IR Basis

class sparse_ir.FiniteTempBasis(statistics: str, beta: float, wmax: float, eps: float = np.float64(2.220446049250313e-16), sve_result: SVEResult | None = None, max_size: int = -1)

Intermediate representation (IR) basis for given temperature.

For a continuation kernel from real frequencies, ω ∈ [-ωmax, ωmax], to imaginary time, τ ∈ [0, beta], this class stores the truncated singular value expansion or IR basis:

\[K(\tau, \omega) \approx \sum_{l=0}^{L-1} U_l(\tau) S_l V_l(\omega),\]

where U are the IR basis functions on the imaginary-time axis, stored in u, S are the singular values, stored in s, and V are the IR basis functions on the real-frequency axis, stored in V. The IR basis functions in Matsubara frequency are stored in uhat.

Example

The following example code assumes the spectral function is a single pole at ω = 2.5:

# Compute IR basis for fermions and β = 10, W <= 4.2
import sparse_ir
basis = sparse_ir.FiniteTempBasis(statistics='F', beta=10, wmax=4.2)

# Assume spectrum is a single pole at ω = 2.5, compute G(iw)
# on the first few Matsubara frequencies
gl = basis.s * basis.v(2.5)
giw = gl @ basis.uhat([1, 3, 5, 7])

property accuracy: Overall accuracy bound.

property beta: Inverse temperature

default_matsubara_sampling_points(npoints=None, positive_only=False): Get default Matsubara sampling points.

default_omega_sampling_points(npoints=None)

Return default sampling points in imaginary time.

Parameters:

npoints (int) –

Minimum number of sampling points to return.

Added in version 1.1.

default_tau_sampling_points(npoints=None): Get default tau sampling points.

property kernel: The kernel used to generate the basis.

property lambda_: Basis cutoff parameter, Λ = β * wmax

rescale(new_beta)

Return a basis for different temperature.

Uses the same kernel with the same eps, but a different temperature. Note that this implies a different UV cutoff wmax, since lambda_ == beta * wmax stays constant.

property s: Vector of singular values of the continuation kernel

property shape: Shape of the basis function set

property significance: Relative significance of basis functions.

property size: Number of basis functions / singular values

property statistics: Quantum statistic (‘F’ for fermionic, ‘B’ for bosonic)

property sve_result: The singular value expansion result.

property u

Basis functions on the imaginary time axis.

Set of IR basis functions on the imaginary time (tau) axis, where tau is a real number between zero and beta. To get the l-th basis function at imaginary time tau of some basis, use:

ultau = basis.u[l](tau) # l-th basis function at time tau

Note that u supports vectorization both over l and tau.

property uhat

Basis functions on the reduced Matsubara frequency (wn) axis.

Set of IR basis functions reduced Matsubara frequency (wn) axis, where wn is an integer. These are related to u by the following Fourier transform:

û(n) = ∫₀^β dτ exp(iπnτ/β) u(τ)

To get the l-th basis function at some reduced frequency wn of some basis, use:

uln = basis.uhat[l](wn) # l-th basis function at freq wn

Note

Instead of the value of the Matsubara frequency, these functions expect integers corresponding to the prefactor of pi over beta. For example, the first few positive fermionic frequencies would be specified as [1, 3, 5, 7], and the first bosonic frequencies are [0, 2, 4, 6]. This is also distinct to an index!

property v

Basis functions on the real frequency axis.

Set of IR basis functions on the real frequency (omega) axis, where omega is a real number of magnitude less than wmax. To get the l-th basis function at real frequency omega of some basis basis, use:

ulomega = basis.v[l](omega)    # l-th basis function at freq. omega

Note that v supports vectorization both over l and omega. In particular, omitting the subscript yields a vector with all basis functions, evaluated at that position:

basis.v(omega) == [basis.v[l](omega) for l in range(basis.size)]

Similarly, supplying a vector of omega points yields a matrix A, where A[l,n] corresponds to the l-th basis function evaluated at omega[n]:

omega = [0.5, 1.0]
basis.v(omega) == \
    [[basis.v[l](t) for t in omega] for l in range(basis.size)]

property wmax: Real frequency cutoff

Piecewise polynomials

class sparse_ir.poly.PiecewiseLegendrePoly(funcs: FunctionSet, xmin: float, xmax: float)

Piecewise Legendre polynomial.

Models a function on the interval [-1, 1] as a set of segments on the intervals S[i] = [a[i], a[i+1]], where on each interval the function is expanded in scaled Legendre polynomials.

__call__(x): Evaluate basis functions at given points.

Note

Additional polynomial classes are currently being refactored. Please refer to the concrete polynomial classes above for the current API.