XTRG¶

Alice's XTRG algorithm computes finite-temperature thermodynamic properties of a Hamiltonian MPO via exponential cooling. The thermal density matrix is repeatedly squared — ρ(β_{n+1}) ≈ compress(ρ_n ⊗ ρ_n) — using a sweep-based variational MPO-MPO compression kernel.

API¶

Symbol	Description
Options	Run options: scheme, τ₀, cooling steps, bond dimension, caching, artifacts
Summary	Thermodynamic history: log Z, free energy, internal energy, specific heat, entropy
Artifact	Density-matrix snapshot `ρ(β)` at one cooling step
run	Top-level entry point

Usage Pattern¶

from alice.network import build_interaction, build_hamiltonian
from alice.network.thermal import thermal_mpo
from alice.algorithm import xtrg

interactions, spc, geo = build_interaction(cfg)
H = build_hamiltonian(interactions, geo.L, spc)

opts = xtrg.Options(scheme='2s', n_steps=20, max_bond=64)
rho0 = thermal_mpo(H, opts.tau_0, opts.taylor_order, spc)
summary, artifact = xtrg.run(xtrg.Artifact(rho=rho0, beta=opts.tau_0, step=0), opts)

for beta, f in zip(summary.betas, summary.free_energies):
    print(f"β = {beta:.4f},  f = {f:.6f}")

run() takes the starting density matrix as an Artifact (rho, beta, step). Resuming an interrupted run is just calling run() again with the Artifact loaded from xtrg.ckpt:

resumed = xtrg.Artifact.load(ckpt_dir / 'xtrg.ckpt')
summary, artifact = xtrg.run(resumed, opts)  # thermal.ckpt auto-loaded from opts.checkpoint_dir

Continuing a finished run¶

A run that already finished continues the same way, starting from any archived artifacts/step_XX.ckpt and new options — here cooling from β = 2²⁰τ₀ to 2²⁵τ₀ at a larger bond dimension:

state = xtrg.Artifact.load(ckpt_dir / 'artifacts' / 'step_20.ckpt')
opts = xtrg.Options(tau_0=2 ** -12, n_steps=25, max_bond=512,
                    checkpoint_dir=str(ckpt_dir))
summary, artifact = xtrg.run(state, opts)

n_steps is the absolute step index to stop at (steps counted from τ₀), not a number of additional steps, and tau_0 must match the τ₀ the recovered history was built on.

Starting from a step before the end of that history is allowed, and is how a segment is re-cooled under different options: loading step_15.ckpt with n_steps=20 truncates the history back to step 15 (with a warning), then recomputes steps 16 … 20, overwriting both thermal.ckpt and the corresponding step_XX.ckpt archives.

Update Schemes¶

Name	Alias	Description
`'1s'`	`'1-site'`, `'one-site'`	1-site direct contraction; preserves bond dimension
`'2s'`	`'2-site'`, `'two-site'`	2-site SVD with truncation; drives bond growth
`'1sp'`	`'1-site-plus'`, `'one-site-plus'`	CBE: 1-site cost with 2-site-like bond flexibility

Temperature Grid¶

XTRG samples an exponentially spaced β grid starting from τ₀:

β_n = 2^n × τ₀,  n = 0, 1, …, n_steps

Thermodynamic observables (u, c_V, S) are derived from log Z using log-β finite differences, which give uniform O((ln 2)²) discretization error across the grid.