Correspondence between symmetry breaking of 2-level systems and disorder in Rosenzweig-Porter ensemble
CCorrespondence between symmetry breaking of 2-level systems and disorder inRosenzweig-Porter ensemble
Adway Kumar Das ∗ and Anandamohan Ghosh † Indian Institute of Science Education and Research Kolkata, Mohanpur, 741246 India (Dated: November 2, 2020)Random Matrix Theory (RMT) provides a tool to understand physical systems in which spectralproperties can be changed from Poissonian (integrable) to Wigner-Dyson (chaotic). Such transitionscan be seen in Rosenzweig-Porter ensemble (RPE) by tuning the fluctuations in the random matrixelements. We show that integrable or chaotic regimes in any 2-level system can be uniquely con-trolled by the symmetry-breaking properties. We compute the Nearest Neighbour Spacing (NNS)distributions of these matrix ensembles and find that they exactly match with that of RPE. Ourstudy indicates that the loss of integrability can be exactly mapped to the extent of disorder in2-level systems.
Introduction : The notion of chaos is somewhat illu-sive in quantum mechanical systems as unlike classicalcounterparts, the calculation of Lyapunov exponents isnot readily applicable due to Heisenberg’s uncertaintyprinciple [1]. However, the spectra of quantum systemsthat exhibit chaos in the classical limit show fluctua-tions similar to random matrix ensembles. For exam-ple, certain real symmetric random matrices constitutethe Gaussian Orthogonal Ensemble (GOE) which cor-responds to systems with time reversal invariance androtational symmetry. This correspondence has led to theBohigas-Giannoni-Schmidt (BGS) conjecture laying thefoundation of
Quantum Chaos [2, 3].On the other hand, the notion of quantum integrability in condensed matter systems described by N × N Hermi-tian matrices also lacks a complete understanding unlikeclassical systems [4]. A classical system with N degreesof freedom is deemed integrable if there exists N num-ber of integrals of motion in involution i.e. their Poissonbrackets vanish [5]. Importantly each integral of motioncorresponds to an underlying symmetry of the dynamicalsystem. An equivalent criteria for integrability is the ex-istence of Lax pairs ( L, B ) such that the equation of mo-tion is dLdt = [
B, L ] [6]. For a quantum mechanical systemdescribed by a Hamiltonian, H , time evolution of an ob-servable, O , with no explicit time dependence is given by d O dt = i (cid:126) [ H, O ] in the Heisenberg picture [7]. Hence, if O ,commutes with H i.e., [ H, O ] = H O − O H = 0, the ob-servable is a conserved quantity. The resemblance of theisospectral evolution of the Lax operator and Heisenbergdescription of quantum mechanics prompts the search forcommuting partners of H . Thus a quantum mechanicalsystem described by a N -dimensional matrix commutingwith a non-trivial set of N − ∗ [email protected] † [email protected] wide variety of spectral statistics ranging from repulsionto clustering and intermediate statistics [11–15]. Therehave been attempts to explain intermediate statisticsby constructing matrix ensembles, namely, Rosenzweig-Porter ensemble (RPE) [16], whereby tuning the rela-tive variance ( σ ) of the off-diagonal matrix elements theregimes of fully ergodic, intermediate, localized states canbe achieved [17, 18]. It is also known that certain ran-dom matrices can be modeled as a Brownian ensemblemodel [19] which are inherently dissipative with lack ofconservation laws. This motivates us to explore the pos-sible connection between the disorder introduced in theconstruction of RPE and the deviation from integrabilityin simple matrix models.We consider the simplest case of 2 × H + ,which obey a symmetry property sufficient for being in-tegrable. We verify that these matrix ensemble woulddemonstrate clustering in NNS distribution, P ( s ). Wealso find the subset of matrices, H − , which specifically vi-olate the symmetry property. Any 2-level system can bewritten as superposition of two matrices from the abovesubsets as H = H + + λH − . Such decomposition readilyindicates the extent of symmetry loss; this in turn givesthe possibility of exploring level clustering (integrable)and level repulsion (chaotic) by tuning λ in arbitrary H .We obtain an exact expression of P ( s ) parametrized by λ and show that it is identical with that obtained in earlierstudies for RPE parametrized by σ [20–22]. Here we havebeen able to demonstrate the correspondence between in-tegrability and disorder in 2 × Real Integrable 2-level systems : In general theHilbert space corresponding to a system is infinite di-mensional with both discrete and continuous energy lev-els. But we are interested in some particular discreteenergy levels (say N of them), then H can be written asa matrix with dimension N . For real and symmetric ma-trices, H , the commuting partner O satisfying [ H, O ] = 0is symmetric (cid:0) O = O T (cid:1) and we also demand that it is or-thogonal, (cid:0) OO T = I (cid:1) . The eigenvalues of such matricesare necessarily λ O = ±
1. In this scenario there exists a a r X i v : . [ c ond - m a t . d i s - nn ] O c t transition matrix Θ containing orthonormal eigenvectorsof O satisfying the similarity condition,Θ T O Θ =
D, D ij = (cid:40) − δ ij , i ∈ [1 , n ] δ ij , i ∈ [ n + 1 , N ] , ≤ n ≤ N. (1)The transition matrix Θ results in block diagonalizingthe Hamiltonian as Θ T H Θ = (cid:18) B B (cid:19) where B , B are two uncoupled matrices with rank n and N − n re-spectively. There can be further symmetries in the sys-tem such that B i ’s can be further reduced into uncoupledblocks. For a N dimensional H , if there exists N − H + and which is integrable. Theabove discussion implies that there exists at least onecommuting partner O such that [ H + , O ] = 0. ThenEq. (1) implies O is similar to a diagonal matrix D with D = − D . Note that D , D both cannot be 1 (or − O = ± I . Without lossof generality, let D = − D = 1 = ⇒ D = σ z .A convenient choice of the change of basis matrix inEq. (1) is the rotation matrix, R ,Θ = (cid:32) cos θ − sin θ sin θ cos θ (cid:33) = R (cid:18) θ (cid:19) (2)resulting in the commutating partner of H + O = Θ D Θ T = (cid:18) cos θ sin θ sin θ − cos θ (cid:19) = ¯ R (cid:18) θ (cid:19) (3)where ¯ R ( θ ) is the reflection matrix, denoting a reflectionabout a line passing through origin making θ ∈ [0 , π ]with X-axis. Starting from a general symmetric matrix, H = (cid:18) x yy z (cid:19) and demanding that it commutes with O in Eq. (3) imposes the constraint x − θy − z = 0.Hence the integrable Hamiltonians are of the form H + = (cid:32) x tan θ ( x − z ) tan θ ( x − z ) z (cid:33) (4)The conservation laws can be understood from the struc-ture of the Hamiltonians. For example if θ = 0 → H + = (cid:18) x z (cid:19) and [ H + , σ z ] = 0, hence Z-component of angu-lar momentum is conserved. Note that tan θ diverges for θ = (cid:0) k + (cid:1) π, k ∈ I . Thus we need to use a differentform of H + then (using z = x − θy ). Correspondingcalculations are given in Appendix A. The matrix, H + can be diagonalized by the similaritytransformation, (cid:18) E E (cid:19) = R (cid:18) θ (cid:19) T H + R (cid:18) θ (cid:19) (5)= 12 (cid:18) x + z + ( x − z ) sec θ x + z − ( x − z ) sec θ (cid:19) and we are interested in obtaining the joint probabilitydistribution of eigenvalues, P ( E , E ). In general for anarbitrary matrix H composed of m -independent elements x , x , . . . , x m each drawn from probability distributions P i ( x i ) we can writeP ( H ) = m (cid:89) i =1 P i ( x i ) . (6)If we consider that the matrix elements of eigenfunctionsΘ are parameterized by { θ , θ , . . . , θ M } then the trans-formation from matrix space to eigenspace requiresP ( x , . . . , x m ) m (cid:89) i dx i = f (cid:16) (cid:126)E, (cid:126)θ (cid:17) | J | N (cid:89) j dE j M (cid:89) k dθ k (7)where, J is the Jacobian of the transformation. TheJPDF of eigenvalues is obtained by integration over θ i ’sP (cid:16) (cid:126)E (cid:17) = (cid:90) dθ · · · (cid:90) dθ M | J | f (cid:16) (cid:126)E, (cid:126)θ (cid:17) (8)and normalized to ensure (cid:82) d (cid:126)E P (cid:16) (cid:126)E (cid:17) = 1. For H + ,P ( E ) can also be obtained from Eq. (6) using Eq. (5) asP (cid:16) (cid:126)E (cid:17) = (cid:90) (cid:90) dxdzδ ( E − x + z x − z cosθ ) × δ ( E − x + z − x − z cosθ ) p ( x, z ) . Let us consider an ensemble of H + where matrix ele-ments are randomly chosen from normal distributions: x ∼ N (0 , , z ∼ N (0 , σ ), then P ( H + ) = P ( x, z ) =12 πσ exp (cid:32) − x − z σ (cid:33) and using Eq. (7), we obtain theJPDF of eigenvalues,P (cid:16) (cid:126)E (cid:17) = | cos θ | πσ exp (cid:32) − (cid:18) − σ (cid:19) ( E − E ) cos θ − (cid:18) σ (cid:19) ( E + E ) + ( E − E ) cos θ (cid:33) (9)Then marginal PDF of energy levels are given as,P ( E i ) = 1 √ πσ E i exp (cid:32) − E i σ E i (cid:33) σ E , = (cid:112) (1 + σ )(1 + sec θ ) ± − σ ) sec θ µ E = µ E = 0 and covari-ance, Cov( E , E ) = −
14 (1 + σ ) tan θ , correspondingcorrelation is, ρ E ,E = Cov( E , E ) σ E σ E = − (cid:32) θσ sin θ (1 + σ ) (cid:33) − . (11)The contour plot of ρ E ,E is shown in Fig. 1(b) for vary-ing θ, σ . It is important to note that E , E are notnecessarily independent even though H + always has acommuting partner and that a symmetry is present.Nearest Neighbour Spacing (NNS) is defined as dif-ference between successive energy levels, S = E − E and for a 2-level system the corresponding density canbe computed as [1]P ( S ) = (cid:90) ∞−∞ d (cid:126)E P (cid:16) (cid:126)E (cid:17) δ (cid:0) S − | E − E | (cid:1) . (12)In order to reveal the universal properties, the gaps arerescaled as s = S/ (cid:104) s (cid:105) satisfying: (cid:90) ∞ P ( s ) ds = 1 (Normalization) (cid:104) s (cid:105) = (cid:90) ∞ P ( s ) s ds = 1 (Unfolding) (13)where unfolding denotes scaling the unit of energy suchthat mean spacing, (cid:104) s (cid:105) = 1 [23]. Using Eq. (9) andEq. (12) and the conditions of Eq. (13), we get the PDFof NNS as, P ( s ) = 2 π exp (cid:32) − s π (cid:33) . (14)P ( s ) is shown for different θ, σ in Fig. 1(a), all of whichmatches with Eq. (14). If we take an ensemble of 2-levelsystems with energy levels drawn from Poisson process,the PDF of NNS takes the same form as Eq. (14) [10].Thus even though E i ’s are not always independent (asevident from Eq. (11)), the NNS distributions show levelclustering. For complete level repulsion in 2-level sys-tems, PDF of NNS is given by Wigner surmise for GOE,P ( s ) = π s exp (cid:18) − π s (cid:19) (15) Breaking Integrability of Real 2-level systems :We again start with a general form of the symmetricmatrix, H = (cid:18) x yy z (cid:19) and find constraints on the elementssuch that O from Eq. (3) is its anti-commuting partner,i.e., { H − , O } = H − O + O H − = 0. Then the Hamiltoniantakes the form, H − = u (cid:18) − tan θ
11 tan θ (cid:19) (16) FIG. 1.
Integrable 2-level Systems: (a) PDF of NNS com-puted for matrix H + in Eq. (4) with x ∈ N (0 , z ∈ N (0 , σ ).The curves correspond to θ = 0 , π , π, π and σ = 10 − , , . P ( s ) given in Eq. (14) is shown in black line. Simulationsare averaged over 10 realizations. The correlation functions ρ E ,E are computed from (b) Eq. (11) (c) Eq. (A4). Again we need to treat θ = (cid:0) k + (cid:1) π separately, whichis given in Appendix A. Our idea is to represent a matrixas superposition of a particular symmetry and its anti-symmetry: H = H + + λH − with λ ∈ [0 , ∞ ). Thus tuning λ it will be possible to break that symmetry, introducingdeviation from integrability. Now we will derive the levelstatistics of H .According to Eq. (4) and (16), H assumes the form, H = (cid:32) x − λu tan θ tan θ ( x − z ) + λu tan θ ( x − z ) + λu z + λu tan θ (cid:33) (17)If we assume that x, z, u ∼ N (0 , H ) = π ) exp (cid:16) − x + z + u (cid:17) . Eigenbasis of H form an orthog-onal matrix, Q = (cid:18) cos φ − sin φ sin φ cos φ (cid:19) such that Q T H Q = diag ( E , E ) and using Eq. (7), we get JPDF of eigen-values as quadratic Rayleigh-Rice distribution [21],P (cid:16) (cid:126)E (cid:17) = cos θ √ πλ | E − E | I (cid:32) c λ,θ ;2 ( E − E ) (cid:33) × exp (cid:32) − ( E + E ) + c λ,θ ;1 ( E − E ) (cid:33) (18)where the coefficients c λ,θ ;1 = (cid:18) λ (cid:19) cos θ c λ,θ ;2 = (cid:18) − λ (cid:19) cos θ s ) = 2 f λ (cid:48) πλ (cid:48) s exp (cid:32) − (cid:18) λ (cid:48) (cid:19) f λ (cid:48) π s (cid:33) I (cid:32)(cid:18) − λ (cid:48) (cid:19) f λ (cid:48) π s (cid:33) (19)here f (cid:48) λ = EllipticE (cid:0) − λ (cid:48) (cid:1) , λ (cid:48) = min (cid:26) √ λ, √ λ (cid:27) .P ( s ) is plotted for different θ in Fig. 2(a), all of whichare in perfect agreement with Eq. (19). Fig. 2(b) showsP ( s ) for different θ for a constant λ = 1 / √
2, verifyingthat level statistics is independent of θ . This impliesonly degree of symmetry breaking affects the energy levelproperties, not individual symmetries themselves.We need to quantify to what extent integrability isbroken in Eq. (19). The limiting cases are complete levelrepulsion, P ( s ) ∼ s exp (cid:0) − s (cid:1) and complete level clus-tering, P ( s ) ∼ exp (cid:0) − s (cid:1) . Thus we can form an interpo-lating function P ( η ; s ) ∼ s η exp (cid:0) − s (cid:1) , similar to Brodydistribution [24]. Normalizing and unfolding this we get,P ( η ; s ) = 2Γ (cid:0) η (cid:1) η Γ (cid:16) η (cid:17) η s η exp − Γ (cid:0) η (cid:1) Γ (cid:16) η (cid:17) s . (20)Fig. 2(c) shows η values as a function of λ with the spe-cific values of λ considered in Fig. 2(a) are shown by themarkers. As η → λ (cid:48) → −∞ ) implies thesame. Thus a system with a particular symmetry or itsantisymmetry will show regular dynamics. Otherwise ifthe system can be thought as an equal superposition ofany symmetry and its counterpart, i.e. no symmetry ispresent at all, then corresponding level statistics is com-pletely chaotic.The generalization of symmetric random matrix en-sembles can be sought by introducing matrix elementswith varying statistical properties by choosing diagonaland off-diagonal elements that are drawn from N (0 , σ d )and N (0 , σ o ), respectively [20, 21]. In our earlier studyof 2-level RPE systems, we have exactly calculated thePDF of NNS for N = 2 given asP ( s ) = 2 f σ πσ s exp (cid:32) − (cid:18) σ (cid:19) f σ π s (cid:33) I (cid:32)(cid:18) − σ (cid:19) f σ π s (cid:33) (21)where, f σ = EllipticE (cid:0) − σ (cid:1) with σ = min (cid:8) ˜ σ, σ (cid:9) andthe ratio of the diagonal and off-diagonal variances de-noted by ˜ σ = σ d √ σ o . We have demonstrated that byvarying σ the eigenvalue statistics can range from levelrepulsion to level clustering. We can clearly see that for˜ σ = √ λ the PDF of NNS given in Eq. (19) and (21) areidentical. Complex Integrable 2-level systems : Now theHamiltonian becomes hermitian, i.e. H = H † and we FIG. 2.
Non-integrable Real 2-level Systems: (a) PDFof NNS computed from H given in Eq. (17) for different valuesof λ and θ = 0. The corresponding analytical expression inEq. (19) are shown in bold lines. (b) PDF of NNS for θ = (cid:8) , π , π , π (cid:9) , while λ = 1 / √
2. (c) Interpolation parameter, η , estimated from Eq. (20) as a function of λ . Markers denotethe λ values mentioned in legend of (a). consider its commuting partner a unitary matrix U suchthat [ H, U ] = 0. Then U must be hermitian as well, whichimplies λ U = ±
1. Then we can find a change of basis Θwhich transforms U into D in Eq. (1). For a complex two-level system, existence of U implies system must be in-tegrable, hence corresponding Hamiltonian is denoted by H + . A convenient choice is Θ = (cid:32) cos θ − sin θ e − iφ sin θ e iφ cos θ (cid:33) ,resulting in the commuting partner of H + given by U = Θ D Θ † = (cid:18) cos θ sin θe − iφ sin θe iφ − cos θ (cid:19) (22)Starting from a general hermitian matrix H = (cid:18) x re iα re − iα y (cid:19) , requiring that it commutes with U inEq. (22) imposes the constraints α = kπ − φ, k ∈ I and 2 r = ( x − y ) tan θ . Hence the integrable complexHamiltonians are of the form, H + = (cid:32) x tan θ ( x − y ) e − iφ tan θ ( x − y ) e iφ y (cid:33) . (23)Again in order to avoid divergence of tan θ for θ = (cid:0) k + (cid:1) π, k ∈ I , we can make a change of variables in H + to obtain terms with cot θ without changing any ofthe results. If we do a similarity transformation of H + via Θ, we will get corresponding eigenvalues, which aresame as those in Eq. (5). Thus eigenspectrum properties FIG. 3.
Non-integrable Complex 2-level Systems: (a)PDF of NNS computed from H given in Eq. (25) for differ-ent values of λ . The corresponding analytical expression inEq. (26) are shown by bold lines. (b) Phase diagram in λ, θ (or λ, φ ) plane showing η values estimated using Eq. (28). (c)Interpolation parameter, η , estimated from Eq. (28) as a func-tion of λ . Markers denote the λ values mentioned in legendof (a). of integrable complex 2-level systems are same as that ofthe integrable real 2-level system. Breaking Integrability of Complex 2-level sys-tems : Starting from a general complex matrix and de-manding that it anticommutes with U i.e. { H − , U } = 0,we find that H − = (cid:18) − u cos ( α + φ ) tan θ v + iwv − iw u cos ( α + φ ) tan θ (cid:19) (24)where u = √ v + w and α = tan − wv . We can ex-press any complex 2-level system as H = H + + λH − by taking matrices from Eq. (23) and (24). Calculat-ing an exact analytical expression for PDF of NNS of H for arbitrary { θ, φ } is difficult. The NNS distributionsdetermined from numerical simulations are presented inFig. 3. Here also we observe that P ( s ) is independentof the values of { θ, φ } and an analytical expression canbe obtained for a very special case θ = 0 = ⇒ U = σ z .Then H assumes a very simple form, H = (cid:18) x λ ( v − iw ) λ ( v + iw ) y (cid:19) (25)where v = u cos α and w = u sin α . If weassume that x, y, v, w ∼ N (0 , H ) =14 π exp (cid:32) − x + y + v + w (cid:33) . The eigenbasis of H form an unitary matrix, Q = (cid:18) cos β − sin βe − iγ sin βe iγ cos β (cid:19) such that Q † H Q = diag ( E , E ) and using Eq. (7), weget JPDF of eigenvalues, P (cid:16) (cid:126)E (cid:17) . From P (cid:16) (cid:126)E (cid:17) usingEq. (12), (13) we get PDF of NNS as,P ( s ) = µ (cid:112) | − λ (cid:48) | s exp (cid:16) − µ s (cid:17) g ( µs ) , λ (cid:48) = √ λµ = √ π (cid:32) λ (cid:48) + cos − λ (cid:48) √ − λ (cid:48) (cid:33) , λ (cid:48) < √ π λ (cid:48) + 1 √ λ (cid:48) − − (cid:32)(cid:114) − λ (cid:48) (cid:33) , λ (cid:48) ≥ g ( s ) = Erf (cid:32)(cid:114) λ (cid:48) − s (cid:33) , λ (cid:48) < (cid:32)(cid:114) − λ (cid:48) s (cid:33) , λ (cid:48) ≥ λ (cid:48) = ˜ σ the above expression is same asthat of Hermitian RPE with N=2 [20]. In Fig. 3(a), wehave shown P ( s ) for different values of λ along with theiranalytical counterpart from Eq. (26). In the same figurewe have also shown PDF of NNS for complete clustering(Eq. (14)), GOE (Eq. (15)) and GUE, which is given by,P ( s ) = 32 π s exp (cid:18) − π s (cid:19) . (27)We need to probe the transition of P ( s ) in Eq. (26) byfitting to a Brody like distributionP ( η ; s ) = 2Γ (1 + η ) η Γ (cid:0) + η (cid:1) η ) s η exp − Γ (1 + η ) Γ (cid:0) + η (cid:1) s . (28)It is easy to see that above equation yields completeclustering, GOE and GUE for η = 0 , . , η as a function of λ in Eq. (26). Wehave numerically estimated P ( s ) from the general formof H , parameterized by θ, φ . Fixing φ (or θ ) to an arbi-trary constant we create a phase diagram in λ, θ (or λ, φ )plane showing η values as a heat map. We observe thatthe nature of P ( s ) is independent of the values of θ or φ and can be described by Eq. (26). Conclusion : In this paper we demonstrate that inan easily tractable 2-level system the notion of symme-try breaking and the effect of disorder are equivalent. Inour decomposition of a Hamiltonian as admixture of in-tegrable and non-integrable matrices we find that levelstatistics is independent of details of the symmetry andexactly corresponds to that of RPE. In one hand the val-ues of λ control the integrability while on the other handthe values of ˜ σ control the degree of disorder and bothof them control the spectral statistics in a remarkablyidentical manner. The clustering to repulsion transitionin RPE can thus be explained from the competing sym-metry and anti-symmetry properties of a random matrix.Studies on 2-level integrable systems indicate that theyalways show complete clustering even though the energylevels may be strongly correlated. In this paper we havestudied 2 × Appendix A: Results for θ = (cid:0) k + (cid:1) π, k ∈ Z Equivalent representations of the Hamiltonians inEq. (4) and (16) are H + = (cid:18) x yy x − θy (cid:19) ,H − = v (cid:18) − cot θ − cot θ − (cid:19) (A1)Via a similarity transformation H (cid:48) + = Θ T H + Θ, using Θin Eq. (2), we can diagonalize H + , H (cid:48) + = (cid:32) x + y tan θ x − y cot θ (cid:33) = (cid:18) E E (cid:19) (A2)Note that even though H + in Eq. (4) and (A1) areequivalent, independent elements are situated in dif-ferent positions. Hence varying variance should leadto a difference in energy level structure. P ( H + ) = πσ exp (cid:16) − x − y σ (cid:17) , if x ∼ N (0 , , y ∼ N (0 , σ ). Us- ing Eq. (7), we get JPDF of E i ’s as,P (cid:16) (cid:126)E (cid:17) = | sin θ | πσ exp (cid:32) − E − E θ − (cid:32) cos θ + sin θσ (cid:33) ( E − E ) − ( E + E ) (cid:33) (A3)Eq. (A3) yields marginal PDF of E i ’s asP ( E i ) = 1 √ πσ E i exp (cid:32) − E i σ E i (cid:33) , where σ E =1 + σ tan θ , σ E = 1 + σ cot θ . Correspondingcovariance is 1 − σ , hence correlation coefficient is givenby, ρ E ,E = (1 − σ ) | sin θ | (cid:113) (1 + σ ) sin θ + 2 σ (1 + cos θ ) . (A4)The contour plot of ρ E ,E is shown in Fig. 1(c) for vary-ing θ, σ . However, using Eq. (12) and (13) upon Eq. (A3),we get complete level clustering described by Eq. 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