[論文レビュー] Continuous unitary transformations using tensor network representations access the full many-body localized spectrum

Qiyu Liu, Jan-Niklas Herre+2 more|arXiv (Cornell University)|Mar 13, 2026

Quantum many-body systems被引用数 0

ひとこと要約

We introduce variational continuous unitary transformations (VCUTs) that couple Wegner-Wilson flow with tensor networks to approximate the full spectrum of disordered spin chains across the ergodic–MBL crossover, yielding access to the diagonalizing unitary and LIOMs for system sizes up to L=48.

ABSTRACT

We develop variational continuous unitary transformations (VCUTs), which integrate Wegner-Wilson flow equations with tensor network techniques to approximately diagonalize many-body localized (MBL) Hamiltonians. The diagonalizing unitary is represented as a matrix product operator whose bond dimension controls the accuracy. For the disordered Heisenberg chain, VCUTs accurately reproduces the full spectrum across the ergodic-to-MBL crossover at small system sizes and scales to $L = 48$ sites. Beyond eigenenergies, the method can track the spatial entanglement structure of the diagonalizing unitary $U(l)$ at each flow step, enabling identification of local integrals of motion deep in the MBL phase.

研究の動機と目的

• Motivate and study many-body localization (MBL) with a focus on the fully many-body localized (fMBL) phase characterized by LIOMs (l-bits).
• Develop VCUTs that diagonalize MBL Hamiltonians by flowing to diagonal form using tensor-network representations.
• Provide access to the full spectrum and to the diagonalizing unitary U, enabling analysis of LIOMs and operator spreading.
• Benchmark VCUTs against exact diagonalization (ED) and Tensor Flow Equations (TFE) across the ergodic-to-MBL crossover.
• Demonstrate scalability to larger system sizes and extract real-space structure of LIOMs and flow dynamics.

提案手法

• Represent the flowing Hamiltonian H(l) as a matrix product operator (MPO) and solve the Wegner flow dH/dl=[η(l),H(l)] using TDVP for MPS evolution.
• Vectorize the MPO to a vectorized Hamiltonian |H(l)⟩⟩ and treat the flow as a Schrödinger-like equation with a superoperator that acts as an SMPO.
• Compute η(l) as the Wegner generator [Hd(l), Hod(l)] and, for stability near degeneracies, also discuss Toda-Mielke variants with masking tensors.
• Construct the diagonalizing unitary U(l) by evolving the identity MPO in the same TDVP framework, yielding |U(l)⟩⟩.
• Obtain LIOMs via τ_j^z = U† σ_j^z U and analyze their locality and resonance structure through the flow.
• Control truncation and accuracy with a bond dimension D_H for H, using adaptive time stepping and convergence criteria based on the off-diagonal variance V(l).
• Assess accuracy by comparing VCUT results to ED and TFE for L ≤ 16 and demonstrate scaling to L = 48, reporting energy errors, variances, and dynamics.

実験結果

リサーチクエスチョン

• RQ1 Can VCUTs reproduce the full spectrum of disordered spin chains across the ergodic–MBL transition with controlled accuracy?
• RQ2 How does the diagonalizing unitary U(l) and the emergent LIOMs reflect the local resonance structure in the MBL phase?
• RQ3 What is the scaling of accuracy and computational cost with bond dimension and system size, and how does VCUTs perform relative to ED and TFE?
• RQ4 Can VCUTs recover real-time dynamics and autocorrelation functions using the LIOM representation?
• RQ5 How does the flow inform about the crossover diagnostics such as entanglement entropy statistics across disorder strengths?

主な発見

• VCUTs accurately reproduce the full spectrum in the localized regime and remain accurate up to L=48, exceeding exact diagonalization limits.
• For L=8 and L=16, VCUTs with D_H up to 48 outperform Tensor Flow Equations in median relative energy error across disorder strengths.
• The final flowed Hamiltonian achieves a final variance much smaller than the initial variance, demonstrating effective diagonalization even at L=48.
• The diagonalizing unitary U(l) shows spatially localized entanglement concentrating at bonds with small local disorder gradient Δh_i, identifying near-resonant bonds that structure LIOMs.
• LIOMs τ_j^z = U† σ_j^z U obtained from VCUTs yield accurate short- and long-time autocorrelations compared to ED, with minor long-time deviations attributed to finite bond-dimension truncation.
• Entanglement entropy statistics across the crossover (mean and sample-to-sample fluctuations) from VCUTs reproduce qualitative features of the MBL transition and agree with ED trends, improving with larger bond dimension.