ( } } – 22 – d q d { { X ,...,i 1 Q G i V { )! , − k d illustrated in appendix G, q 5 1) N , cross links and 3 ) for a definition -variables containing -variables with , comes from “vertices” and “edges” of the corresponding intersection − ˜ L d d is even G k M ! E ( r D.5 ... qE | i G =2 s ! a j 1) V k and and d -variables with even number of Latin-letter subscripts, N − . 3 d ...β P , ( V q -variables whose subscripts denote intersecting chords, 1 ··· 3 j c X + ˜ β ]. In those references the letter M r G − i 66 q = sum of all = sum of all qV ...a
! = product of the factorials of all 1 k − i δ ... } i d a see equation ( β a d N ] and [ L =1 -variables with odd number of Latin-letter subscripts (and an arbitrary number d ( }{ i Y { = number of chords in = number of chord intersections in = set of all = multiplicity factor due to partitioning a 62 d { ) = sum of } = G G ··· d ∆ = product of Kronecker deltas enforcing the constraints d X ) = ) = G } M V E } { ( } } d d 3 to one vertex is The of Greek-letter subscripts) must be set toKronecker deltas zero. are needed to enforce that the total number of indices corresponding c d d { { Y { { The choice of letters ( • • 3 ∆( M nately convoluted. We encourageto the the readers examples to of look into the application ofgraphs, see ( [ but it should not cause confusion in the present context. As one can see, the most general description of the double trace combinatorics is unfortu- in which More specifically, if there are where More explicitly, a double-trace chord diagram ber of elements common andare only all common different to from the each sets other in this definition). The additional rules are as follows: regions in the Venn diagram of overlapping indices. In other words, JHEP07(2020)193 with (6.14) (6.15) (6.13) m,n ˜ illustrates M 5 is odd, in which ). , ]. This implies (or both), such q 62 1 − 4.11 4 α . C Γ q i γ or 1 ... , or more generally )). Figure − + 1 n q i 2 − C 3.2 k γ . ˜ α M , n Γ n n ··· . This means each term con- k 2 of ( 2 2 α 1 ˜ i η q M times another integer (and times − Γ N γ 1 m i 2 q N q γ N ˜ Tr 2 M / 2 / = 1) = 0 for all 1) − mn n must be even for the trace to be nonzero. q − 2 ,n 2 ( q 1 k 1 q ˜ ( M ˜ − M kq 2 1) ˜ − 1) M q N − – 23 – is odd, − = ( q n 1 2 = i , 2 ) = ( γ 2 k will factorize into ˜ i M α (2) m,n i γ Γ n ˜ M ··· H ... , but when 1 2 Tr − kq q α i , this implies the following reflection identity m 1) Γ γ T i 1 − q H is odd. Hence we only need to concern ourselves with the cases of i γ α γ Tr n h , this implies won’t make a difference: a potential difference only arises when + = N Tr (Γ αβ denotes the sum over all contractions that contribute to − − δ m C C i , there exists a charge conjugation matrix, either or (2) m,n γ ) = N 1 ˜ + β being even. M − C = 0 if Γ ). Analogous to the single trace situation, this implies a further factorization = 0 even before taking average, so C n α n + + H m,n m ˜ double-trace diagram with thisof single-trace link two removed chords and with asuch one single-trace an intersection diagram example. (whose value is every contraction’s value must containtributing a to 2 factor of σ property of a special class ofwith double-trace exactly contractions: one cross if a link single-trace and link nothing intersects else, then this chord diagram factorizes into a This reflection, together with the cyclicdihedral permutations group of the action Γ’s, on gives rise the to traces. a natural of the remaining unsummed subscript, see appendix D of reference [ where we also used that where exactly two cross links. This gives a precise meaning to equation ( M m Tr(Γ For any Tr If all but one subscript in the traces are summed, the result is a constant independent Choosing 4 (i) (v) (ii) (iv) (iii) case we have an extra factor ( There are a few otherating properties double-trace of averages: double-trace contractions that will be useful for evalu- 6.2 Some useful properties of double-trace moments JHEP07(2020)193 (7.1) given , m , 5 2 , = 32 and − 2 = 0 ˜ 10 N M i η 7 × . c 0 c = 4 the following h and spectral form 11629 , q , . × 7.2 = 0 = 1 = 32 i i 2 6 c c N 0 2 c c h h we discuss the number variance. These numbers agree reasonably , , 6 , , . 3 . General expressions for low-order 7.1 5 5 6 5 − − dy. − − − 4835943 i 10 58721678190449900000 10 10 ) 13063777128020000 10 10 10 72657269900 201028717157105439 2710088957667107403387 y ) = 0 for every realization. × ( × × × × H ρ = = = ) h c 5 6 6 2 , , , γ / 2 3 4 6 / 4397 ˜ ˜ ˜ . M M M = 6, we obtain for ∆ +∆ – 24 – 6 78799 74962 154203 55032 . . . . x − − x = Z = 32. In section = = 1 = 2 = 6 = 4 = i i i i i N 6 6 4 5 6 2 ¯ n c c c c c is normalized to unity and spectral density is normalized to unity . i, j, m, n h 0 2 3 4 6: c c c c . The factorization of a chord diagram. E h h h h QH ≤ ρ , , , , 5 6 6 6 i, j = 0 even before averaging. − − − − ) up to = 0 is because Tr ((1 + . 1 1701 1 8812289619 10 10 10 10 c c Figure 5 6.5 2906290796000 20902043404832 161640200 14912736088383 × × × × 7.3 = = = ), we also need the following results: 5 3 4 , , , 5 3 4 47581 68918 59628 11476 ˜ ˜ ˜ , . . . . = 1 and M M M 6.13 0 c = 1 = 3 = 3 = 9 = 2 = 1 is simply because i i i i i 0 is the average spectral density, then the average number of levels in an interval of 3 2 4 2 5 2 0 2 2 2 c i c c c c c ) h h h h h x ( That ρ 5 h = 4 which is the case we study numerically later. Apart from the moments width ∆ is given by for every realization. That ization of the SYK HamiltonianThe for fluctuations due tofactor ensemble is averaging evaluated are in analyzed in section 7.1 Number variance If moments are given in appendix 7 Numerical analysis ofIn spectral this section correlations we analyze the spectral correlations of the eigenvalues obtained by diagonal- Note that well with the numerical results presented in figure Then using equation ( nonzero covariances up to 6.3 Low-order covariancesIn of this Q-Hermite section expansion we coefficients giveq explicit results for the covariancesby up equation to ( sixth order for JHEP07(2020)193 . x (7.5) (7.6) (7.7) (7.8) (7.2) (7.3) (7.4) ]. Even if 67 . i ) y ( ρ h defined as , ) dy k k dy. x ) x i −∞ ) − Z y . ( dy. +1 dy, i ≈ } ρ ) k ) ) y y x dx. δn 1 ( ( ( i − h { − i ρ ρ ) p ) ) h 2 x k y x / ( 2 var ( x x / ρ ( ρ −∞ − h , we speak of spectral ergodicity [ ∆ ( ≡ +∆ . In particular, this may happen in many-body ρ Z p – 25 – 2 h x − ) ). = / 2 x x N / n 0 ( x Z = (¯ i ( ∆ ) = +∆ ) in coordinates where the level density is constant. ) mode number function is known as the number variance, 2 → ∞ 0 k dx x − x 1 = N x Σ ( x n − ), to the spectral density of a single disorder realiza- 7.5 n Z p N N − x x ( ( = ρ +1 h 0 k smo 1 δn x ρ − =1 as a function of the number of levels with energy less than p k X x = , it can give a large contribution to the number variance for intervals 0 k x /N as a function of ¯ 1 δn i ∼ ) x ( ρ is no longer much smaller than − h n ) x ( To calculate the number variance as a function of the average number of levels in an smo This procedure is usually referrednecessary to for as the unfolding. calculation We of emphasizeto the that number invert this variance, the procedure but mode is it not number makes it function much more convenient and the new level sequence is obtained by adding the differences, formation, it is simplestIf to the invert constant ( is equal to one, the transformation is particularly simple: The spacing of the levels in the new coordinates is thus given by In other words, we have to invert the Since the number of eigenvalues in an interval does not change under a coordinate trans- ρ where systems where the number of levels increases exponentially withinterval the we number need of to particles. know The average eigenvalue densityerage can of be the obtaineda in spectral two fit density ways. of for ation. a First smooth as very If the function, large the ensemble ensemble. two av- are the Second, same for for large matrices, as The variance of so that the deviation from the average number is given by The actual number of levels in the interval is equal to JHEP07(2020)193 , 2 2 / 1 200 (7.9) − 268) = 32. . The (7.10) (7.11) 2 q N n/ N GOE SYK 293) 150 n/ (¯ ∼ ) = 3. The Thouless n q (¯ n 100 (which gives (¯ 2 q N 2 / 50 2 = 4 and n q ) (black points) versus the average . , n ,8hodrQ-Hermite order 8th q=4, 32, = N nebeUnfolding Ensemble 2 2 (¯ ] for 2 ¯ ¯ n n 0 . 1 1 2 3.0 2.5 2.0 1.5 1.0 0.5 0.0 / − − 16 , 1 (n) 15 2 q q ]. However, this is only the first term in a N N Σ q N 53 1 2 1 2 , – 26 – 0000 10 ∼ ∼ 52 , ) = ) Th 2 n n n 17 (¯ (¯ where we plot Σ 2 2 – 8000 6 GOE cn a+ SYK Σ Σ 15 , due to the relative error in the spectral density of 4 6000 n . In the left figure we also show a quadratic fit (red curve) and the grey n 4000 in an interval for an ensemble of 400 SYK Hamiltonians with = 4). n q 2000 40 we see excellent agreement with the GOE (red curve) (see right figure), the number variance behaves as ,8hodrQ-Hermite order 8th q=4, 32, = N nebeUnfolding Ensemble n < 0 n 0 . The number variance of the eigenvalues of an ensemble of 400 SYK Hamiltonians versus 600 400 200 800 1200 1000 = 32 and To calculate the number variance, we unfold the spectrum by means of a fit to the en- As was argued in section (n) N 2 Σ This contribution can be subtractedensemble by according rescaling to the its eigenvalues“multi-pole” width of expansion, each [ realization and of influctuations the this that section give we risemodel systematically and to study random the the matrix long-wavelength discrepancy theory. between spectral correlations in the SYK 7.2 The effectAs of mentioned ensemble before, averaging the ensemble fluctuations contribute to the number variance as number of levels ¯ For small ¯ but for large distances,latter the result number is variance in grows goodfor quadratically, Σ agreement with analytical result of ¯ semble average of the spectralals. density including The up results to are eighth shown order in Q-Hermitian figure polynomi- which agrees with resultsenergy obtained scale previously can [ thus be estimated as number variance of the GOE (red curve). for large ¯ Figure 6 the length of the interval band depicts the error bars. The right figure which is a blow-up of the left figure also shows the JHEP07(2020)193 , n (7.14) (7.12) (7.13) statistic 3 0000 10 and roughly n 1)¯ 8000 , the ∆ SYK GOE − 7 2 √ 6000 n 4
. =
-th order: 4000 q
# l ) 32,
x =
( N norm excluding a small fraction , = 8 the systematic dependence η k ) 2 2000 l ¯ n . H L ) u k 3 ( c 2 u ) 0 l =1 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 7 + )Σ k X ) u n u ( ( with a Thouless energy that is about 2000 3 2 Δ 1 + n − " duK ) 1.0 1 x – 27 – ( 0 4. Indeed, as is shown in figure Z n/ QH ) = 2(1 ) due to the scale fluctuations is projected out in the ρ 0.8 u ) = ( . It is clear that for n 8 (¯ 7.11 , K ). ) = 3 statistic (left) projects out the quadratic part of the number x ∆ ( QH 0.6 3 3.7 ) given by (see figure ρ u x ( . QH,l l ¯ ρ K and ¯ 0.4 6 , = 0, in case where the number variance depends only weakly on ¯ QH 0.2 can be calculated by minimizing the ρ du k , ¯ ) c 4 , u ( we show the difference of the cumulative spectral density of the SYK model 0.0 QH 8 K 2.0 1.5 1.0 0.5 0.0 ρ -0.5 1 , ¯ − . The kernel of the ∆ ) statistic in essence only measures level fluctuations up to ( 2 2 , K(x) 1 n √ (¯ R QH 3 ρ In figure Next we explore the origin of other contributions to the discrepancy between the SYK This quadratic contribution ( statistic defined by 3 density obtained in equation ( and ¯ is of the sameincluding order higher as values of the statistical fluctuations, and no further gains can be made by The coefficients of the eigenvalues in bothare tails nonvanishing with of an the ensemble spectrum. average that Generically, agrees all with even the and fit odd to coefficients the average spectral of eigenvalues moves togetherculating relative the to average the mode ensemble numbera average. by single fitting configuration. This a The is smoothway by smoothened function achieved expanding to by spectral it the cal- density in spectral can Q-Hermite density be polynomials, of obtained truncated in at a systematic agrees with the GOE tolevel much spacings. larger values of ¯ model and the Wigner-Dyson ensembles.ations We of do the this eigenvalues by of eliminating each the realization of “collective” fluctu- the ensemble in which a macroscopic number Since the ∆ corresponds to the number variance at ¯ ∆ with smoothening kernel Figure 7 variance resulting in agreement with random matrix theory to much larger distances (right figure). JHEP07(2020)193 1.0 . In 6000. 10 2000 ≈ fluctuate 4 n
= 7
c q
0.5 GOE SYK in the interval. 32,
becomes of the =
) in Q-Hermite 1500 n N and x n ( 5 c N E 0.0 order n 1000 Y,6th SYK, order 8th SYK, 0.5 16 = 2048. Second, because - / 500 2 N/ to determine the number of eigen- oa Unfolding Local ,8hodrQ-Hermite order 8th q=4, 32, = N 1.0 6 4 2 0 2 4 6 - - - - 0 QH,l ) 1.5 1.0 0.5 0.0 3.0 2.5 2.0 E ρ ( N (n) 4000. The right figure is a blow-up of the left 2 Δ Σ ¯ n < 1.0 – 28 – < 0000 10 GOE Constant SYK 8000 0.5 = 8. The Thouless energy is about 2000 level spacings, but l 6000 fluctuate about a nonzero value. 8 E n 0.0 Y,2dorder 2nd SYK, order 4th SYK, c 4000 and 2. / 6 2 c . In both figures, the GOE result is depicted by the red curve. -0.5 n 2000 N/ oa Unfolding Local ,8hodrQ-Hermite order 8th q=4, 32, = N the number variance saturates to a constant and decreases beyond ¯ 0 we show results for 0 n -1.0 2.5 2.0 1.5 1.0 0.5 0.0 3.0 20 40 we show the number variance when an increasing number of coefficients has . Number variance (black points) versus the average number of levels . The difference between the mode number function of the SYK model and the Q-Hermite 9 -40 -20 (n) 2 11 Σ Next we study the dependence of the deviation of the number variance from the GOE The coefficients of the expansion of the mode number function We now calculate the number variance using ¯ ΔN(E) agreement with our analyticalabout results, zero, while the first four coefficients and result on the numberfigure of Q-Hermite polynomials that have been taken into account. In nated fluctuations with wavelengththe larger total than number of about eigenvaluessame 2 is order fixed, as the 2 variance is suppressed when ¯ polynomials as a function of the ensemble realization number are shown in figure values in the interval forto each configuration. reduce We the use statistical bothIn fluctuations. ensemble figure and For spectral the averaging for latter large we choose ¯ half-overlappingThis intervals. has two reasons. First, by unfolding configuration by configuration, we have elimi- The average number offit eigenvalues to in the the modea interval number constant is density determined calculated of by fromfigure the each for an data configuration. smaller eighth for ¯ order In 2500 the Q-Hermite left figure we compare the data to Figure 9 Figure 8 approximation to order 2the (left, systematic black), fluctuations 4 are (left, of red), the 6 same (right, order black) as and the 8 statistical (right, fluctuations. red). At order 8, JHEP07(2020)193 has 2 c 400 400 corresponding n lower left) when 300 300 is still quite accu- 6 8 5 7 11 c c c c n n n 200 200 r r ) in the first eight Q-Hermite have been put equal to their x ( 8 c N 100 100 and illustrates that the Thouless energy 7 c n 0 0 , 6 0.015 0.010 0.005 0.000 0.015 0.010 0.005 0.000 c -0.005 -0.010 -0.015 -0.005 -0.010 -0.015 , k k 5 c c c for an ensemble of 400 realizations. 400 400 – 29 – n of the mode number r k c 300 300 2 4 1 3 c c c c -dependence, but with a much smaller coefficient than in the have been put to the their ensemble-averaged values. In the n n n 200 200 r r 8 c have been fitted and 4 c until 100 100 3 and c 3 c , 0 0 2 c 0.015 0.010 0.005 0.000 0.015 0.010 0.005 0.000 -0.010 -0.015 -0.005 -0.010 -0.015 -0.005 . The expansion coefficients k k c c -dependence. n For second-order local unfolding, the number variance for large ¯ rately given by a quadraticcase ¯ of ensemble averaging,negligible For and fourth the order numberlinear and variance ¯ beyond, beyond the the Thouless quadratic energy contribution is is very well fitted by a a blow-up of the figuresincreases in gradually the with left the columnaccount number for in of small fitting Q-Hermite ¯ the polynomials localto that spectral the have density, until been shortest it wavelength taken saturates used into at for a unfolding. value of ¯ the Q-Hermite approximation isis given sensitive by to a whether sixth thezeros order length of the of polynomial, polynomial. the the interval Thatthe number is is number variance commensurate why of we with observe intervals the large usedof distance jumps the to of (see interval calculate figure the becomes the smallerorder number than Q-Hermite variance about polynomial, changes. half this the When effect distance is the between no length the longer zeros visible. of The the right sixth column which shows been fitted while second row, ensemble-averaged values. In the third (fourth)obtained row, the by first fitting five while (six) coefficients theensemble-averaged have remaining been values. ones up Because to the order bulk eight have of been the put deviation equal of to their the spectral density from been fitted to the spectral density of each configuration. In the upper row only Figure 10 polynomials versus the realization number JHEP07(2020)193 ). In 6 (7.15) c ( , 5 . 2000 2000 2000 2000 3 , c ) 4 i +cn ) 2 in the interval. , c y 3 ( a+bn a+bn GOE GOE SYK SYK n ρ , c GOE SYK GOE a+bn+cn 2 SYK 1500 1500 1500 ih 1500 ) , c x 1 ( c ρ n n n n 1000 1000 1000 1000 i − h ) y ( ρ ) 500 500 500 500 x ( ρ h ( ,6hodrQ-Hermite order 6th q=4, 32, = N oa Unfolding Local oa Unfolding Local ) ,2dodrQ-Hermite order 2nd q=4, 32, = N oa Unfolding Local oa Unfolding Local ,5hodrQ-Hermite order 5th q=4, 32, = N ,4hodrQ -Hermite order 4th q4, = 32, = N 0 0 0 y 0 3.0 2.5 2.0 1.5 1.0 0.5 0.0 3.0 2.5 2.0 1.5 1.0 0.5 0.0 6 5 4 3 2 1 0 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -dependence, and in the next two rows to a − x n ( (n) (n) (n) (n) 2 2 2 2 it Σ Σ Σ Σ – 30 – 3 dxdye 0000 10 0000 10 000 10 000 10 +cn 2 Z a+bn GOE GOE a+bn SYK SYK 8000 8000 8000 8000 GOE GOE SYK a+bn+cn SYK ) = x, y ( 6000 6000 6000 6000 c 2 n n ρ n n , and in the third (fourth) row the same for are fitted to the mode number for each configuration, in the second ) 4 y 2 4000 , c 4000 − c 4000 4000 3 x ( , c it 2 and , c 2000 2000 1 1 2000 2000 c c oa Unfolding Local dxdye ,2dodrQ-Hermite order 2nd q=4, 32, = N ,6hodrQ-Hermite order 6th q=4, 32, = N oa Unfolding Local 0 0 0 ,4hodrQ -Hermite order 4th q4, = 32, = N Q-Hermite order 5th q=4, 32, = N oa Unfolding Local Unfolding Local 40 20 80 60 values is shown in the right column. 0 0 Z 0.0 3.0 2.5 2.0 1.5 1.0 0.5 5 4 3 2 1 0 6 5 4 3 2 1 0 120 100 6 n . Number variance (black points) versus the average number of levels (n) (n) (n) (n) 2 2 2 2 ) = Σ Σ Σ Σ t ( c K Alternatively spectral correlations can befactor studied defined by as means of the connected spectral form row the same for the first row thelinear SYK one data (blue are curve). fittedfor The to smaller GOE a result quadratic is ¯ depicted by the red curve. A blow-up of7.3 the left figures Spectral form factor Figure 11 In the upper row JHEP07(2020)193 w = t ( c (7.18) (7.19) (7.17) (7.20) (7.16) = 4000 K = 1 and w i ) π , the results x 2 ( . t/ ) 12 unf → ρ u h
of moments. x, y ( c 2 1 /N ) we need to perform ρ +1 − 2 , limit is thus given by a u u w ) 2 2 2 i 7.15 / ) N 2 x ) πw ( ¯ log E 2 ρ u − h √ y ) ( − y 2 − e − 2 x w 1) . 2 . Its large (( 2 / i − 2 ) w ) u . 2 πw x ( ¯ t unv ) E ( , 2 θ t − 2 ρ − ( e h √ x ρ δ ( i , when we can use the diagonal approximation, N ) t/ N − t ))+ y e u – 31 – ( ) πb ρ y 2 πwb − ih √ x ) ( 2 x it ( -function ρ log(1+2 δ h u − dxdye ) = u ], but in this paper we only consider the connected spectral Z 12 x, y )(2 ( we show the spectral form factor for the unfolded spectra that u c πw ], 2 − ρ √ 14 13 (1 θ ) = 2 and = 0 [ t = 0 measures the fluctuations of the number of levels inside the Gaussian ) = ( we show the form factor for small times on a linear scale. Then the constant t c ¯ π E 12 0. Note that without the Gaussian cut-off the spectral form factor is the universal random matrix result. Therefore, the spectral form factor 2 K 13 this converges to the t/ ( → unv , t 2 ρ → ∞ GOE K In figure In figures We calculate the spectral form factor for the eigenvalues of the Hamiltonian of the w part of the two-point correlator gives the contribution For zero for 0) = 0 because ofform the factor normalization at of thewindow. spectral It density. becomes With only the zero cut-off, when the the spectral window is larger than the spectral support. is represented by the redshort curve times. in Comparing both the two figures. figures,ensemble we average It observe result agrees that in the with a spectral peak thecontribution fluctuations at SYK from due small spectra the to times. the disconnected up part The to of peakmagnitude very the should larger. two-point not correlator For be local which confused unfolding is with there many the orders is of no peak, and spectral form factor approaches We used four different values forthe the cutoff, width the of results the arewe Gaussian affected observe cutoff. considerable significantly For by finite still finite size larger size effects. values of effects. The Already GOE for result given by it is equal to unity. were used to calculate the numberfor variance unfolding in with the the previous ensemble section.figuration average In by are figure configuration given in with the eighth left order figure, Q-Hermite and polynomials, for unfolding in con- the right figure. with centroid We will evaluate the spectral formhave chosen factor the for prefactor such the that unfolded for spectrum large with becomes universal in termsdouble of scaling the limit variable which receives contributions from all ordersSYK in model. 1 In ordersome degree to of have smoothening a which well-behaved we sum will in do equation by introducing ( a Gaussian cutoff of width without this subtraction [ form factor. Sincespacing, we spectral have correlations are universal on thewhere scale of the average level The second term is the disconnected part. One could also study the spectral form factor JHEP07(2020)193 ), 5 7.21 (7.24) (7.21) (7.22) (7.23) 0 GOE 1000 = w SYK, 2000 = w SYK, 3000 = w SYK, 4000 = w SYK, 3000, the fitted ≤ w log(t) -5 . 2 0.0020 )) Fit . using the ensemble averaging for
2) 5 -10 t 2 4.8 − oa Unfolding Local ,8hodrQ-Hermite order 8th =4, q 32, = N . ) nt/ 10 0 ¯ ¯ n -2 -4 -6 -8 nt/ SYK Gaussian 0.0015 -10 -12 / 4 ×
. sin(¯ =
(1 2 (t)) q
c ) versus 39 ¯ n . t K ) 32, ( 1000
t N c = =
( b obtained by using the ensemble average of the t to the SYK data. For log(K N w K = 1 0.0010 t →∞ lim 1 ¯ w n limit of the number variance is given by 5 ) = − dtK ¯ n – 32 – n n (¯ ∞ 2000 ≈ 2
q and −∞ N =
) Σ Z w 3000
N n 0.0005 = 2
1 2 (¯ b π is close to the theoretical value given in equation ( w 2 ¯ n 2 4000 0
2 = Σ =
w GOE N 1000 = w SYK, 2000 = w SYK, 3000 = w SYK, 4000 = w SYK, ) = 282 0 the large ¯ b 0 / n 0.1 0.2 0.0000 (¯ 1 0.15 0.05 → 2 ) t t ≈ log(t) Σ ( K -5 5 ]: − 68 10 × ) contribution to the spectral form factor leads to a quadratic dependence can be approximated by (see equation ( t ( 26 . N δ -10 b 1 ,8hodrQ-Hermite order 8th =4, q 32, = N nebeUnfolding Ensemble = 4000 it is 15 % smaller. , we fit the parameters ≈ 0 . The spectral form factor for small . The connected spectral form factor -2 -4 -6 -8 w -10 -12 N 13 b (t)) ) remains finite for c t ( The number variance is related to the spectral form factor by a an integral over a K log(K of the number variance, If smoothening kernel [ Therefore, the In figure value of while for spectral density for unfolding (black points). The results are fittedThe to a constant Gaussian (red curve). Figure 13 Figure 12 unfolding (left) and unfoldingto configuration eighth by order configuration (right). including Q-Hermite polynomials up JHEP07(2020)193 ). 5 5 (8.1) 7.22 0 (which is = 3000 and n 0 0 ˜ w → M GOE GOE 1000 = w SYK, 2000 = w SYK, 3000 = w SYK, 4000 = w SYK, 1000 = w SYK, 2000 = w SYK, 3000 = w SYK, 4000 = w SYK, t m ˜ M − log(t) log(t) -5 -5 = 2000, m,n w ˜ M we discuss the calculation of n ) ! -10 -10 is 8.1 n ( = 1000, oa Unfolding Local Unfolding Local ,4hodrQ-Hermite order 4th =4, q 32, = N Q-Hermite order 6th q=4, 32, = N ! m 0 0 w m ) -2 -4 -6 -8 -2 -4 -6 -8 -10 -12 -10 -12 it ( we evaluate the subleading corrections for (t)) (t)) c c 4 8.2 m,n X − log(K log(K 10 iyt − 5 5 × e – 33 – 0 . ixs − and 3 . We conclude that a seemingly insignificant deviation 0 0 dsdte 4 − 15 GOE GOE 1000 = w SYK, 2000 = w SYK, 3000 = w SYK, 4000 = w SYK, 1000 = w SYK, 2000 = w SYK, 3000 = w SYK, 4000 = w SYK, ∞ −∞ 10 Z ) it is clear that this will result in a linear dependence of the × ∞ 7 log(t) log(t) . −∞ -5 7.22 -5 Z -dependence of the number variance. , 5 2 4 n 1 π − 4 10 = we show the spectral form factor for an increasing number of fitted coeffi- × c -10 -10 i 1 ) . 14 oa Unfolding Local ,5hodrQ-Hermite order 5th q=4, 32, = N oa Unfolding Local ,2dodrQ-Hermite order 2nd =4, q 32, = N y which is in good agreement with the spectral form factor for ( 0 0 , 7 ρ -8 -2 -4 -6 -8 -2 -4 -6 . Double logarithmic plot of the spectral form factor for increasing order of the Q-Hermite 4 4 -10 -12 -10 -12 ) − − x ( (t)) (t)) c c 10 10 ρ h 0. From equation ( In order to see the consistency of the calculation we compare two different ways to In figure × × = 4000, respectively.) log(K log(K 9 7 → . . to estimate the convergence to the RMT8.1 result. The calculationThe of connected high-order two-point moments correlator is given by 8 Double-trace moments andIn the this validity section of we explore random theconvergence matrix question to whether correlations random double-trace matrix moments correlations. canhigh-order shed In double-trace light moments, subsection on and the in section calculate the number variance:The directly results from are eigenvalues shown andof in from the figure using form equation factor ( matrix from result the for the GOE ¯ result gives rise to a large deviation from the random t number variance. The3 coefficient of the linear5 term for fifthw order unfolding is equal to polynomial used to fit the smoothened spectral density for each configuration. cients while the remaining onesFor up a to fourth eighth order or are fifth kept order equal fit, to the the ensemble spectral average. form factor seems to approach a constant for Figure 14 JHEP07(2020)193 kq − (8.4) (8.2) (8.3) 5000 5000 N 1 4000 4000 a Γ ] on the nested ... 1 12 3000 3000 − GOE SYK 40,fo K(t) from 4000, = w K(t) SYK, from 6000, = w SYK, k a n n Γ k a 2000 . Γ 2000 p 1) − ) Tr ( 1000 k N a 1000 SYK GOE 30,fo K(t) from 3000, = w K (t) from 4000, = w − Γ 1 m p ,8hodrQ-Hermite order 8th q=4, 32, = N nebeUnfolding Ensemble − ] proved that − ... 0 0 ,4hodrQ-Hermite order 4th q=4, 32, = N = 2 oa Unfolding Local q 2 0 12 , yet a naive subscripts counting when 5 4 3 2 1 0 400 300 200 100 , N a 1 2 N Γ ... ) , 1 (n) (n) 1) y 2 2 k a Σ Σ ) − p m k − ( 1 ): k x | ( N, q – 34 – Tr (Γ 2 ( q 0.010 0.010 =0 k 5.3 ...k p X m D ). The deviations in the number variance from the GOE 1 ...a T X 1 − 1234 a t 7.22 k m N 0.008 0.008 q − N →∞ lim k q =0 GOE 4000 = w SYK, 6000 = w SYK, N N GOE 3000 = w SYK, 4000 = w SYK, X m := 0.006 0.006 peak in the spectral form factor. N N m t t − − t T = 2 0.004 0.004 := 2 1 ... 1) 0.002 − 0.002 k nebeUnfolding Ensemble ,8hodrQ-Hermite order 8th q=4, 32, = N ( k | ,4hodrQ-Hermite order 4th q=4, 32, = N oa Unfolding Local was first defined in equation ( . Number variances in the right figures are obtained from the spectral form factor in the ...k is the number of cross links. 0.000 m 0.000 k T 0.0010 0.0005 0.0000 0.05 0.00 0.0030 0.0025 0.0020 0.0015 0.20 0.15 0.10 1234 t An important observation was first made in the appendix F of [ (t) (t) c c K K The observation was that where where cross-linked-only chord diagrams. The authors of [ Since the universal random matrix result scales as high-order moments have toWick-contracting be Γ’s suppressed would by suggest 2 double traces are suppressed by a power law Figure 15 left figures by evaluating theresult integral are ( due to the small JHEP07(2020)193 ) we (8.6) (8.7) (8.5) ... 8.7 , both of ,N ), and in turn ], however, this
= 0 · · · 8.3 62 m m µ term is exponentially , i , then the completeness m N µ n , ’s by the number of Dirac H ... µ n jk ). m + δ X l µ . il σ 8.4 δ is the set of all linearly indepen- X jk l δ . } m N = il µ H ). The dashed chord represents the m δ 2 m kl X ) µ 8.7 { = = N/ m m | , whereas the : X 2 µ kl µ m for illustration. From equation ( ) 5 X µ µ m N /N | 1). This shows the naive expectation for ( Tr 16 of the multi-index h X ij ( | < ) =0 N µ ) in to double-trace moments, we have P ij | m | m ) m N µ – 35 – m µ N 8.5 X T X | − ( elements because they form a basis for the vector ( m , namely µ m X N m µ µ = 2 X X =0 N 2 =0 X m N X m has 2 N/ N 2 − } − 2 ... µ N/ − X = 2 2 { 2. is a multi-index containing all the subscripts of the Dirac matrices l,n µ N/ ˜ matrices. It will be useful to organize
M
∼ 2 · · · N/ , and m 2 1 = 1 (other terms have × = m 2 . Chords can intersect in arbitrary ways in the shaded regions. 2 µ T when m is a multi-index of length . Diagrammatic representation of equation ( N/ X µ N ... m X ’s are as defined at the beginning of section µ µ X Inserting the completeness relation ( In this section we offer a proof ofThe a starting generalized point version of of our equation proof ( is the completeness relation grams with one extra chordcan inserted, further see calculate each figure single tracedoes by not the combinatorics warrant developed a inthis [ straightforward approximation has application errors of in polynomials thelarge of in Q-Hermite 1 approximation because thus reducing every double trace tothis a sum means over every single traces. double-trace chord At the diagram level can of chord be diagrams, written as a sum of single trace dia- where such that in the product.space The of set 2 matrices in the product, thatrelation is, can the be length rewritten as The dent matrices formed by products of Dirac gamma matrices, with appropriate prefactors moments is wrong andfrom may a moment serve method as perspective. a starting pointoffer for an understanding interesting the generalization RMT of the ramp limiting scenario ( Figure 16 inserted for the simple reasonwhich that give this limit is dominated by only two terms JHEP07(2020)193 q ’s. µ (8.8) (8.9) X (8.10) (8.11) (8.14) (8.12) (8.13) ). The . , 8.4 ) 1 a ) for even independent Γ m , so the limit k 8.3 aµ N c ... ) and ( k 1) = a 8.3 − ]. Γ ( k m m 62 µ q N . m X k , m m k ) aµ T a c m N − Γ . ) is whatever remains of the 2 − kmq N = 0 and , )] q ... N/ 1) m 2 1 N, m, q m µ a − ( Γ have in common) X which in terms of a single trace is m N, m, q a m m m Γ rest( µ 1 N, N, q aµ m N ) reproduces equation ( c µ ... m X × 1) aµ are odd because both formulas give zero. 8.7 k c m − =0 . k =0 Tr ( + ), we arrive at N k and T ( X m q . m m k k | a mq X T ) + rest( 8.8 m N aµ kmq c 1) T ...k and − ,...,a , q – 36 – X mq 1) 1 0 − q a mq mq 1) − 1234 = 2 ( t m 1) 1) N, − = ( 1 µ X − − a ... m N Γ 1) = ( = ( m − [rest( m N k µ ) ( N =0 k a N X | − X m ) = single-trace diagram with dashed chord removed Γ 2 =0 N ) into equation ( ...k m N X m q N µ − N 2 X 1234 8.10 a − t m N N, N, q Γ ) to derive a modest generalization to equations ( that 2 m = 2 µ 8.7 N/ 1 m the number of indices that X 2 ... m 1) Tr ( − ) = rest( aµ This property states that the chord intersections that do not form a closed loop k c ( , q , and trivially for when both k 6 a | 0 k X ]. is the number of nested cross links and rest( ...k m 62 N, k µ X We thus have demonstrated that equation ( Let us calculate the contraction 1234 The adjective “cut-vertex” is understood at the level of intersection graphs [ t 6 rest( Note that where single-trace chord diagram after takingnested out the links. intersections We between remind the the dashedAgain reader chord in that and this the dashed limitis chord the represents simply the sum insertion is of dominated by two terms diagrams we consider are theof ones nested with links. a Repeating numberof of the the intersecting previous form cross analysis, links we and conclude a that number these contributions are factorize into products of individualintersections, intersections. each giving In a this factor instance of we ( have and even We will now use ( This result is antraces application [ of the so-called “cut-vertex factorization” property for single Substituting equation ( we have for fixed given by Since (with JHEP07(2020)193 ) q 6.12 ), the , hence )–( N 6.13 instead, as 6.9 , as we have N N − ) and ( 4.12 . We first summarize N × N − 1 2 and is independent of q = ··· ) for certain diagrams. we conclude that a diagram with a large 7 N, N, q ··· for a concrete example. times the corresponding single-trace diagram , we give a partial explanation to this phe- N 17 8 – 37 – rest( − 1 − is even, ), and the chirality matrix insertion is equivalent ). The ellipses on the left represent infinite number of q ) = gives a completely combinatorial formula ( 8.7 8.13 , q 0 6.1 terms correspond to the insertions of identity matrix and N, N = m = 0 and m it is possible that rest( q : the method of section . An example of equation ( ··· 6.1 The approach developed in this section can be viewed as complementary to that of For odd 7 The mode numbers correspond to the numbers of crossThe links double-trace in contractions converge double-trace rapidly contractions. tolinks the increases. RMT result as the number of cross ··· contractions stop being suppressedwould by be the the case number fordouble of an traces cross RMT theory. from links As diagrams butscale we fluctuations with by have — seen precisely 2 the in lowest two long-wavelength equations mode. cross ( For links higher modes can and larger be number entirely attributed to • • Here by “RMT result” we simply mean the weak statement that after a certain point the move exponentially many long-range modesWith to get the very techniques closenomenon to discussed and the in estimate random how matrix section the result. the number gist of of subtractions this scales with section: behaviors amenable to discussion forand a will large see more number of soon. nested cross links8.2 as we have Long-wavelength seen fluctuations Previously we saw a somewhat mysterious numerical phenomenon: we do not need to re- in which the number ofis summations useful scales for as a calculatingseen. power exact in However, numerics for for exactly low-orderbehaviors, the moments neither same with is reason large it itof is helpful cross not for links very discussing due useful the toa for its asymptotic symbolically discussing complexity. behavior simple the On with formula large the large in other number terms hand, the of approach single of traces, this hence section gives makes various asymptotic to the identity matrix insertionnumber of when nested cross linkswith is all equal to nested 2 links removed — see figure section nested cross links. Onsingle the trace right diagram. we have same contraction with all thebecause nested links the removed, as a Dirac chirality matrix in equation ( Figure 17 JHEP07(2020)193 is For 1 must ... 8 . (8.17) (8.18) (8.19) (8.20) (8.16) (8.15) . i 1) ) relative − N 1) n k k 1 − 2 ( . q a k H N 4 k | m ( c Γ T γ ) log N k ...k q , ... + k m (2 1 1 m N T 1234 ) (Tr q − 1) − t 4 k ] m N a 2 − m Γ m N q H , − k c 4 N 1) 1 a N γ ] ( k < N/ Γ − − m c 2 N γ log to be sharply peaked near . Tr ( 1) h / + − k 1) ). In fact m [1 + ( 2 1 ] = 2 N ), it was pointed out that for T 1 1 2 2 − ) Tr q 8.4 − q k ... 2 4 m N a 6.8 4 log N 1) [1 + ( − =2 ( Γ N − X 1 2 − m N k N ( − 1 2 ... k =0 | N 1 2 k > are suppressed, and to obtain the RMT + X m a ...k 1 Γ 1 = N N 1 q − a − − 4 1 2 1234 N Γ , ) gives c u N k m ) log 1 γ ]. Since the goal of this section is humble, this q = 2 T – 38 – + ... − 8.20 ≈ 1 1 m 1) (2 69 Tr ( k ...... − m 1) k k 1) T 1) = 1 ( − k − − | ...a ( 2 k k X 1 ( ( − a m N k k ...k | k > N/ | N k m N ] T terms contribute. It is clear − 1, and the remaining largest terms are ...k ...k 1234 m = m < u q 1) =0 2 N | N 1234 1234 X + t m T − u m [ 1 T N N | + ... − − , we would expect the summand 1 →∞ 1) lim k k [1 + ( ... − k 1 2 = 2 1) estimate. Using equation ( ( := 2 k 2 − | k 1 N − =2 ( ... k X ...k m N | = 4, we subtracted the first eight long-range modes to get a decent agreement , we have 1) 1 2 q 2. Hence − ...k 1234 k ( ,N t − k | ). Hence we have 1234 t N 6= 0 ...k ,N 8.5 − 1 2 We will estimate speed of convergence of equation ( m This is a large- 2 1234 = 32 and = 2 8 u So we see that contractionsresult with we only have toN subtract long wavelength fluctuations with So for large enough m For and this is the quantitydeviation whose from speed the of limiting convergence result we is wish to study. Its finite- and so we see only the even where the secondtion equality ( can again be proved by the insertion of completeness rela- higher moments (large number ofbe cross included. links) the The contribution completely nested diagram contribution in this double trace is both probe finer structures ofa the correspondence correlations. can In be thelevel made case of of precise understanding actual [ should RMT suffice ensembles, as such a starting point. not quite the right quantity to study. Near equation ( of cross links, we cannot make the correspondence as precise, but it is intuitively clear that JHEP07(2020)193 , 6.1 ]. The 14. This 72 , ≈ 71 -dimensional N 2 N/ ) log q (2 N/ , implying that very few matrix entries of α J lattice). However, it was soon realized that 4 – 39 – ] were the number variance was calculated from the 70 model parameters q N ). 8.13 In the same vein of understanding QCD Dirac operator’s spectral statistics through The situation with the SYK model is similar. Comparing the estimate of the scale For a more complete analysis, we would need to get a handle on the cases with linked and it has been usedapply to to calculate general the high-order first moments. few We moments, hope but to it investigatechiral is perturbation this complicated theory, in we and studied future the hard works. nonlineardeterminant to sigma of model formulation the of SYK the spectral model. Although it is reassuring that the large energy expansion of scales between the long-wavelengthhave modes gained and some scale insight of into universalof this RMT scale double-trace fluctuations. separation chord by diagrams We looking towardtheir those at early the of convergence RMT, convergence of and behavior ato analytically is class have demonstrated consistent analytically that with controlled estimatescomparisons. our of numerical We did all results. develop chord a diagrams combinatorial It to formula is for make all desirable more chord quantitative diagrams in section modes. This can be partlywhereas explained there by are the only fact thatthe the Hamiltonian Hilbert can space fluctuate is independently.random 2 This matrix scenario ensembles, is where all to matrix bestraints) entries contrasted can (barring with Hermiticity fluctuate and actual independently. symmetry con- However, this does not quite explain the separation ations from the RMTfluctuations. predictions This can is mostlylong-wavelength further modes be confirmed are accounted by subtracted for realization ourtral by by numerical form realization such study, of factor where long-wavelength the becomes after ensemble,has the RMT-behaved RMT the spec- spectral until first rigidity very few until short the energy times scale and given by the the number wavelength of variance the subtracted scale where deviations from randomthe matrix corresponding theory pion occur Compton is wave the length quark is mass equal scale to forwhere the which spectral size correlations of start thespectral box. deviating density from from RMT one with realization statistical of fluctuations the of the ensemble to the next, we see that the devi- dom matrix theory was(which found is up about to 100when as level we spacings many compare for level toalready a spacings the 12 ensemble visible as average, allowed on which by theunderstood is the and what scale can statistics should of be be obtained a done, analytically couple from deviations chiral of are perturbation level theory [ spacings. The deviations are now well 9 Discussion and conclusions How can we analytically understandtive, the we observations first of this discussThe paper? spectral first To correlations results get of were somequenched the obtained perspec- Dirac eigenvalues in of eigenvalues [ of the QCD a Dirac single operator. lattice QCD configuration. Agreement with ran- analysis also applies to the slightlynear broader class equation of ( diagrams that include the ones described intersections and the unfolding, which is outside the scope of this paper. to the RMT result, which is in the right ball park of our estimate JHEP07(2020)193 (A.2) (A.1) -model σ ] for the complex 14 ! L α Γ α -model: J σ α P . ) U α } 1 Γ ] = 0 we have the Hamiltonian in α − 1 J − ,H 2 α c {z = 2 case is not qualitatively different γ N/ ,..., P 2 q 1
− | , = } 1 1 α − Γ 2 ]. Since [ {z α – 40 – N/ J ,..., 73 = 2, which has Poisson statistics, also gives RMT 2 | -model for the SYK model, see [ q α σ X = -model is not complete. Similar issues arise in the calcu- ! = diag(1 -model approach is inherently flawed, but they do suggest σ L c σ γ H -model σ U H
= ] for the Majorana SYK model. Our derivation follows a different route . 15 H q N = 4 case. We hope to address some of these questions in future work. q = 4, it has not been shown that all other corrections in the loop expansion are that the loop expansion has to break down in this case. q small — in factnaive we application do of the not result knowspectral to the correlations range albeit of with validity a of much the smaller loop Thouless expansion. energy, Since and we conclude branch cut. It is notaction. known how to In perform a particular, resummationwhen we at the have level not of the been able to obtain a Gaussian spectral density Although the correction terms to the RMT result explain the numerical results for The most straightforward calculation gives a one-point resolvent that has the wrong 2. 1. in the chiral basis where In this section we derive theSYK nonlinear model, and [ using the Fierz transformation, seethe also block-diagonal [ form Altland, Dmitry Bagrets,discussions. Steven Shenker and Douglas Stanford are thankedA for useful Derivation of the Acknowledgments We acknowledge partial support from thanked U.S.Garc´ıa-Garc´ıais for DOE collaboration Grant No. in DE-FAG-88FR40388. the Antonio early stages of this project. Alexander Neither issue implies that the that our understanding oflation the of nested cross-linkedfrom diagrams the where the wide correlator is given by thewith sum numerical of the results, GUE only resultnot and few corrections clear low-order that how are correction the in termsIn agreement contribution could our of be view all the calculated otherof following and corrections the two it spectral to issues is density have the and to two-point spectral be function correlations addressed cancels. in to terms complete of our the understanding of the sigma model correctly reproduces the moments in a nontrivial manner, and that the JHEP07(2020)193 ” µ · X (A.6) (A.8) (A.9) (A.3) (A.4) (A.5) (A.10) , so after 2 ) and “ v U . H , , kj 2 ) k , U µ 2 pk φ φ · X kj · ) ) ) U µ U α il H Γ X ( ) by half by noticing that with variance + · ( U µ . y . ( ) ∗ pk il α · X ) φ k J A.9 p ∗ 2 U µ µ η ) φ x, y X X k ( U µ ) (A.7) p,k U α 2 one-to-one by multiplication of { Z )( P X , y Γ i P µ U U α d ( − U dy µ X α n k Γ }| H 1 α ··· X U d P µ φ as also having an implicit index of the < N/ , dx · Γ U α + ) 2 | z y, 2 µ X 2 v φ Γ U +1 y µ 1 U α | { p X − D H , x, α n 2 + pk 1) 1 x φ − n n · ( ). Restricting this equation to the left-upper − }| · Tr(Γ ··· 0 k ∗ pk ∗ 1 Tr(Γ – 41 – det φ → 8.5 φ z x, = ( p n ) = 2Tr k η even µ µ p U | can be viewed as the diagonal entries of X η pk P = lim X µ X z i | H i N α e pk ) µ, ∗ z − p,k Γ y + φ ( µ N := diag( P x − i D 2 are related to G X = 2 1 z ) φ and α n φe x kl D ( − . We also introduce the sign factor ) D = 2 α G i > N/ Z h Tr(Γ det kl Z | − (Γ µ , . . . , n | U α ij 2 = ) Γ ) = , ) = α ) = ’s with even string length because only a product of even number of y ij µ (Γ with x, y = 1 x, y ( ( U α µ k Z Γ Z ’s are defined in equation ( X 2, µ dimensional Hilbert space (the index of the block Hamiltonian , and Im( X 1 − i is the replica index. We should think of 2 = 1 . We also used that k p N/ U µ ) = x X = 2 by We can further reduce thegenerators number of of generators in equation ( We summed over Dirac matrices has a non-zero left-upper block, and we denote the left-upper block of where the block, we get where defined in the replica space. Next we apply the Fierz transformation Recall that the disordered averageaveraging is we over obtain Gaussian variables which will be inserted inconvergent. the The definition correlation of function the of partition two function resolvents such is that given the by integrals are where D indicates summation over theIm( Hilbert space index. The integrals are convergent because we focus on the upper block and define the generating function as Taking into account the unitary symmetries as required for universal RMT correlations, JHEP07(2020)193 ), 2 . q ql N/ φ · A.12 (A.14) (A.15) (A.16) (A.11) (A.12) (A.13) U µ X terms of · ∗ pk , φ q N pk terms of this D 1 φ 2 · √ i U µ (there are D q , N ql ; X 2 α · 2 is odd, and also n pk µ , each with weight kj ∗ ql . D a α φ ) 2 0 µ N/ U ql µ U µ n . φ ). This is the partition P · X X q µ U µ η a 2 if 5.1 p X il · D η . 1 ) √ ∗ pk U µ 0 µ φ U µ pkql µ X < N/ T P X P | 0 µ + U α ) ) consists of 4 + = constant resulting in the partition function µ z | Γ 2 µ U P µ 2 ∗ a ) q U µ φ X η tr φ · p A.12 X µ U α η U µ and Tr log( T U α Γ 1 2 X | − · and σ U µ 2 µ ) are actually canceled by the other terms gives equation ( 2 ∗ µ pkql a | φ φ 0 µ . However, the expression before the Fierz X µ tr P Tr(Γ D ]. 1 U α D µ P 1 √ D T A.12 σa 2 1 2 − α µ 15 2 / 2 σ e T X σ 2 – 42 – 2 such terms from the diagonal Γ 2 → 1 Tr(Γ in ( − σ pk 0 µ 0 iσ − µ φ 2 · pk 2 is even. After the Fierz transformation we thus − P a j pk = 0 term in ( q φ µ µ · ∗ pk φ N/ − X a q φ ∗ N ( µ N/ ∗ pk e 2 is related to the other half in the same way. On the j ql ’s with even 2 such terms from the off-diagonal Γ φ µ pk µ 1 D φ µ z T a D 1 i pk 2 2 p 1 i ql N/ z D η σ D n 2 if p 2 2 φ = η q pk = ∗ − = i pk Z N/ e N/ | kl P p,k φ µ µ i µ T | e P = U α − , and ∗ i ) = da Γ 2 | e φ have only one nonzero matrix element in each column and in each each with weight ∗ µ q N Z D = 0 term in this sum is exactly the GUE result. This derivation | ) which is an exponentially smaller number than in equation ( D α φ x, y ij ( φ pk j h ) has only 4 = D µ φ is the multi-particle variance and Z U α D ’s with φ A.6 2 ∗ D µ µ Γ ql j 2 2 − D A.6 a v φ X n N is the identity matrix when restricted to upper block, so within the upper D ql i ’s with Z q N c φ µ γ Z denotes the sum over ∗ pk i ) = = 0 µ φ 2 ) = P σ x, y ( Note that the Now we use that x, y Z 2. Note that all Γ ( / , and half of Z 2 c σ row in the representationform we in are equation working ( in.and most So terms in of the total form obtained we after have the 4 Fierz transformation. Indeed, the dominance of the GUE contribution raises an important concern.the form The transformation ( of them) and 4 A rescaling of the integrationfunction variable obtained by Altland and Bagrets [ This results in We can now perform the Gaussian integral over where where over half of obtain the generating function other hand block the above-mentioned pairsgenerators to are 2 identical. This allows us to reduce the number of γ JHEP07(2020)193 (B.4) (B.1) (B.2) (B.3) (A.17) (A.18) (A.19) | , p s µ | η = 4 | s µ ν , . X − | | − µ pk ν | φ X ∗ pk N . = 4. · ν φ . 1 q · 2 U X − µ µ | X qp/ U α µ X N · q | − N Γ . For the SYK model we do not have ql Tr N s · pk ν φ /β φ T dependence in the saddle-point equation 1) ∗ ql ∼ = 2 and pk p · = 4 µ φ p − φ q µ β T · 2 ( µ T U α √ T + U µ s ν Γ | X X X | pk – 43 – µ N · | · µ φ N = | | · ∗ pk ν ν ∗ pk N | φ φ X =0 U α | N µ =0 -model for µ | X µ N Γ | T σ X µ | X =0 0 µ | 2 N · ν ν X 2 1 | D 1 matrices are real and the spectral correlations are in the X ∗ pk D X µ | fixed we have φ γ X µ | N | ν 1 2 pkql X = 2, where the eigenvalues are uncorrelated. Yet, there is no | Tr q =0 | N fixed X µ | | X ν | 3 1 ν, D 1 the sum behaves as D p mod 8 = 0 the N -independent spectral density for the Wigner-Dyson ensemble, we have to scale the ) β is summed over with For A.6 ν Using this we obtain the identity If For arbitrary One can easily prove the identity variance of the randomsuch matrix rescaling Hamiltonian and as we 1 and expect the an corresponding additional resolvent and semi-circular spectral density. B Some combinatorial identities which is the so-calledget Cooperon a contribution to the GOE result. Note that in order to can be written as After applying the Fierz transformation, we obtain additional terms of the form qualitative difference between the universality class of thein Gaussian ( Orthogonal Ensemble (GOE). In this case, the term has to break down for JHEP07(2020)193
9 ) (B.5) (C.5) (C.2) (C.3) (C.4) (C.1) σσ
5 ) tr(¯ σ σσ tr , ,
) tr(¯ z σ 4 s . 9 . To obtain the ) 2 N +˜ ν z − q + 2 k σσ X
m ν 7 + ) tr(¯ tr log( X 1 q σ ν N for large m σσ T tr − T − k 2
2 ˜ /z σ tr(¯ 2 m N 2 σ N 20 T N , 1 − tr . 2
+ m σ k σe
T 7 2 m 2 N 6 N s 5.1 tr˜ ) − ˜ σ 1) + e in terms of a derivative with respect . Using that the replica limit of the d
σσ ) − σ k 5 k z ( . σ ) σ 2 Z m k tr(¯ σ 3 + 1 1 n − σσ s − ) σ, = 0 tr¯ z 0 1 m 1 q k ( − N z tr(¯ + σσ → N − 2 σ n =3 σ 1 + ∞ k tr(¯ m m tr that asymptotes as 1 N ˜ σ det – 44 –
P = ¯ σ − ) = lim σ k + tr ˜ 5 σ z 2 + N − dσ
( σ q 1 ) ˜ σ 2 + s Z 2 Z 1) m ¯
σ 18 , N 3 replicas is given by − − ) ( log + (1 ··· Hermitian matrices n ) =
1 2 =0 z q N σσ d 4 + dz ( n N X ) m m − Z tr(¯ 1 × N
q σσ σe =0 σ 3 1 1 n k X n ) . The saddle point equation is given by tr tr -model calculation we need the identity =0 3 2 m 0 tr(¯ s X
) σ 3 − → dσ ) /N 3 lim n N =0 σσ 2 N . Z q X σσ N − 6 m 1 1 n n T (tr(¯ tr(¯ 0 0 4 =0 σ σ 1 N ) = → → X tr z = 4 ≡ n n m tr (
3 G 2 )+ lim )+ lim and partially integrated these variables. We evaluate the resolvent in powers of z z N 12 162 N ( ( denotes the value of the single-trace diagram with three chords all intersecting kk ¯ ¯ + G G + 6 σ T = and in powers of 1 ) = z ( /z G and using the saddle point equation we obtain the expansion Expanding around the physical solution ¯ where we have expressed theto derivative with the respect ˜ 1 where the integral is over partition function is equal to one, we find that the resolvent is given by C Replica limit ofC.1 the GUE partition One-point function function The GUE partition function for Here with each other, see the footnote near the end of section sixth moment from the which can also be shown by applying the Fierz transformation to JHEP07(2020)193 (C.8) (C.7) (C.6) (C.9) , (C.11) (C.10) , , ··· ··· 105 different + + ··· 9 × 1 9 z + 1 z 9 1 z . + 11 + 11 7 1 7 z + 47 1 ··· z 7 1 + 2 z 2 1 , 9 N 1 1 j z N , 2 + 8 2 = σ 5 = 2 , 1 2 4 z N 1 , , , . 9 ) 3 N 1 z − ) 2 + 1 N 2 e ¯ 2 4 σ 7 j = = 0 = 0 = 0 = 0 ) ¯ σ 7 1 ¯ σ 21 N N σ − ¯
σ 5 − 3 7 6 5 ) 2 + j = = σ σ σ σ p (1 ¯ vanish and we have not included them in σ (1 9 z σσσσ 2 σσσσ. 1 tr tr tr tr 3 5 ( ¯ σ 2 1 ) ) 3 4 σ σ σ tr − 21 of them do not vanish in the replica limit. tr N 1 2 2 N σ σ p =0 N tr tr ¯ ¯ σ σ 5 9 j X × (1 tr tr
= ¯ ¯ σ σ det – 45 – = σ σ 2 σσσ − − σσσ 1 1
n n 1 4 1)!! tr tr tr N tr 0 0 ) (1 (1 dσ 4
σ σ ) → → − 2 4 1 1 = lim lim n n tr n n 1 tr Z σσ p 21 N N σσ 0 0
3 → → ) Tr(¯ but only 9 = = lim lim n n ) = 3 3 Tr(¯ = (2 i ) z ) 3
correspond to two different contraction patterns, namely, 9 ( ) 9 5 p σ ) ) σ 2 3 4 ). σσ Z p 2 σ σσ tr A M ]) σσ σσ σ (tr(¯ tr(¯ tr(¯ 75 tr tr(¯ tr(¯ σ σ σ NM h and σ σ ( tr tr tr / 3
tr tr
) for the moments is in agreement with the general formula obtained p
2 A 3 2 2 9 5 12 12 M N C.7 N N N 162 N 1 1 1 1 1 n n n n n 0 0 0 0 0 ] (see also [ → → → → → 74 n n n n n = lim = lim = lim = lim = lim 3 4 6 1 2 A A A A A This result ( and A few comments on the evaluationdiagrams of contribute the to contractions are in order.The expression In for total 9 This leaves us with the following nonvanishing contributions The terms with anthe odd above. number One of can factors easily of show that the following terms vanish in the replica limit where the expectation values are given by the sum over all corresponding Wick contractions. C.2 Two-point function The generating function for the two-point function of the GUE is given by by Mehta [ by comparing to JHEP07(2020)193 (C.19) (C.14) (C.15) (C.16) (C.17) (C.18) (C.12) (C.13) . 21 σ 2 σ 12 2 σ N − e ))tr )) ) y . x σ ( , ( ). The corresponding blocks σ 2 , + )¯ 3 } ¯ σ ) c 4 , y x z . 1 i ) ( ( k − ¯ n σ σσ ) 21 {z N (¯ ··· σσ σ 2 − σσ . | 22 ))(1 det tr (¯ tr(¯ replicas. Note that for the two-point connected 2 , y, σ x
. 1 4 1 k σ } ! . 1 ( ) )) 18 n 12 , x ) 2 σ z 22 y σ σ ( y ¯ ( σ =1 + 2(1 ∞ P 22 ( = 0 k X n σ Z 11 22 {z ··· ··· P σ )¯ tr − 2 22 σ σ P x σ σ ) 0 tr = 0 has the replica-diagonal form x, + ( tr log tr ) = x | σ + 11 0 ¯ σ ( 3 σ σ 12 22 ))(1 (¯ P )Tr 11 y d ¯ – 46 – σ σ = ( σ σ y dy 1 2 11 3 P + ( tr ) 1 z x tr
P + σ y d z tr dx )¯ ( have the same sign. dσ tr z 11 = + ¯ x 2 σ σ ( z 2 2 ¯ σ Z N ¯ tr block. The differentiation gives other contributions but σ 1 − 2 y 2 h 1 2 N 1 2 tr log( 1 2 n N − n 1 2 x 2 kk n 0 0 − two diagrams contribute to the two-point function: N n 1 (1 + (1 0 → 2 → 0 2 2 2 matrix and n n → n 2 11 1 1 lim → n N N 0 n σ lim n = lim /N 2 → ) = lim lim = = )tr n × 2 ) n x, y x ( ( have an opposite infinitesimal imaginary part, while for the one-point ¯ σ C y − (1 and and both with the same infinitesimal increment, this becomes the generating 2 x the projection on the N y matrix are referred to as the 11 block, the 12 block, the 21 block and the 22 block. a Hermitian 2 − kk = σ P σ x To leading order in 1 The two-point correlation function is given by and The connected part of the first diagram is given by The propagators and the vertices follow from the expansion of the logarithm This results in the following quadratic part of the action In a 12 block notation, the solution with with they do not contribute tosaddle the point connected approximation. two-point function. The saddle We evaluate point the equation integral is by given a by of the For function for the one-point functionfunction, but now with 2 function, the imaginary parts of all with JHEP07(2020)193 1 , 1 M 1 for ) will (D.1) (D.2) (D.3) (C.20) (C.21) 1. This . → 18 . ) 2 y | ( ) 3 . σ a )¯ 2 ]) and the | x Γ ) = 0 or 1 ( 2 3 14 . For example let a a σ 2 | β Γ Γ ) 1 2 a m a a (see [ Γ c Γ 1 and i 2 a )) )) 12 Tr(Γ x α ) in this limit. | x ... ( 2 σ /N ( 3 2 ) can only be 0 or ) 2 a 2 a 21 y ¯ 1 σ Tr(Γ m ¯ σ | a σ Γ a c 3 1 − − Γ − 22 + a ,a x 2 σ 2 , a ( ), the shuffling (see figure ... 1 ,a α 2 X tr 2 a 1 ))(1 2 ))(1 Γ a c a Tr(Γ 2 21 x a N 3 | β x Γ ( ( , σ 1) ( Γ )) 3 Γ 2 = / N )) 1 2 a y 12 ¯ − σ a y − 2 ¯ ( σ αβ Γ ( | ( σ c 3 1 σ ) − σ a )¯ − 3 + ··· 11 ,a )¯ q a 2 x σ x Tr(Γ (1 ( Γ + ) = 2 (1 ,a ( X 2 1) 2 tr . 2 1 1 σ 2 σ 3 a a (¯ a , and hence − )) (¯ – 47 – 22 N/ )) y )Tr(Γ 3 Γ Γ y 2 a 3 3 σ y − 1 ( 1 a ( a x = ( a σ tr ) = ··· Γ σ c ... )¯ 2 )¯ 2 β 11 1 + + a x ) by 3-cross-linked examples a + x 2 Γ ( − σ Γ ( 2 a Γ Tr(Γ 3 α ¯ 3 σ m 1 | 1 ¯ y tr σ a y a a 1 Γ a 3 h 2 1 c 3 a − Γ Γ x − 6.12 2 1 x 1 1) a m a c (1 a N 1 (1 − + 2 2 2 )–( 2 2 2 1 1 n a 1 1 N N 1 N N 0 a )Tr(Γ 6.9 c )Tr(Γ 3 → = = lim = = a n m 1) a Γ is the number of common elements in sets 2 − Γ | ( a 3 β Γ ,a 1 ... ∩ 2 a 2 ,a a X α 1 | Γ a 1 Tr(Γ a = ) we work out two double-trace diagrams with three cross-links in this appendix. moments. Note that in the normalization of this appendix, ¯ 3 ,a 2 , αβ 2 6.1 2 c Tr(Γ , so that the two-point function behaves as 1 ,a X 1 y M N The starting point is the fact that 2 a − 2 → However we want tothe illustrate phase factors a in general our point discussion. beyond We see this in simple general example, each intersection so among we the keep chords introduce a phase factor of ( In fact in thisthe particular Γ’s case in we the second can trace, get so rid we of must the have phase factor by cyclically permuting The shuffling will introduce phase factors due to the relation where us consider the contraction Tr(Γ implies This particular simplicity motivates us to shuffle the Γ’s to the above “canonical” ordering. D Illustration of ( To illustrate thesection general prescription for the evaluation double-trace diagrams (see which gives the correct result forand two-point correlator to order 1 x The sum of the two contributions is equal to The connect part of the second diagram can be evaluated as JHEP07(2020)193 . 2 9 } k i k a 3 i a Γ ...a , ...a 2 , d (D.4) (D.5) (D.6) 1 Dirac i 3 . The 2 a i a 3 ]. Here a a 1 a i 2 N = 0. We i a 62 , d a , but they 2 elements. d 1 (naturally, 3 | 1 3 i a a ) a k q a 1 d 3 i a d a and Γ { c } or Γ + 2 2 2 2 2 a ,i a a 1 Γ 1 i Γ , a a , , . . . , a 1 c 1 1 / 2 ∈{ a i a a k X -variables: 1) d , a , no less and no more. − 1 2 ,...,i ( i equal to 0 and when is it Tr(Γ : a 3 | a i × 2 | 1 + ) 19 a counts the number of Dirac 3 and a ··· . There are also 2 6= 0, Γ 2 a } 1 3 + 2 1 . 3 a a a a a 4 a 3 i 2 ; ; . 2 a d Γ a a 3 3 3 a 2 3 1 1 3 a a a 1 a i a a a 2 2 2 a a d d a a a 2 1 1 1 i , d + a a a a 3 1 d d d Tr(Γ 3 a i 3 | 2 a a . Here 1 a a + + + 2 d N a | 2 3 3 } − ) d , d 2 a a a is thus 3 2 different Dirac matrices. The necessary and 3 1 1 2 ,i a a a + a a a 1 3 1 – 48 – i Γ q . Hence if a d d d 2 a 2 3 a Γ / 1 a ∈{ . The total number of Dirac matrices that appear a X 1 . We would still like to get rid of the trace in our = = = , 4 , d 3 a -variables have played an important role in calculat- a Γ Γ 2 2 a ,i 1 2 3 3 d , d a a 2 3 αβ ) to be nonvanishing is that every one of the a a a a i 2 c 1 a Γ 1 1 2 3 a a d = + = a a a , a + d q Γ 2 1 c c c 3 Γ 2 , { i a a 2 d counts the number of Dirac matrices that appear exactly 1) 1 a Tr(Γ a | a 2 and i − 3 Γ 3 ). a a 1 3 1 a 2 a i a ) and the a a 1 2 1 D.7 i a d a 1 a d } d a 1 2 i -variables a ,i 1 c d i X 3 / ∈{ 3 a i + 2 2 i ) in favor of a purely combinatorial term, so that it can be effectively handled a a . Shuffling a double trace to its canonical ordering (subscripts not summed over). 1 i 1 -variables ( a D.3 a are all different from each other in this definition). In the case of three index sets c d . k 3 = a 2 i On the other hand This suggests that a useful perspective will be provided by the Recall that each Γ is a product of For example see equation ( a = 3), we have the Figure 18 9 1 i , . . . , i a k 1 three times in the totality of Γ We come back tomatrices the of case which of theSo subscripts there appear the exactly Diracsame matrices in can appear sets be exactly said twiceexactly about in twice the in totality the of totality Γ of Γ In general, c ing single-trace contractions, and thewe cite relations one between diagram them that were discussed makes their in relation [ clear, see figure i ( which count the number of subscriptsare that not occur independent exactly variables onceBoth in due the only to the constraint that each index set has sufficient condition for Tr(Γ matrices occurs exactly evenand number Γ (including zero) of times in theis totality the number of of Γ elements common and only common to the sets introduces a phase factorequation of ( ( by computers. The key questionequal is, to when 1? is 2 JHEP07(2020)193 3 a (D.7) (D.8) (D.9) 2 (D.10) a d by a sum . ) 3 3 2 a a q. 2 , a , but we have q, a 2 3 d a 2 ! , a a − 3 1 1 a a ; . ; 3 a 2 3 3 3 a 3 d a 1 a a a a d 2 2 2 a for even ! a a a ].) d 3 1 1 1 a a a a 1 62 − d d d a 1 q d 3 a ! ( − − − ) 2 ! δ 3 3 3 a ) ) a a a 1 3 q 2 ! is nonzero. Synthesizing every- 2 2 1 a a a a a 2 , d ! (( N } ! d d d a q 2 3 ) )! N d a Red region has cardinality q 2 2 − − − 3 d = − 2 3 2 , d − a a a 3 2 2 + 1 1 1 a N a a a a a 2 ( 1 q 0 for odd a d d d a , d 3 d 2 1 d | a − − − ) − d = − 3 = q q q a { 3 q N 2 | Γ a ( ( = = = ) ) 1 2 – 49 – δ 3 2 3 a a 2 3 1 ) a a a a a a d 3 1 Γ Γ a a d d d Γ 1 3 1 constraint is satisfied: d 2 a = : : : a a a + δ d 2 1 2 3 Γ 2 3 Γ a a 1 a a a a 1 1 1 a 2 − a a a Tr(Γ a | d 2 d c 3 a . In principle there is also a sum over a 1 1) 3 1 a a Tr(Γ a − 2 d )Tr(Γ 2 | c ( a 3 3 a + a a − 2 1 =0 , d Γ a a q 3 3 = 0 case contributes. It is clear that in this case there is only one 1 2 c q ( a a a a 2 X + 3 1 δ c = 0 also if any of the a Γ 2 a a d 2 a 1 2 1) × 3 | 1 a a , d ) a a 1 − Exactly once in Exactly once in Exactly once in 2 =0 3 c a ( a a 3 3 q d 1 a 1) Γ a 1 X ,a Tr(Γ 2 a 2 − d a 3 d 1 ( ,a X elements. The box is the set of all possible values an index can take, which has 3 . The box is partitioned into eight regions. (Taken from [ Γ a ,a 1 1 2 a ,a =0 N q . Venn diagrams with three index sets. Each index set is represented by a circle, a 2 2 ,a X q a N 1 ,a X 1 X a 1 − a a d N Tr(Γ | Red region has cardinality N − Note for the second equality we replaced the sum over index sets = = 2 2 − 2 which implies In more general cases no such drastic simplification can be expected. over numbers argued only the scenario where the three Kronecker And thing discussed so far, we arrive at the formula also want to know how many Dirac matrices appear exactly once: Figure 19 containing cardinality JHEP07(2020)193 . ! 3 α )! 18 2 (D.11) (D.12) (D.13) 3 α α d 2 ! α 3 1 α α 1 . d α } d β ! − 3 2 a 3 α 2 α 1 . a 2 1 2 α 1 | α a d ) d 3 )! , d a 3 β − ]: ! Γ α 3 2 2 2 a are vanishing. This means N 62 a 2 α α ) 1 1 a 3 Γ β α α 1 Γ , d d d a , a β ), so we may say similar things 2 β 2 β a 3 3 − a a Γ , a + 2 1 2 2 q D.8 1 β Tr(Γ a ( a 2 2 a | a 1 Γ d α β a 3 )! , d 2 a 3 β a + β , d α c 2 3 Γ 3 2 a q 1 + α a 1 α a 3 3 2 a 1 3 a a α 1 1 a 1 − . a , d d ) with the single trace contraction that has a c α 3 )! a + N 3 1 − , d )Tr(Γ 2 ( 2 α 3 a 3 a β a D.8 3 2 3 a 1 3 – 50 – 1 α α ), using the results of [ α α a a a 1 Γ 1 1 c 1 2 α α α 2 ) + , d , d a d d 20 3 q 2 d β β α Γ + α a 3 2 1 2 − 1) a a − , just as in equation ( Γ a α 1 2 3 2 1 − 3 2 | , d , d α α α ( α α ) a 1 β β d 2 3 Γ ,β 2 1 α 1 α a 1 3 Tr(Γ a a 1) d d α 1 Γ ,a d ,β 2 2 − a , d Γ − 3 a X ( − 3 β ,a 3 3 ,a 1 q 1 Γ α 3 α α 2 a ( a 1 1 α Γ 2 X a ! ,a 1 α 2 α X 3 c N , d 1 α d α a 3 α + − d 3 a 2 Γ 2 α 2 1 1 N α α Tr(Γ 1 . A chord diagram with three cross links and one single-trace link. a − 1 α 1 | − α X = 2 α α q d 2 , d c ( d 2 3 a α 1) 1 1 Tr(Γ × × a α − X 3 d ( 3 ,α , we would need to calculate , d 3 2 -variables are 2 . The single trace chord diagram that has the same intersection structure as that of α ,α Figure 21 a 2 d 2 ,α X 1 21 α 1 , note here all chords are attached to a single backbone as opposed to two in figure a ,α 1 X X α 1 α d d α { 18 N It is instructive to contrast equation ( − = = 2 However the constraint ofdeltas the arising sum from on theabout right-hand the side conditions on is the stillsubscripts contributing appear given summands: odd by number the of the traces times with Kronecker in the Dirac totality matrices of whose Now the How about cases with bothin single-trace figure links and cross links? Let us consider the example the same intersection structure (figure Figure 20 figure JHEP07(2020)193 (D.18) (D.14) (D.17) (D.15) (D.16) ! β 3 a 2 a d ! β β 3 3 a a 2 1 , , . a . a d β β β } d 3 3 3 ! + β a a a 3 β β 1 2 1 2 a 3 a a a 2 a a . 1 1 d d d a a β a 1 d 3 d , d − − − a ! . , + , β 2 3 , , 3 β β β β ) ) ) 3 3 a a 2 2 3 a 3 a β β a β β β 2 d 2 a a a 1 a 2 3 1 3 3 3 a a 1 1 2 a 2 a a a a a a d a a a a d 1 1 d 1 2 1 + ! a a d d d d a a a + , d 3 β 2 d d d d d + a β 3 a − − − 1 + 2 a 1 3 a + + a − − − 1 a 3 3 3 a β 1 d a d 1 a a a 2 2 3 ! β β β a 1 2 2 a d a a a + 2 2 2 3 a a a 1 1 1 d q a a a a , d a a a d d d 1 1 1 2 + – 51 – 3 a d d d 1) a a a + a β d − − − d d d 2 2 2 − a = = = 2 2 3 a a ( ) is equal to )! − − − 1 1 a a a β 2 3 3 β 1 1 1 a a , d 3 a a 3 3 3 2 d a a a a 3 d d 1 1 a a a a 2 a d d d a a 1 2 2 ) is shortened to X c a 1 D.12 a a a c c d = = a ). + 2 − − − , and they are counted respectively by d d d β 2 3 3 2 3 q q q , d d d a a − − − ! 2 d D.13 1 a 2 2 3 X 6.12 = = = a 1 N a a a + d or a 1 2 3 1 1 1 β q d a a a a a a 2 )–( 2 4 { d d d d d d a a 1 , X 6.9 a − − − − 1 d a q q q 3 N ( ( ( a ( δ δ δ 2 a X d × × × × 3 a 1 a X d 2 a 1 a X d These examples enable usmarized to in see equations how ( the general situation can be handled, which is sum- where We conclude then that equation ( So these must also be zero. We also have exclusively in either We still need to take into account the Dirac matrices whose subscripts only appear once must all be zero. So the list ( JHEP07(2020)193 = 4 (E.6) (E.4) (E.5) (E.1) (E.2) (E.3) α q 1 . There are a , . 2 6.2 = 32 and the following few α 3421 . We only present 42531 | 2 | N 462531 / 10 | 1) α 3421 4 1234 6.1 | α − t 21 12345 q 536421 31 t ( α | β q 123456 α 1234 t 34 3 + | 41 β t 42 1) + 16 | β 31 − 23 + 3 β 123456 | 1234 465321 , t | t , 1234 + 16 42 35421 4321 34 | discussed in section t | | 4 α α 34 1 + ( + 12 3421 α 31 | + 12 436521 4321 (iv) 1234 -variables. α | | α 123456 α t 12345 12 3 t t t t α 8 21 42 | t 1234 α + + 5 1234 t ) t , β 34 2 + + 5 | 563421 η 123456 + 12 8 | 1234 t 321 31 β | t 321 β | 2 + 16 21 1234 ) 1 + 24 β 123 + 2 t | 45321 η t | 123 α 123456 34 2 t 564321 34 | t + | ) – 52 – 4321 , 21 α | η 34 + 6(1 + α α 2 12 α 12345 / 6 + 9 321 21 356421 34 α t | | | 12 1234 ˜ 1) t α t 123456 M α . Definition of the − t (1 + 4 t 43 123 . We are interested in the case of q | 1 + 5 t ( +2 + 36(1 + ˜ 1234 q t M + + 8 2 2 6 456321 123456 1 + 6 22 | / 2 2 t 1) β 1234 4 ˜ + − / t 2 1) M ˜ − 21 54321 1 α 1) M | − β ) 1 q − − q +12 21 ( N η q 43 − 123456 q 4 | 654321 α ( Figure 22 t | q 1 + ( 12345 q 1) 34 43 N | t q 1) α N +3 − ) 3 − 12 8(1 + η + 12 × α 123456 1234 3421 | t t t 6 × ˜ 1 + ( + 18 + 8 M 1 + ( 2 2 1234 6 4 4 4 t ˜ ˜ ˜ M M M + 36 + 12 + 36 + 24 3 = = = 15(1 + = = 75 = 3 6 4 5 6 5 3 , , , , , , 5 6 3 4 3 4 2 ˜ ˜ ˜ ˜ ˜ ˜ M M M M M M -variables with subscripts denote chord diagrams, whose meaning should be evident t 1 The sorting is mostly based on the dihedral action property 10 moments can be calculatedthe using final the values formulas of developed these in terms section in the next subsection of this appendix. The from the two examplesfor in comparison figure with the numerical results. Every term in these low-order double-trace few groups of diagramsstructures, which hence are the not same related values. by the dihedral action but they have the same intersection nontrivial low-order double-trace moments: E Low-order double-trace moments Using the properties of double traces discussed so far, we can sort out JHEP07(2020)193 ), 6 (E.8) (E.7) ,M 4 η, M = 4) , , , , , , , , , q , , , . , , , , , = 32 , , , , N , . , 469773425523599200000 , 20205025 469773425523599200000 469773425523599200000 234886712761799600000 / 234886712761799600000 234886712761799600000 , / / 469773425523599200000 469773425523599200000 / / , / / / / 234886712761799600000 104510217024160000 104510217024160000 234886712761799600000 104510217024160000 / 234886712761799600000 104510217024160000 / / / 4495 234886712761799600000 / 104510217024160000 / / 104510217024160000 104510217024160000 4495 / / / 7576990734251600000 / / 23250326368000 / – 53 – / 234886712761799600000 23250326368000 / / / : 323280400 / ]: = 1191 = 10181 = 137227959 19709513199 368559700939 60665945079 687382239 555243 711893157 462480963 6933507 6.1 4 6 η 62 − − − − − − − − ˜ ˜ M M = 31234591 = 89686592361 = 3173529 = = 5189715001 = 589707393 = 7519992447797 = 520655935797 = = 120146887221 = 4043901987381 = 6135081997081 = = = 7168277 = = 5928671067 = 518354267 = = 8116630557 = = 160002717 = = 567 β β β β β α α α 321 21 21 31 31 41 21 21 31 | 4321 3421 β β β β β | | α α α 54321 45321 35421 42531 | | | | 456321 563421 536421 356421 436521 462531 654321 564321 465321 43 34 42 24 23 | | | | | | | | | 123 43 34 42 | | | | | | | | t 1234 1234 34 34 34 34 34 t t α α α α α 12345 12345 12345 12345 1234 1234 1234 t t t t t t t 12 12 12 12 12 123456 123456 123456 123456 123456 123456 123456 123456 123456 t t t t t t t t t α α α α α t t t t t The rest of theby terms the are method the laid connected out double-trace in contractions section that can be calculated In this appendix wedouble -trace list moments. the Some explicit ofand the values contributing are terms of thoroughly are the discussed purely in single-trace terms ( [ that appeared in the low-order E.1 Values for low-order double-trace contractions ( JHEP07(2020)193 09 ]. 94 Phys. 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