thermal_mpo¶

Approximate the thermal density matrix \(\rho(\beta) = e^{-\beta H}\) as a NormalMPO via a truncated Taylor series.

thermal_mpo ¶

thermal_mpo(
    H: MPO,
    beta: float,
    order: int,
    spc: Index,
    trunc: Optional[dict] = None,
    coeff_thresh: float = 1e-15,
) -> NormalMPO

Approximate the thermal density matrix via a Taylor expansion.

Computes

\[\rho(\beta) = e^{-\beta H} \approx \sum_{n=0}^{N} \frac{(-\beta)^n}{n!} H^n\]

where \(H^0 = I\) (identity). The computation is performed in MPO arithmetic, keeping the underlying MPO normalized at each step (via explicit compact() calls) to prevent numerical blow-up.

Parameters:

Name	Type	Description	Default
`H`	`MPO`	Hamiltonian as an MPO (will not be modified).	required
`beta`	`float`	Inverse temperature \(\beta \geq 0\).	required
`order`	`int`	Maximum truncation order \(N\) of the Taylor series. The loop terminates early once the actual contribution of the \(n\)-th term — measured as \(\|\beta^n / n!\| \times \\|H^n\\|_F\) — drops below `coeff_thresh`.	required
`spc`	`Index`	Physical space index used to build the identity MPO (\(H^0 = I\)).	required
`trunc`	`Optional[dict]`	Truncation options forwarded to each `compact()` call during the Taylor series accumulation. Defaults to `{'thresh': 1e-14}`.	`None`
`coeff_thresh`	`float`	Early-stopping threshold on the actual contribution magnitude of the \(n\)-th Taylor term, \(\|\beta^n / n!\| \times \\|H^n\\|_F\). Once this falls below `coeff_thresh` the remaining contributions are negligible and the loop exits. Defaults to `1e-15`. Note that checking the bare coefficient \(\|\beta^n / n!\|\) alone is insufficient for Hamiltonians with large operator norm, because \(\\|H^n\\|_F\) can be much larger than 1.	`1e-15`

Returns:

Type	Description
`NormalMPO`	Approximation of \(e^{-\beta H}\) with physical scale stored in `_scale` (which equals \(Z = \operatorname{Tr}[\rho]\) up to the MPO norm).

Notes

compact() is called after every addition and every power step to keep bond dimensions under control. Bond arrow directions are restored by capcup inside compact(), so the result is always in a well-defined state for subsequent trace() or observe() calls.

Notes¶

Algorithm. The Taylor expansion

\[\rho(\beta) = \sum_{n=0}^{N} \frac{(-\beta)^n}{n!} H^n\]

is accumulated iteratively. At each step the running sum rho and the current power H^n are compacted — bond dimension is truncated and the norm is folded into _scale — preventing exponential growth of virtual bonds.

Convergence. The expansion converges when \(|\beta \lambda_{\max}| \ll N\), where \(\lambda_{\max}\) is the spectral radius of \(H\). In practice order=20–25 is sufficient for \(\beta \|H\| \lesssim 4\).

Partition function. After the call, rho.trace() returns \(Z(\beta) = \operatorname{Tr}[e^{-\beta H}]\).

Thermal expectation values. Pass rho to observe(rho, O) to compute \(\langle O \rangle_\beta = \operatorname{Tr}[\rho O] / \operatorname{Tr}[\rho]\).

thermal_mpo¶

thermal_mpo ¶

Notes¶

See Also¶