H ψ = E ψ. Introduction to Exact Diagonalization. Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden

Transcription

1 H ψ = E ψ Introduction to Exact Diagonalization Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden laeuchli@comp-phys.org Simulations of Quantum Many Body Systems - Lecture Spring 2010

2 Outline of Today s Lecture (part 1 of 3) Overview of Lecture (Short) Introduction to Quantum Lattice Models Introduction to Exact Diagonalization (part 1) Overview of the ED Algorithm Basis construction First aspects of symmetries

3 Quantum Lattice Models kagome lattice Volborthite

4 Models Single particle lattice models Many body lattice models Constrained lattice models

5 Single Particle Lattice Models Disordered potentials and hoppings Anderson Localization

6 Single Particle Lattice Models Disordered potentials and hoppings Anderson Localization Hopping phases (Gauge fields) Hofstadter Butterfly

7 Single Particle Lattice Models Disordered potentials and hoppings Anderson Localization Hopping phases (Gauge fields) Hofstadter Butterfly Jordan-Wigner transformation (maps XX spin chains onto free fermions) Disordered XX-spin chains

8 Many-Body Hamiltonians Spin models: S=1/2: two states per site Heisenberg couplings (2 sites): Modifications: easy plane, easy axis anisotropies higher order spin interactions (biquadratic, bicubic, ring exchange,...)

9 Many-Body Hamiltonians Fermionic models: Hubbard model: The c/c + operators satisfy anticommutation rules i.e. Fermi sign comes in! There also exists a Bose-Hubbard models, with bosons replacing the fermions.

10 Many-Body Hamiltonians Fermionic models: t-j model: ( large U limit of the Hubbard model) The c/c + operators satisfy anticommutation rules i.e. Fermi sign comes in!

11 Constrained Models Hardcore-dimer models:

12 Constrained Models Hardcore-dimer models: Vertex / Ice models:

13 Lattices Infinite Lattices are made out of: A Unit Cell, containing n atoms

14 Lattices Infinite Lattices are made out of: A Unit Cell, containing n atoms Spanning vectors defining the Bravais lattice: T 2 T 1

15 Lattices II Some Examples: Square Lattice single site unit cell

16 Lattices II Some Examples: T 2 T 1 Square Lattice single site unit cell Checkerboard Lattice two sites in unit cell

17 Finite Lattices T 2 Extent is given by multiples of the Bravais vectors T 1 and T 2 T 1 For the square lattice: Extent =6x6 36 sites

18 Finite Lattices T 2 T 1 Extent given by spanning vectors which are not parallel to the simple axes For the square lattice: F 1 =(6,2)=6T 1 +2T 2 F 2 =(-2,6)=-2T 1 +6T 2 40 sites The discrete k-vectors are also tilted

19 Total Hilbert space of a finite many body system

20 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density)

21 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density) For N spin 1/2 particles, the total Hilbert space has dim=2 N states

22 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density) For N spin 1/2 particles, the total Hilbert space has dim=2 N states 10 spins dim=1 024

23 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density) For N spin 1/2 particles, the total Hilbert space has dim=2 N states 10 spins dim= spins dim=

24 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density) For N spin 1/2 particles, the total Hilbert space has dim=2 N states 10 spins dim= spins dim= spins dim=

25 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density) For N spin 1/2 particles, the total Hilbert space has dim=2 N states 10 spins dim= spins dim= spins dim= the same for spinless fermions and hardcore bosons.

26 Total Hilbert space of a finite many body system The Hilbert space of a quantum many body system grows exponentially in general (eg at constant density) For N spin 1/2 particles, the total Hilbert space has dim=2 N states 10 spins dim= spins dim= spins dim= the same for spinless fermions and hardcore bosons. for actual bosons it is even worse...

27 Introduction to Exact Diagonalization H ψ = E ψ

28 Exact Diagonalization: Main Idea H ψ = E ψ

29 Exact Diagonalization: Main Idea Solve the Schrödinger equation of a quantum many body system numerically H ψ = E ψ

30 Exact Diagonalization: Main Idea Solve the Schrödinger equation of a quantum many body system numerically H ψ = E ψ Sparse matrix, but for quantum many body systems the vector space dimension grows exponentially!

31 Exact Diagonalization: Main Idea Solve the Schrödinger equation of a quantum many body system numerically H ψ = E ψ Sparse matrix, but for quantum many body systems the vector space dimension grows exponentially! Some people will tell you that s all there is.

32 Exact Diagonalization: Main Idea Solve the Schrödinger equation of a quantum many body system numerically H ψ = E ψ Sparse matrix, but for quantum many body systems the vector space dimension grows exponentially! Some people will tell you that s all there is. But if you want to get a maximum of physical information out of a finite system there is a lot more to do and the reward is a powerful:

33 Exact Diagonalization: Main Idea Solve the Schrödinger equation of a quantum many body system numerically H ψ = E ψ Sparse matrix, but for quantum many body systems the vector space dimension grows exponentially! Some people will tell you that s all there is. But if you want to get a maximum of physical information out of a finite system there is a lot more to do and the reward is a powerful: Quantum Mechanics Toolbox

34 Exact Diagonalization: Applications

35 Exact Diagonalization: Applications Quantum Magnets: nature of novel phases, critical points in 1D, dynamical correlation functions in 1D & 2D

36 Exact Diagonalization: Applications Quantum Magnets: nature of novel phases, critical points in 1D, dynamical correlation functions in 1D & 2D Fermionic models (Hubbard/t-J): gaps, pairing properties, correlation exponents, etc

37 Exact Diagonalization: Applications Quantum Magnets: nature of novel phases, critical points in 1D, dynamical correlation functions in 1D & 2D Fermionic models (Hubbard/t-J): gaps, pairing properties, correlation exponents, etc Fractional Quantum Hall states: energy gaps, overlap with model states, entanglement spectra

38 Exact Diagonalization: Applications Quantum Magnets: nature of novel phases, critical points in 1D, dynamical correlation functions in 1D & 2D Fermionic models (Hubbard/t-J): gaps, pairing properties, correlation exponents, etc Fractional Quantum Hall states: energy gaps, overlap with model states, entanglement spectra Quantum dimer models or other constrained models: nature of novel phases

39 Exact Diagonalization: Applications Quantum Magnets: nature of novel phases, critical points in 1D, dynamical correlation functions in 1D & 2D Fermionic models (Hubbard/t-J): gaps, pairing properties, correlation exponents, etc Fractional Quantum Hall states: energy gaps, overlap with model states, entanglement spectra Quantum dimer models or other constrained models: nature of novel phases Full Configuration Interaction in Quantum Chemistry reference and benchmark results

40 Exact Diagonalization: Present Day Limits low-lying eigenvalues, not full diagonalization

41 low-lying eigenvalues, not full diagonalization Exact Diagonalization: Present Day Limits Spin S=1/2 models: 40 spins square lattice, 39 sites triangular, 42 sites honeycomb lattice (64 spins or more in elevated magnetization sectors) up to 1.5 billion(=10 9 ) basis states with symmetries, up to 4.5 billion without

42 low-lying eigenvalues, not full diagonalization Exact Diagonalization: Present Day Limits Spin S=1/2 models: 40 spins square lattice, 39 sites triangular, 42 sites honeycomb lattice (64 spins or more in elevated magnetization sectors) up to 1.5 billion(=10 9 ) basis states with symmetries, up to 4.5 billion without t-j models: 32 sites checkerboard with 2 holes 32 sites square lattice with 4 holes up to 2.8 billion basis states

43 low-lying eigenvalues, not full diagonalization Exact Diagonalization: Present Day Limits Spin S=1/2 models: 40 spins square lattice, 39 sites triangular, 42 sites honeycomb lattice (64 spins or more in elevated magnetization sectors) up to 1.5 billion(=10 9 ) basis states with symmetries, up to 4.5 billion without t-j models: 32 sites checkerboard with 2 holes 32 sites square lattice with 4 holes up to 2.8 billion basis states Fractional quantum hall effect different filling fractions ν, up to electrons up to 3.5 billion basis states

44 low-lying eigenvalues, not full diagonalization Exact Diagonalization: Present Day Limits Spin S=1/2 models: 40 spins square lattice, 39 sites triangular, 42 sites honeycomb lattice (64 spins or more in elevated magnetization sectors) up to 1.5 billion(=10 9 ) basis states with symmetries, up to 4.5 billion without t-j models: 32 sites checkerboard with 2 holes 32 sites square lattice with 4 holes up to 2.8 billion basis states Fractional quantum hall effect different filling fractions ν, up to electrons up to 3.5 billion basis states Hubbard models 20 sites square lattice at half filling, 21 sites triangular lattice sites in ultracold atoms setting up to 160 billion basis states

45 Structure of an Exact Diagonalization code

46 Ingredients

47 Ingredients Hilbert space

48 Ingredients Hilbert space Basis represention, Lookup techniques

49 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries

50 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix

51 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk)

52 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free)

53 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation

54 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization

55 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization Lanczos type diagonalization (needs only v = H u operations)

56 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization Lanczos type diagonalization (needs only v = H u operations) More exotic eigensolver techniques, real oder imaginary-time propagation,

57 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization Lanczos type diagonalization (needs only v = H u operations) More exotic eigensolver techniques, real oder imaginary-time propagation, Observables

58 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization Lanczos type diagonalization (needs only v = H u operations) More exotic eigensolver techniques, real oder imaginary-time propagation, Observables Static quantities (multipoint correlation functions, correlation density matrices,...)

59 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization Lanczos type diagonalization (needs only v = H u operations) More exotic eigensolver techniques, real oder imaginary-time propagation, Observables Static quantities (multipoint correlation functions, correlation density matrices,...) Dynamic observables (spectral functions, density of states,...)

60 Ingredients Hilbert space Basis represention, Lookup techniques Symmetries Hamiltonian Matrix Sparse Matrix representation (memory/disk) Matrix recalculation on the fly (matrix-free) Linear Algebra : Eigensolver / Time propagation LAPACK full diagonalization Lanczos type diagonalization (needs only v = H u operations) More exotic eigensolver techniques, real oder imaginary-time propagation, Observables Static quantities (multipoint correlation functions, correlation density matrices,...) Dynamic observables (spectral functions, density of states,...) Real-time evolution

61 Hilbert Space

62 Basis representation States of the Hilbert space need to be represented in the computer. Choose a representation which makes it simple to act with the Hamiltonian or other operators on the states, and to localize a given state in the basis Simple example: ensemble of S=1/2 sites in binary coding [ ] 2 = 13 detection of up or down spin can be done with bit-test. transverse exchange can be performed by an XOR operation: S + S + S S + [ ] 2 XOR [ ] 2 = [ ] 2 initial config bit 1 at the two sites coupled final config For S=1, one bit is obviously not sufficient. Use ternary representation or simply occupy two bits to label the 3 states.

63 Basis representation For t-j models at low doping it is useful to factorize hole positions and spin configurations on the occupied sites. For Hubbard models one can factorize the Hilbert space in up and down electron configurations. For constrained models - such as dimer models - the efficient generation of all basis states requires some thought. One of the key challenges for a fast ED code is to find the index of the new configuration in the list of all configurations (index f in Hi,f). Let us look at the example of S=1/2 spins at fixed S z

64 Basis lookup procedures (Lin tables) One of the key problems for a fast ED code is to find the index of the new configuration in the list of all configurations (index f in Hi,f). [ ] 2 = But is 11 the index of this configuration in a list of all S z =3/2 states? no! Use Lin tables to map from binary number to index in list of allowed states: (generalization of this idea works for arbitrary number of additive quantum numbers) Two tables with 2 (N/2) [=sqrt(2 N )] entries, one for MSBs and one for LSBs [0 0] = X [0 1] = 0 [1 0] = 1 [1 1] = 2 MSB [0 0] = X [0 1] = 0 [1 0] = 1 [1 1] = 0 LSB Ind([ ]) = = 0 Ind([ ]) = = 1 Ind([ ]) = = 2 Ind([ ]) = = 3

65 Basis lookup procedures (Lin tables) Lookup can therefore be done with two direct memory reads. This is a time and memory efficient approach (at least in many interesting cases). An alternative procedure is to build a hash list [const access time] or to perform a binary search [log access time]. This is all you require to write to code the Hamiltonian for a S=1/2 system with Sz conservation, but without spatial symmetries. This becomes somewhat more involved when using spatial symmetries...

66 Symmetries H = i,j J xy i,j (Sx i S x j + S y i Sy j )+J z i,js z i S z j

67 Symmetries Consider a XXZ spin model on a lattice. What are the symmetries of the problem? H = i,j J xy i,j (Sx i S x j + S y i Sy j )+J z i,js z i S z j

68 Symmetries Consider a XXZ spin model on a lattice. What are the symmetries of the problem? H = i,j J xy i,j (Sx i S x j + S y i Sy j )+J z i,js z i S z j The Hamiltonian conserves total S z, we can therefore work within a given S z sector This easily implemented while constructing the basis, as we discussed before.

69 Symmetries Consider a XXZ spin model on a lattice. What are the symmetries of the problem? H = i,j J xy i,j (Sx i S x j + S y i Sy j )+J z i,js z i S z j The Hamiltonian conserves total S z, we can therefore work within a given S z sector This easily implemented while constructing the basis, as we discussed before. The Hamiltonian is invariant under the space group, typically a few hundred elements. (in many cases = Translations x Pointgroup). Needs some technology to implement...

70 Symmetries Consider a XXZ spin model on a lattice. What are the symmetries of the problem? H = i,j J xy i,j (Sx i S x j + S y i Sy j )+J z i,js z i S z j The Hamiltonian conserves total S z, we can therefore work within a given S z sector This easily implemented while constructing the basis, as we discussed before. The Hamiltonian is invariant under the space group, typically a few hundred elements. (in many cases = Translations x Pointgroup). Needs some technology to implement... At the Heisenberg point, the total spin is also conserved. It is however very difficult to combine the SU(2) symmetry with the lattice symmetries in a computationally useful way (non-sparse and computationally expensive matrices).

71 Symmetries Consider a XXZ spin model on a lattice. What are the symmetries of the problem? H = i,j J xy i,j (Sx i S x j + S y i Sy j )+J z i,js z i S z j The Hamiltonian conserves total S z, we can therefore work within a given S z sector This easily implemented while constructing the basis, as we discussed before. The Hamiltonian is invariant under the space group, typically a few hundred elements. (in many cases = Translations x Pointgroup). Needs some technology to implement... At the Heisenberg point, the total spin is also conserved. It is however very difficult to combine the SU(2) symmetry with the lattice symmetries in a computationally useful way (non-sparse and computationally expensive matrices). At S z =0 one can use the spin-flip (particle-hole) symmetry which distinguishes even and odd spin sectors at the Heisenberg point. Simple to implement.

72 Spatial Symmetries Spatial symmetries are important for reduction of Hilbert space Symmetry resolved eigenstates teach a lot about the physics at work, dispersion of excitations, symmetry breaking tendencies, topological degeneracy, (more on that in later lectures) sites square lattice T PG =40 x 4 elements Icosidodecahedron (30 vertices) Ih:120 elements

73 Spatial Symmetries Symmetries are sometimes not easily visible, use graph theoretical tools to determine symmetry group [nauty, grape]. In an ED code a spatial symmetry operation is a site permutation operation. (could become more complicated with spin-orbit interactions and multiorbital sites) 40 sites square lattice T PG =40 x 4 elements Icosidodecahedron (30 vertices) Ih:120 elements

74 Spatial Symmetries: Building the basis Build a list of all allowed states satisfying the diagonal constraints, like particle number, total S z,... for each state we apply all symmetry operations and keep the state as a representative if it has the smallest integer representation among all generated states in the orbit. Example: 4 site ring with cyclic translation T, S z =3/2 sector T 0 ([ ]) [ ] T 1 ([ ]) [ ] T 2 ([ ]) [ ] T 3 ([ ]) [ ] keep state T 0 ([ ]) [ ] T 1 ([ ]) [ ] T 2 ([ ]) [ ] T 3 ([ ]) [ ] discard state...

75 Spatial Symmetries: Building the basis r = 1 N χ(g) g(r) G N = g G g G,g(r)=r χ(r)

76 Spatial Symmetries: Building the basis For one-dimensional representations χ of the spatial symmetry group: 1 Bloch state r = N χ(g) g(r) G Norm of the state is given as: N = The norm (and therefore the state itself) can vanish if it has a nontrivial stabilizer combined with a nontrivial representation χ. g G g G,g(r)=r χ(r)

77 Spatial Symmetries: Building the basis For one-dimensional representations χ of the spatial symmetry group: 1 Bloch state r = N χ(g) g(r) G Norm of the state is given as: N = The norm (and therefore the state itself) can vanish if it has a nontrivial stabilizer combined with a nontrivial representation χ. Example: 4 site S=1/2 ring with cyclic translations: g G g G,g(r)=r χ(r)

78 Spatial Symmetries: Building the basis For one-dimensional representations χ of the spatial symmetry group: 1 Bloch state r = N χ(g) g(r) G Norm of the state is given as: N = The norm (and therefore the state itself) can vanish if it has a nontrivial stabilizer combined with a nontrivial representation χ. Example: 4 site S=1/2 ring with cyclic translations: g G g G,g(r)=r χ(r) S z =2 S z =1 S z =0 K = , N = , N = , N = , N =1

79 Spatial Symmetries: Building the basis For one-dimensional representations χ of the spatial symmetry group: 1 Bloch state r = N χ(g) g(r) G Norm of the state is given as: N = The norm (and therefore the state itself) can vanish if it has a nontrivial stabilizer combined with a nontrivial representation χ. Example: 4 site S=1/2 ring with cyclic translations: g G g G,g(r)=r χ(r) S z =2 S z =1 S z =0 K = , N = , N = , N = , N =1 K = ±π/ , N = , N =1

80 Spatial Symmetries: Building the basis For one-dimensional representations χ of the spatial symmetry group: 1 Bloch state r = N χ(g) g(r) G Norm of the state is given as: N = The norm (and therefore the state itself) can vanish if it has a nontrivial stabilizer combined with a nontrivial representation χ. Example: 4 site S=1/2 ring with cyclic translations: g G g G,g(r)=r χ(r) K =0 K = ±π/2 K = π S z = , N =2 S z = , N = , N = , N =1 S z = , N = , N = , N = , N = , N =1

81 Spatial Symmetries: Building the basis For one-dimensional representations χ of the spatial symmetry group: 1 Bloch state r = N χ(g) g(r) G Norm of the state is given as: N = The norm (and therefore the state itself) can vanish if it has a nontrivial stabilizer combined with a nontrivial representation χ. Example: 4 site S=1/2 ring with cyclic translations: g G g G,g(r)=r χ(r) K =0 K = ±π/2 K = π 2 4 =16 S z = , N =2 1+1 S z =1 S z = , N = , N = , N = , N = , N = , N = , N = , N =

82 The Hamiltonian Matrix

83 The Hamiltonian Matrix Now that we have a list of representatives and their norms, can we calculate the matrix elements of the Hamiltonian? s H r =? Let us look at an elementary, non-branching term in the Hamiltonian: h α r = h α (r) s We can now calculate the matrix element the Bloch states: s h α r without double expanding s h α r = N s N r χ(g )h α (r) key algorithmic problem: given a possibly non-representative, how do we find the associated representative s, as well as a symmetry element g relating s to s? s

84 The Hamiltonian Matrix key algorithmic problem: given a possibly non-representative, how do we find the associated representative s, as well as a symmetry element g relating s to s? Brute force: loop over all symmetry operations applied on and retain s and. This is however often not efficient (many hundred symmetries). g Prepare a lookup list, relating each allowed configuration with the index of its representative, and also the associated group element linking the two. Gives fast implementation, but needs a list of the size of the non spatiallysymmetrized Hilbert space. For specific lattices and models (Hubbard models) clever tricks exist which factorize the symmetry group into a sublattice conserving subgroup times a sublattice exchange. They give s fast, then a hash or binary search is needed to locate in the list of representatives in order to get its index. s s s

85 Hamiltonian Matrix Storage Different possibilities exist: Store hamiltonian matrix elements in memory in a sparse matrix format Fast matrix vector multiplies, but obviously limited by available memory. Store hamiltonian matrix elements on disk in a sparse matrix format. In principle possible due to the vast disk space available, but I/O speed is much slower than main memory access times. Difficult to parallelize. Recalculate the hamiltonian matrix elements in each iterations on the fly. Needed for the cutting edge simulations, where the whole memory is used by the Lanczos vectors. Can be parallelized on most architectures.

86 Thank you!