Continuous unitary transformations using tensor network representations access the full many-body localized spectrum

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.

Figures (7)

• Figure 1: Tensor network representation of the VCUTs method. (a) Vectorization of an MPO into an MPS by combining the bra and ket physical indices at each site into a single index of dimension $d^2$. (b) Construction of the Wegner generator $\eta(l) = [\mathcal{H}_\text{d}(l), \mathcal{H}_{\text{od}}(l)]$ as the commutator of the diagonal and off-diagonal parts of the Hamiltonian. (c) The local superoperator $\mathcal{U}^{j}(l)$ from Eq. \ref{['eq:superoperator']}.
• Figure 2: Mean of the median relative energy error with respect to ED as a function of disorder strength $W$ [panels (a) and (b)], and variance suppression as a function of system size $L$ [panel (c)]. (a) $L=8$: TFE (triangles), VCUTs with $D_H = 16$ (diamonds) and $D_H = 48$ (squares), averaged over 50 disorder realizations. (b) $L=16$: TFE (triangles) and VCUTs with $D_H = 48$ (squares), averaged over 30 disorder realizations. (c) Initial variance $V(0)$ (black circles, solid line) and final variance $V(l_{\mathrm{final}})$ (open markers, dashed lines) as a function of system size $L$ for disorder strengths $W=1, 2, 6$. For $L=8$, $D_H = 32$, averaged over 49 disorder realizations; for $L=16$, $D_H = 48$, averaged over 30 realizations; for $L=32$ and $48$, $D_H = 32$, a single realization is shown.
• Figure 3: Comparison between exact eigenenergies (blue lines) and energies obtained via VCUTs (red lines) for a system with $L=8$ and $W=6$. The left panel shows all 70 eigenstates in the zero magnetization sector. The right panel displays a magnified view of the energy levels near zero.
• Figure 4: Spatially resolved bond entropy $S^U$ of the diagonalizing unitary $U(l)$ during the VCUTs flow for $L=48$, $W=12$ (single disorder realization, $D_H=48$). Main panel:$S^U$ on a logarithmic color scale as a function of bond index (horizontal) and flow step (vertical). The red contour encloses the high-entropy core ($S^U \geq S^U_{\min}$, where the threshold is set adaptively from the minimum final $S^U$ among bonds with $\Delta h_i < 2$). Blue shading marks bonds where $\Delta h_i < 2$. Top panel: local disorder gradient $\Delta h_i = |h_i - h_{i+1}|$ at each bond; scatter points are colored red (below $\Delta h$ threshold), orange (above threshold but inside the entropy core), or blue (above threshold and outside the core). The entropy accumulates preferentially at low-$\Delta h$ bonds, revealing the resonance-driven spatial structure of the flow.
• Figure 5: Disorder-averaged dynamical autocorrelation $\langle C(t) \rangle$ for $L=8$ and three disorder strengths $W=1, 2, 6$, averaged over 50 realizations. Solid lines: ED (exact). Dashed lines with square markers: VCUTs ($D_H = 48$). Dotted lines with triangle markers: TFE. Different colors correspond to different disorder strengths. A moving average is applied for $t > 1000$ to reduce oscillations. The autocorrelation is computed from the LIOM $\tau^z_{L/2} = U^\dagger \sigma^z_{L/2} U$ located at the middle site.