4 Brute-force methods Flashcards
(22 cards)
What is the general idea behind brute-force methods in computational quantum physics?
- Brute-force methods aim to calculate properties of model systems.
- They are used for finding the ground state (or a few lowest states) and for time evolution of an initial state.
- For system size N, the Hilbert space dimension is 2^N, making the Hamiltonian a 2^N x 2^N matrix. This limits applicability to small systems, typically in one dimension. They are useful for benchmarking more advanced methods. They sometimes remain the only available method. Results must be interpreted carefully in relation to the thermodynamic limit.
What is exact diagonalization (ED), and what are its limitations?
Exact diagonalization fully diagonalizes the Hamiltonian matrix. If the full spectrum is needed, systems with N ~ 20 can be handled. If only the ground state is needed, iterative solvers (especially the Lanczos algorithm) can be used for N ~ 40. ED provides very accurate results but is limited to small systems.
What is the power method, and how does it motivate the Lanczos algorithm?
The power method starts from a random vector v = sum_n c_n |n>, where H|n> = lambda_n |n>. Applying H repeatedly yields vectors v_k = H^k v = sum_n c_n lambda_n^k |n>. For large M, v_M converges to the eigenvector with the largest eigenvalue (assuming c_n0 ≠ 0). This convergence is slow; the Lanczos algorithm optimizes the process.
What is the Krylov space in the Lanczos algorithm?
The Krylov space K_M is the span of normalized vectors: { v/||v||, Hv/||Hv||, H^2v/||H^2v||, …, H^M v/||H^M v|| }.
What is the Gram-Schmidt orthogonalization process in the Lanczos algorithm?
Basis vectors r_n are constructed iteratively with orthogonalization. The formula includes r_0 = v/||v||, followed by β_{n+1} r_{n+1} = H r_n - α_n r_n - β_n r_{n-1}. The α_n and β_n are computed as projections. The resulting r_n vectors are orthonormal.
What is the structure of the tridiagonal matrix H^{(M)} in the Lanczos algorithm?
The Hamiltonian in the Lanczos basis is a tridiagonal matrix with α_n on the diagonal and β_n on the off-diagonals. Its extremal eigenvalues converge quickly. The matrix is (M+1)x(M+1), easy to diagonalize.
What are the computational advantages of the Lanczos algorithm?
Memory: only 3 vectors of size 2^N need to be stored. Full Hamiltonian matrix is not required; only the matrix-vector product H v is needed. Efficient matrix-vector multiplication possible for spin Hamiltonians. Eigenvectors of H^{(M)} can be transformed to the original basis via a second Lanczos run. Ghost states can appear due to round-off errors.
Where can efficient Lanczos algorithm implementations be found?
Open-source packages: EIGEN library (C++), scipy.linalg (Python).
How can a spin Hamiltonian be implemented using tensor products?
Operators are applied via tensor products with identities on other sites. Example for an Ising term: S^z_i S^z_{i+1} = 1⊗…⊗1⊗S^z⊗S^z⊗1⊗…
How are spin-1/2 basis states represented and manipulated?
Basis states are represented as N-bit strings, integers from 0 to 2^N-1. Each bit represents a spin orientation (0 = down, 1 = up). Example for N=3: |↓↓↓> = 000_2, |↑↑↑> = 111_2, etc.
How can diagonal terms be computed efficiently in the Heisenberg model?
Use XOR between state s and right-shifted state s̃. Aligned spins result in bit 0, antialigned in bit 1. Counting bits in s̃ gives the diagonal matrix value.
How can off-diagonal terms like S^+i S^-{i+1} + h.c. be implemented?
Use bitwise operations: if a bit is set at position r, flip neighboring spins via XOR: s_new = s XOR (3 «_space;r). Matrix element between s and s_new is J/2.
How can symmetries like parity be used to simplify Hamiltonians?
Symmetries allow block-diagonalization of the Hamiltonian. Example: TFIM parity operator P = product of sigma^x_i. Commutation [P, H] = 0. Eigenvalues of P are ±1. Matrix elements between different parity sectors vanish.
How can symmetry eigenspaces be constructed and used in implementation?
Find orbits of basis states under symmetry operator S. Apply S repeatedly until the original state is reached. Construct eigenfunctions from orbits. Use efficient lookup tables to work within symmetry sectors.
What is the goal of time evolution in computational quantum physics?
Solve the time-dependent Schrödinger equation: i ħ ∂/∂t |ψ(t)> = H(t) |ψ(t)>, for both static and time-dependent Hamiltonians.
What is the exact solution for time-independent Hamiltonians?
Exact solution: |ψ(t+Δt)> = exp(-i H Δt / ħ) |ψ(t)>. The matrix exponential is dense and computationally expensive.
What is the Trotter-Suzuki decomposition?
Split H = sum h_k. First-order: product of exponentials with error O(Δt^2). Second-order: symmetric product with error O(Δt^3).
How is the Trotter-Suzuki decomposition applied to the TFIM?
Split H into Ising term and local field. Ising term is diagonal in z-basis. Local field is site-wise exponentiated using 2x2 matrices.
How is the Trotter-Suzuki decomposition applied to the XXZ model?
Split H into even and odd terms. Each contains commuting terms, which can be individually exponentiated. Results in block-diagonal 4x4 matrices.
What is imaginary-time evolution, and how does it find ground states?
Imaginary-time evolution replaces i t → τ: |ψ(τ)> = exp(-τ H) |ψ(0)>. Non-unitary, changes norm. Converges to ground state as excited state contributions decay exponentially.
What are the challenges of time evolution with time-dependent Hamiltonians?
For time-dependent H(t), the evolution operator requires time-ordering: U(t, t’) = T_s exp(-i/ħ ∫ H(s) ds). Cannot simply use exp(-i H(t) Δt / ħ) if H(t1), H(t2) do not commute.
What is the Magnus expansion?
Approximate U(Δt) = exp(-i/ħ Δt H̄_t), with H̄_t = H̄^{(1)} + H̄^{(2)} + … . First term is the average Hamiltonian over the interval. Higher terms involve commutators of H at different times.
