Many-Body Fermions and Riemann Hypothesis
aa r X i v : . [ m a t h . G M ] N ov Many-Body Fermions and Riemann Hypothesis
Xindong Wang and Alex Shulman SophyicsTechnology,LLCNovember 11, 2020
We study the algebraic structure of the eigenvalues of a Hamiltonian that corresponds to amany-body fermionic system. As the Hamiltonian is quadratic in fermion creation and/orannihilation operators, the system is exactly integrable and the complete single fermion exci-tation energy spectrum is constructed using the non-interacting fermions that are eigenstatesof the quadratic matrix related to the system Hamiltonian. Connection to the Riemann Hy-pothesis is discussed. iemann Hypothesis has long been conjectured to be related to the eigenvalues of a Hamil-tonian since Hilbert in early twentieth century. In this paper, we show that the eigenvalues of ananti-symmetric real matrix that arises from the off-diagonal paring matrix elements of a many-bodyfermionic Hamiltonian seems to provide the necessary link between the Berry-Keating Conjec-ture and the final proof of Riemann Hypothesis. This work points to the importance of RiemannHypothesis to the understanding of intricate quantum entanglement of a many body system.
We study the following 1-dimensional spin-half many-body fermionic Hamiltonian ˆ H = X i,σ σ { ˆ p † iσ ˆ p iσ − ˆ h † iσ ˆ h iσ } − (cid:8) X i>i ′ ,σ t ( i − i ′ )ˆ p † iσ ˆ h † i ′ − σ + h.c. (cid:9) = (cid:20) ˆ ξ †↑ ˆ ξ †↓ (cid:21) T ↑ T ↓ ˆ ξ ↑ ˆ ξ ↓ i ∈ { , , ..., N } , σ ∈ {↑ , ↓} , N ≥ (1)and we further assume t ( i + N ) = t ( i ) , i.e., the system is a closed loop. ˆ ξ † σ = (cid:20) ˆ p † σ ˆ p † σ ... ˆ p † Nσ ˆ h − σ ˆ h − σ ... ˆ h N − σ (cid:21) (2) T σ = σ I N σ ∆ σ ∆ † − I N (3)and ∆ = 12 t (1) t (2) ... t ( N − − t (1) 0 t (1) ... t ( N − ... − t ( N − − t ( N − − t ( N − ... (4)1s an anti-symmetric matrix, due to the anti-commutative relation of the fermionic operators.The total charge operator for this system is defined as ˆ N c = X iσ (ˆ p † iσ ˆ p iσ − ˆ h † iσ ˆ h iσ ) (5)and the total spin operator ˆΣ = X iσ σ (ˆ p † iσ ˆ p iσ + ˆ h † iσ ˆ h iσ ) = σ ˆ N cσ (6)where ˆ N c σ = X i (ˆ p † iσ ˆ p iσ + ˆ h † i − σ ˆ h i − σ ) (7)And one can show that the total charge operator commutes with the Hamiltonian (1) [ ˆ H, ˆ N c ] = 0 , [ ˆ H, ˆΣ ] = 0 (8)We will focus on the case where the total charge as well as the total spin of the system is zero, i.e.,zero chemical potential and zero external magnetic field.Since the two spin channels are completely decoupled and degenerate, we will only need todiscuss the energy spectrum of T ↑ below.For a Hamiltonian of quadratic form, it can be exactly diagonalized in the subspace of zerocharge as ˆ H = X ( nσ ) ∈{ ( nσ ) | ε n ≤ } ε n {| V ac h ih V ac h | + | V ac p ih V ac p |} + X nσω ∈{ h,p } | ε n | ˆ γ † nσω ˆ γ nσω (9)2here ˆ γ nσp = X i u σn,i ˆ p iσ + X j v σn,j ˆ h † j − σ ˆ γ nσh = X i u σn,i ˆ h i − σ + X j v σn,j ˆ p † jσ (10)and the coefficients and ε n are eigenvectors and eigenvalues of the following Hermitian matrix Tdefined in Eq.(3) T u n v n = ε n u n v n (11)Note that we have explicitly retained the two time reversal symmetry related degenerate vacuumstates | V ac p i and | V ac h i , representing the two degenerate ground states of filled Fermi sea of p-fermions or h-fermions, and ˆ γ p , ˆ γ † p and ˆ γ h , ˆ γ † h represent the Majorana fermions corresponding totheir respective vacuum states.Note that the two time reversal symmetry related sets of solutions can be considered decou-pled to each other at the thermodynamic limit, since the only common eigenstate for each set is theabsolute empty vacuum where all the filled Fermi sea fermions in the 2 vacuum states are all ex-cited. Thus in the thermodynamic limit, we can consider the two sets of solutions two time-reversalsymmetry related universe.Expand explicitly the Eq.(11), we have (1 / − ε n ) I N u n + ∆ v n = 0∆ † u n + ( − / − ε n ) I N v n = 0 (12)which leads to { ( ε n − / ε n + 1 / I N − ∆ † ∆ } v = 0 (13)3hat is ε n are roots of the following polynomial P ( z ) = det (cid:20) ( z − (1 / ) I N − ∆ † ∆ (cid:21) (14)The eigenvalues ε n are thus ε n = p (1 / + t n (15)where t n are singular values of ∆ † ∆ To make the connection to Riemann Hypothesis, we first develop the mathematical theory for thealgebraic structure of eigenvalues of anti-symmetric matrices.
Lemma 1.
For any given anti-symmetric matrix ∆ , it can be unitarily diagonalized. And when ∆ is anti-symmetric real, all its eigenvalues are imaginary.Proof. Since ∆ = ∆ r + i ∆ i is anti-symmetric, ∆ r , ∆ i are real anti-symmetric, ∆∆ † = ∆ † ∆ = − ∆ r + ∆ i (16)Thus both ∆ and ∆ † are normal matrices, i.e., they can be unitarily diagonalized ∆ = U DU † , ∆ † = U D ∗ U † (17)where D is a diagonal matrix.When ∆ is anti-symmetric real matrix, i ∆ is a Hermitian matrix, thus it can be diagonalizedwith all eigenvalues being real, i.e., iD is a real diagonal matrix. This completes the proof.4ext we show that following Lemma concerning the rank of an anti-symmetric matrix Lemma 2.
If M is an anti-symmetric matrix of size N , denote its rank as r M , then r M is an evennumber and all non-zero eigenvalues of N come in pairs of ± z i .Proof. This is because every eigenvalue of M is also an eigenvalue of M T and M T = − M , so if λ is an eigenvalue of M , then − λ is also an eigenvalue. Thus, all non-zero eigenvalues of M comein pairs. This completes the proof.We note that both Lemmas have been established in the literature . Lemma 3.
For any anti-symmetric matrix M , there exists an anti-symmetric real matrix ˜ M , suchthat M M † and ˜ M ˜ M † have the same eigenvalues and they are related by a unitary transformation U or an anti-unitary transformation A = KU , i.e., M = U ˜ M U † = U K ˜ M KU † (18) where K is the complex conjugate operator.Proof. M = U DU † by Lemma 1, and due to Lemma 2, D can be arranged as the following blocks5f pairs D = λ − λ ... λ − λ ... ... ... λ r M / − λ r M /
00 0 0 ... (19)Define the following phase factors ϕ i λ i = iε i e iϕ i (20)where ε i = | λ i | . Thus we have D = Φ( { ϕ i } ) E Φ † ( { ϕ i } ) (21)where Φ( { ϕ i } ) = e i ϕ I ... e i ϕ I ... ... ... e i ϕrM / I
00 0 0 ... (22)6s a unitary matrix and E is E = iε − iε ... iε − iε ... ... ... iε r M / − iε r M /
00 0 0 ... (23)And we have iε i − iε i = v i ε i − ε i v † i where v i v † i = 1 , as v i = √ − i √ √ i √ v † i = √ √ i √ − i √ Thus we have D = Φ( { ϕ i } ) V · E · V † Φ † ( { ϕ i } ) (24)where E is an anti-symmetric real matrix and Φ( { ϕ i } ) V is a unitary matrix since product of twounitary matrix is still a unitary matrix. Thus M M † = U DD † U † = U Φ( { ϕ i } ) V E E † V † Φ † ( { ϕ i } ) U † = (cid:0) U E U † (cid:1) ( U E † U † (cid:1) where U U = U Φ( { ϕ i } ) V (25)7s unitary as products of unitary matrices are unitary. And for the case of anti-unitary transforma-tion, we observe that both E and E † are real, we have KE K = E KE † K = E † Thus, any non-degenerate anti-symmetric real matrix defines an equivalence relationship perLemma 3, if
M M † have the same set of pairs of eigenvalues. All members of the equivalence classare related by a unitary or an anti-unitary transformation. Definition 1.
The real anti-symmetry matrix of an equivalent class of anti-symmetric matrices iscalled the characteristic of the class. We use C ( M ) to denote the characteristic of an anti-symmetricmatrix.Intuitively, since the anti-symmetric matrices arise from the off-diagonal paring block ofa many-body fermionic system, the topological effect of the anti-symmetric matrix is intricatelyrelated to the quantum entanglement of a many-body fermionic system and any periodicity in theoff-diagonal matrix elements in real space will imply some harmonic resonances in the eigenvaluesof the original Hermitian Hamiltonian, but in a way through the imaginary eigenvalues of thecharacteristics of the off-diagonal anti-symmetric matrix.8 Connection to Riemann Hypothesis
Next we solve for the eigenvalues of the following anti-symmetric real matrix for a given p , where p is a prime number that corresponds to the period of the gauge field.We will solve the eigenvalue problem with each k ∈ [0 , /p ] . Let the anti-symmetric matrix ∆ N ( p, k ) be given explicitly as ∆ N ( p, k ) = 12 t ( p, k ) t ( p, k ) ... t ( N − ( p, k ) − t ( p, k ) 0 t ( p, k ) ... t ( N − ( p, k ) ... − t ( N − ( p, k ) − t ( N − ( p, k ) − t ( N − ( p, k ) ... (26)and t l ( p, k ) , k ∈ [0 , /p ] is t l ( p, k ) = p · l Z / − / dq · q · sin (2 π ( q + k ) · l ) = ( − l π · cos (2 πl · k ) , l ∈ { , , ..., N − } (27)And the matrix element for ∆ N ( p, k ) is anti-symmetric real: ∆ l,l ′ ( p, k ) = − ∆ l ′ ,l ( p, k ) (28)Once the eigenvalues λ n ( p, k ) = iε n ( p, k ) are solved for all k ∈ [0 , /p ] , then the followingspectral function can be calculated G N ( p, z ) = p N X n Z /p dk z − λ n ( p, k ) = Z ∞−∞ dε · ρ N ( p, ε ) 1 z − iε (29)9here the normalized density of state ρ N ( p, ε ) is given by ρ N ( p, ε ) = 2 N p X n Z p/ dkδ ( ε − ε n ( p, k )) (30)and it has the following sum rule Z ∞−∞ dε · ρ N ( p, ε ) = 1 (31)Note that G N ( p, z ) contains the periodicity of p of the underlying gauge field, thus poles ofthis function are harmonic resonances, that is when N → ∞ , we have the density of states ρ ( ε ) defined in Eq.(30) diverges at those resonance frequencies.The poles of G N ( p, z ) are all expected to be imaginary since all λ n ( p, k ) are imaginary. Thuspoles of the following function G N ( z ) = Y p This work is supported by Sophyics Technology, LLC.1. Berry, M. V. & Keating, J. P. Supersymmetry and trace formulae:Chaos and disorder (Kluwer Academic/Plenum, New York, 1999). URL http://link.springer.com/chapter/10.1007%2F978\protect\unhbox\voidb@x\hbox{-}1\protect\unhbox\voidb@x\hbox{-}4615\protect\unhbox\voidb@x\hbox{-}4875\protect\unhbox\voidb@x\hbox{-}1_19 .2. Bender, C. M., Brody, D. C. & M ¨uller, M. P. Hamiltonian for the zeros of the riemann zetafunction. Phys. Rev. Lett. , 130201 (2017).3. Youla, D. A normal form for a matrix under the unitary congruence group. Can. J. Math.13