To find the discrete symmetries of a Hamilton operator $\hat H$ is of central
importance in quantum theory. Here we describe and implement a brute force
method to determine the discrete symmetries given by permutation matrices for
Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi
systems are considered as examples. A computer algebra implementation in
SymbolicC++ is provided

Hardy, Yorick

Steeb, Willi-Hans

International Journal of Modern Physics C

English

arXiv

To find the discrete symmetries of a Hamilton operator ˆH is of central importance in quantum theory. Here we describe and implement a brute force method to determine the discrete symmetries given by permutation matrices for Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi systems are considered as examples. A computer algebra implementation in SymbolicC++ is provided

University of Johannesburg Institutional Repository

arXiv:1208.4721v1  [cs.MS]  23 Aug 2012Hamilton Operators, Discrete Symmetries, Brute Force andSymbolicC++Willi-Hans Steeb† and Yorick Hardy∗† International School for Scientific Computing,University of Johannesburg, Auckland Park 2006, South Africa,e-mail: steebwilli@gmail.com∗ Department of Mathematical Sciences,University of South Africa, Pretoria, South Africa,e-mail: hardyy@unisa.ac.zaAbstract To find the discrete symmetries of a Hamilton operator Hˆ is of centralimportance in quantum theory. Here we describe and implement a brute forcemethod to determine the discrete symmetries given by permutation matrices forHamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermisystems are considered as examples. A computer algebra implementation in Sym-bolicC++ is provided.1 IntroductionIn quantum mechanics the system is described by a self-adjoint (Hamilton) opera-tor Hˆ acting in a Hilbert space H. Here we consider the finite dimensional Hilbertspace Cn where Hˆ is a hermitian matrix [1, 2, 3, 4, 5]. One of the main tasks is tofind the n × n unitary matrices U such that U∗HˆU = Hˆ , where U∗ = U−1. Then×n unitary matrices form the compact Lie group U(n). Note that if U∗HˆU = Hˆand V ∗HˆV = Hˆ then (UV )∗Hˆ(UV ) = Hˆ , where (UV )∗ ≡ V ∗U∗. Thus the set ofmatrices that keep Hˆ invariant form a group themselves [6].An important finite subgroup of the group U(n) are the n×n permutation matrices.The number of n×n permutation matrices is n!. For a given hermitian n×n matrixHˆ we want to find all the n× n permutation matrices P such thatP T HˆP = Hˆwhere P−1 = P T . These permutation matrices form a subgroup of all n × n per-mutation matrices. Obviously the n × n identity matrix In satisfies InHˆIn = Hˆ .Here we describe and implement in SymbolicC++ a brute force method to findthese permutation matrices.After this finite group has been found one determines the conjugacy classes. Nowfor a finite group G the number of conjugacy classes is equivalent to the number ofnon-equivalent irreducible matrix representations. From the conjugacy classes andthe permutation matrices we can construct projection matrices to decompose theHilbert space into invariant sub Hilbert spaces [6]. For example, if the permutationmatrix P satisfies P 2 = In, then Π1 = (In + P )/2, Π2 = (In − P )/2 are projectionmatrices (with Π1Π2 = 0) which can be utilized to decompose the Hilbert spaceCn into invariant subspaces.2 ExamplesWe consider four examples: two Fermi systems and two spin systems.Example 1. Let c†j , cj (j = 1, 2, 3) be Fermi creation and annihilation operators,respectively. Consider the Hamilton operatorHˆ = t(c†1c2 + c†2c1 + c†2c3 + c†3c2 + c†1c3 + c†3c1) + k1c†1c1 + k2c†2c2 + k3c†3c3and the number operator Nˆ = c†1c1+c†2c2+c†3c3. Then [Hˆ, Nˆ ] = 0. Since [Hˆ, Nˆ ] = 0we find Nˆ is a constant of motion, i.e. the total number of Fermi particles remainsconstant in the sense that if |n〉 is an eigenstate of the number operator Nˆ witheigenvalue n at time 0, then |n〉(t) = e−iHˆt/~|n〉 remains an eigenstate of Nˆ witheigenvalue n for all times. Given a basis with two Fermi particlesc†1c†2|0〉, c†1c†3|0〉, c†2c†3|0〉 .Then we find the matrix representation of HˆHˆ = k1 + k2 t −tt k1 + k3 t−t t k2 + k3 .The matrix representation of Nˆ is the diagonal matrix 2I3, where I3 is the 3 × 3identity matrix. For k1 6= k2, k1 6= k3, k2 6= k3 no non-trivial symmetry is found.Also for k1 6= k2, k1 6= k3, k2 = k3 no non-trivial symmetry is found. For k = k1 =k2 = k3 we obtain the permutation matrixP = 0 0 10 1 01 0 0 .Thus using the projection matrices Π1 = (I3 − P )/2, Π2 = (I3 − P )/2 the Hilbertspace C3 can be decomposed into invariant subspaces. For the case k = k1 = k2 =k3 we find the eigenvalues 2k − 2t and 2k + t (twice).Example 2. Consider the Hamilton operator (two-point Hubbard model)Hˆ = t(c†1↑c2↑ + c†1↓c2↓ + c†2↑c1↑ + c†2↓c1↓) + U(n1↑n1↓ + n2↑n2↓)where nj↑ := c†j↑cj↑, nj↓ := c†j↓cj↓. The operators c†j↑, c†j↓, cj↑, cj↓ are Fermi opera-tors. The Hubbard Hamilton operator commutes with the total number operatorNˆ and the total spin operator Sˆz whereNˆ :=2∑j=1(c†j↑cj↑ + c†j↓cj↓), Sˆz :=122∑j=1(c†j↑cj↑ − c†j↓cj↓) .We consider the subspace with two particles N = 2 and total spin Sz = 0. A basisin this four dimensional Hilbert space is given byc†1↑c†1↓|0〉, c†1↑c†2↓|0〉, c†2↑c†1↓|0〉, c†2↑c†2↓|0〉 .We find the matrix representation of Hˆ for this basis. Using the Fermi anti-commutation relations and cj↑|0〉 = 0, cj↓|0〉 = 0 for j = 1, 2 we obtain the matrixrepresentation of Hˆ with the given basisH =U t t 0t 0 0 tt 0 0 t0 t t U .We find the four permutation matrices P0 = I4 = I2 ⋆ I2 = I2 ⊗ I2,P1 = I2 ⋆(0 11 0), P2 =(0 11 0)⋆ I2, P3 =(0 11 0)⋆(0 11 0)where we define the star product ⋆ of the two 2× 2 matrices A, B as [6](a11 a12a21 a22)⋆(b11 b12b21 b22):=a11 0 0 a120 b11 b12 00 b21 b22 0a21 0 0 a22 .We note that the star product of two 2×2 permutation matrices is a 4×4 permu-tation matrix. Here P1 is the swap gate. The four permutation matrices P0, P1,P2, P3 form a commutative group with P2j = I4 for j = 0, 1, 2, 3. If P is an n× npermutation matrix with P 2 = In thenΠ1 =12(In + P ), Π2 =12(In − P )are projection matrices with Π1Π2 = 0n, where 0n is the n× n zero matrix. Usingthe permutation matrices P0 and P3 (which form a subgroup) these projectionoperators can now be used to find the invariant subspaces1001 ,0110,100−1 ,01−10.These four vectors (after normalization) are known in quantum computing as theBell basis [1, 2, 3, 4, 5]. This leads to the two invariant sub Hilbert spaces{1√2(c†1↓c†1↑|0〉+ c†2↓c†2↑|0〉),1√2(c†1↓c†2↑|0〉+ c†2↓c†1↑|0〉)}{1√2(c†1↓c†1↑|0〉 − c†2↓c†2↑|0〉),1√2(c†1↓c†2↑|0〉 − c†2↓c†1↑|0〉)}.Example 3. Let σ1, σ2, σ3 be the Pauli spin matricesσ1 :=(0 11 0), σ2 :=(0 −ii 0), σ3 :=(1 00 −1).Consider the Hamilton operators [7]Hˆ = ~ω1σ3 ⊗ I2 + ~ω2I2 ⊗ σ1 + ǫ(σ3 ⊗ σ1)Kˆ = ~ω1σ3 ⊗ I2 + ~ω2I2 ⊗ σ1 + ǫ(σ1 ⊗ σ3)where for the second Hamilton operator Kˆ the interaction term is swapped around,i.e. σ3 ⊗ σ1 → σ1 ⊗ σ3. This provides symmetry breaking. For the Hamiltonoperator Hˆ we find the symmetries (permutation matrices)P0 = I4, P1 = I2 ⊕ σ1, P2 = σ1 ⊕ I2, P3 = σ1 ⊕ σ1where ⊕ denotes the direct sum. The four matrices form a commutative groupunder matrix multiplication. All satisfy P 2j = I4. Thus we can use the projectionmatrices Π1 = (I4 + Pj)/2, Π2 = (I4 − Pj)/2 to decompose the Hilbert space intotwo invariant subspaces. On the other hand for the Hamilton operator Kˆ we onlyfind the identity matrix P0 = I4, i.e. no non-trivial symmetry is admitted.Example 4. Consider the Hamilton operator for triple spin interactionHˆ = σ1 ⊗ σ2 ⊗ σ3 .The eigenvalues of this hermitian and unitary 8 × 8 matrix are +1 (four-fold de-generate) and −1 (four-fold degenerate). Owing to this degeneracy one expects a“large” number of symmetries. Applying the SymbolicC++ code we find 24 permu-tation matrices listed P0, P1, . . . , P23 with P0 = I8. They form a non-commutativegroup under matrix multiplication and are a subgroup of the group of 8 × 8 per-mutation matrices. We note that the Kronecker product ⊗ and the direct sum ⊕of two permutation matrices is again a permutation matrix [8]. Now we can listthe ones with P 2j = I8. We haveP1 = I2 ⊕((0 11 0)⊗(0 11 0))⊕ I2P2 = (I2 ⋆ σ1)⊕ (σ1 ⋆ I2)P5 = I2 ⊗ I2 ⊗(1 00 0)+ σ1 ⊗ I2 ⊗(0 00 1)P6 = (σ1 ⋆ I2)⊕ (I2 ⋆ σ1)P8 = I2 ⊗ σ1 ⊗ σ1P13 = I2 ⊗ I2 ⊗(0 00 1)+ σ1 ⊗ I2 ⊗(1 00 0)P15 = σ1 ⊗ I2 ⊗ I2P23 = σ1 ⊗ σ1 ⊗ σ1 .The other ones can be found by multiplication of these permutation matrices, forexample P3 = P1P2 etc. Thus the 24 matrices form a subgroup of the permutationgroup of 8× 8 matrices.Another spin Hamilton operator studied is [9]Hˆ = a4∑j=1σ3(j)σ3(j + 1) + b4∑j=1σ1(j)with cyclic boundary conditions, i.e. σ3(5) ≡ σ3(1). Here a, b are real constantsand σ1, σ2 and σ3 are the Pauli matrices. Thus the underlying Hilbert space isC16. Recall thatσk(1) = σk ⊗ I2 ⊗ I2 ⊗ I2, σk(2) = I2 ⊗ σk ⊗ I2 ⊗ I2σk(3) = I2 ⊗ I2 ⊗ σk ⊗ I2, σk(4) = I2 ⊗ I2 ⊗ I2 ⊗ σkwhere k = 1, 2, 3. We obtain the symmetric 16× 16 matrix for Hˆ4a b b 0 b 0 0 0 b 0 0 0 0 0 0 0b 0 0 b 0 b 0 0 0 b 0 0 0 0 0 0b 0 0 b 0 0 b 0 0 0 b 0 0 0 0 00 b b 0 0 0 0 b 0 0 0 b 0 0 0 0b 0 0 0 0 b b 0 0 0 0 0 b 0 0 00 b 0 0 b −4a 0 b 0 0 0 0 0 b 0 00 0 b 0 b 0 0 b 0 0 0 0 0 0 b 00 0 0 b 0 b b 0 0 0 0 0 0 0 0 bb 0 0 0 0 0 0 0 0 b b 0 b 0 0 00 b 0 0 0 0 0 0 b 0 0 b 0 b 0 00 0 b 0 0 0 0 0 b 0 −4a b 0 0 b 00 0 0 b 0 0 0 0 0 b b 0 0 0 0 b0 0 0 0 b 0 0 0 b 0 0 0 0 b b 00 0 0 0 0 b 0 0 0 b 0 0 b 0 0 b0 0 0 0 0 0 b 0 0 0 b 0 b 0 0 b0 0 0 0 0 0 0 b 0 0 0 b 0 b b 4a.The Hamilton operator Hˆ admits the C4v symmetry group. The order of thisnon-commutative group is 8. One finds the following set of eight symmetries [9]E : (1, 2, 3, 4)→ (1, 2, 3, 4) C2 : (1, 2, 3, 4)→ (3, 4, 1, 2)C4 : (1, 2, 3, 4)→ (2, 3, 4, 1) C34 : (1, 2, 3, 4)→ (4, 1, 2, 3)σv : (1, 2, 3, 4)→ (2, 1, 4, 3) σ′v : (1, 2, 3, 4)→ (4, 3, 2, 1)σd : (1, 2, 3, 4)→ (1, 4, 3, 2) σ′d : (1, 2, 3, 4)→ (3, 2, 1, 4)which form a group isomorphic to C4v. The symmetries can be found by calculatingthe 16× 16 permutation matrices such that Hˆ = P T HˆP .3 Code DescriptionAlgorithms for finding all permutations of a sequence of objects are described byKnuth [10]. For a given n the permutation matrices are generated with the follow-ing algorithm. The algorithm implements the nested loopsFor j0 = 0, 1, . . . , n− 1 doFor j1 = 0, 1, . . . , n− 1 do...For jn−1 = 0, 1, . . . , n− 1 doIf j0 6= j1 6= · · · 6= jn−1 thenuse the permutation (0, 1, 2, . . . , n− 1)→ (j0, j1, j2, . . . , jn−1).End loop...End loopEnd loopAlgorithm to find all permutation matrices.1. Create an array (j0, j1, . . . , jn−1) of loop variables.2. Initialize jk := −1 for k = 0, 1, . . . , n− 1.3. Initialize the loop variable index i to i := 0.4. While i ≥ 0(a) Iterate.Set ji := ji + 1.(b) Termination condition.If ji = n terminate this loop:i. Set ji := −1.ii. Exit the nested loop.Set i := i− 1.iii. Goto 4.(c) If jk = ji for some k ∈ {0, 1, . . . , i− 1} then goto 4.(d) Enter the next nested loop.Set i := i+ 1.(e) Innermost loop completes.If i = n then use the permutation (0, 1, 2, . . . , n−1)→ (j0, j1, j2, . . . , jn−1)i.e. the permutation matrix P is given by(P )uv ={1 if v = ju0 otherwise.The SymbolicC++ program [11] utilizes the vector class of the Standard TemplateLibrary. The Hamilton operator refers to example 2 in the text (two point Hubbardmodel).// permutation.cpp#include <iostream>#include <vector>#include "symbolicc++.h"using namespace std;int total;Symbolic H;void commutes(const Symbolic &P){if(P*H==H*P) cout << "P[" << total++ << "] = " << P << endl;}void find_perm(int n,void (*use)(const Symbolic&)){int i, k;Symbolic P;vector<int> j(n,-1);i = 0; j[0] = -1;while(i >= 0){if(++j[i]==n) { j[i--] = -1; continue; }if(i < 0) break;for(k=0;k<i;++k) if(j[k]==j[i]) break;if(k!=i) continue;++i;if(i==n){P = 0*Symbolic("",n,n);for(k=0;k<n;k++) P(k,j[k]) = 1;use(P);--i;}}}int main(void){using SymbolicConstant::i;Symbolic sqrt2 = sqrt(Symbolic(2));Symbolic U("U"); Symbolic t("t");H = ((U,t,t,Symbolic(0)),(t,Symbolic(0),Symbolic(0),t),(t,Symbolic(0),Symbolic(0),t),(Symbolic(0),t,t,U));cout << "H = " << H << endl;total = 0;find_perm(H.rows(),commutes);return 0;}A Maxima implementation is available from the authors.4 ConclusionWe applied a brute force method to find all possible permutation matrices thatprovide symmetries for given Hamilton operators in a finite dimensional Hilbertspace. With growing size of the Hamilton operators matrix representation findingthe permutation matrices becomes very time-consuming. A more efficient approachwould be to find only the generators of the group of permutation matrices thatprovide symmetries for a given Hamilton operator. Another open question is howthis method can be extended to find other classes of symmetries.AcknowledgmentThe authors are supported by the National Research Foundation (NRF), SouthAfrica. This work is based upon research supported by the National ResearchFoundation. Any opinion, findings and conclusions or recommendations expressedin this material are those of the author(s) and therefore the NRF do not acceptany liability in regard thereto.References[1] M. A. Nielsen and I. L. Chuang, Quantum Computing and Quantum Infor-mation, Cambridge University Press, Cambridge (2000).[2] Y. Hardy and W.-H. Steeb, Classical and Quantum Computing with C++ andJava Simulations, Birkhauser Verlag, Basel (2002).[3] M. Hirvensalo, Quantum Computing, second edition, Springer, New York(2004).[4] N. D. Mermin, Quantum Computer Science, Cambridge University Press,Cambridge (2007).[5] W.-H. Steeb and Y. Hardy, Problems and Solutions in Quantum Computingand Quantum Information, third edition, World Scientific Publishing (2011).[6] W.-H. Steeb, I. Tanski and Y. Hardy, Problems and Solutions for Groups, LieGroups, Lie Algebras with Applications, World Scientific, Singapore (2012).[7] W.-H. Steeb, Y. Hardy and J. de Greef, Z. Naturforsch. 67a, 551 (2012).[8] W.-H. Steeb and Y. Hardy, Matrix Calculus and Kronecker Product, secondedition, World Scientific, Singapore (2011).[9] W.-H. Steeb, Problems and Solutions in Theoretical and MathematicalPhysics, Third edition, Volume II: Advanced Level, World Scientific, Sin-gapore (2009).[10] D. E. Knuth, The Art of Computer Programming, Volume 4A: CombinatorialAlgorithms, Part 1, Addison-Wesley (2011).[11] Y. Hardy, Kiat Shi Tan and W.-H. Steeb, Computer Algebra with Symbol-icC++, World Scientific, Singapore (2008).

Hamilton operators, discrete symmetries, brute force and symbolicC++

 To find the discrete symmetries of a Hamilton operator Ĥ is of central importance in quantum theory. Here, we describe and implement a brute force method to determine the discrete symmetries given by permutation matrices for Hamilton operators acting in a finite-dimensional Hilbert space. Spin and Fermi systems are considered as examples. A computer algebra implementation in SymbolicC++ is provided. </jats:p

Hamilton Operators, Discrete Symmetries, Brute Force and SymbolicC++

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