Ergodicity breaking in matrix-product-state effective Hamiltonians

Abstract

Thermalization and its breakdown in interacting quantum many-body systems are governed by mid-spectrum eigenstates, which are typically accessible only in small system sizes amenable to exact diagonalization. Here we demonstrate that the density-matrix renormalization group (DMRG) effective Hamiltonian, an object routinely used to variationally approximate ground states, encodes detailed information about the dynamics far from equilibrium. In the random-field XXZ spin chain, the spectrum of the effective Hamiltonian is shown to capture the transition from thermal to many-body localized regimes, including spatially resolved probes of ergodic bubbles. Furthermore, the same approach also captures weak ergodicity breaking associated with quantum many-body scars. Our results establish the DMRG effective Hamiltonian as a versatile spectral probe of quantum thermalization and its breakdown in large systems beyond exact diagonalization.

Figures (10)

• Figure 1: (a) The DMRG effective Hamiltonian $H_\mathrm{eff}$, supported on a subsystem of $M$ lattice sites. $H_\mathrm{eff}$ is constructed from the MPS left- and right-canonical tensors, $A_L$ and $A_R$, with maximum bond dimension $\chi$ and physical dimension $d$, and the MPO of the Hamiltonian, $\mathcal{H}$. (b) The low-lying spectrum of $H_\mathrm{eff}$ was shown to encode the conformal field theory (CFT) describing the critical point Chepiga2017Eberharter2023cocchiarella2025excitedstateslocaleffective. (c) Beyond low-lying excitations, in this work we show the full spectrum of $H_\mathrm{eff}$ captures ergodicity breaking phenomena, such as many-body localization and quantum many-body scars, which manifest as weakly-entangled nonthermal eigenstates.
• Figure 2: Many-body localization in the effective DMRG Hamiltonian for the random-field XXZ model, Eq. (\ref{['eq:Heisenberg']}). (a) The mean level spacing ratio $\langle r \rangle$ as a function of disorder strength $W$ with increasing bond dimension $\chi$. The GOE (Poisson) value is marked by a dashed (dotted) line. We average over 125--500 disorder realizations and ten central sites; error bars show the 95% confidence interval. (b) The spectral form factor (SFF), $K(\tau)$, against the rescaled time $\tau = t / t_H$, where $t_H$ is the Heisenberg time. The familiar dip-ramp-plateau structure is observed for small $W$ (dashed line shows the GOE prediction), while the dip is absent for sufficiently large disorder. (c) The average entanglement entropy $S_E$, normalized by the maximum entropy (see text), as a function of $W$. The inset shows raw data. (d) The standard deviation of $S_E$ between eigenstates in each disorder realization. The inset shows a finite-size scaling collapse of the data, along with calculated critical exponents $\nu$, $\zeta$; the critical disorder $W_c$ (marked by a dashed red line in the main panel); and their 95% confidence intervals. All data are for system size $N = 50$.
• Figure 3: Quantum many-body scarring in the effective DMRG Hamiltonian for the PXP model, Eq. (\ref{['eq:PXP']}). We target the eigenstate at $E_{\mathrm{target}}(n)=-1.331(N/2-n)$ via DMRG-S and use it to construct $H_{\mathrm{eff}}$. The kinetic constraint is encoded as an additional energy penalty term SOM. The main plot shows the overlaps between the eigenstates of $H_{\mathrm{eff}}$, $|\psi(A,v_i)\rangle$ and the $|\mathbb{Z}_2\rangle$ state for $n\in[4,10]$, while the inset shows the same data convolved with a Gaussian filter $e^{-(E-E_i)^2/\sigma^2}$ with $\sigma=0.1$. All results are for $N=40$ and $\chi=150$.
• Figure 4: (a) A rare low-disorder region forms an ergodic grain that threatens the stability of the MBL phase. The thermal grain can grow by thermalizing nearby degrees of freedom: this may arrest, or continue forever in a thermal avalanche. We simulate a rare region by setting $W = W_\text{rare} = 0.5$ for $n_\text{rare}$ sites. (b) Disorder-averaged bipartite entanglement entropy $\langle S_E \rangle$ in a system with a rare region of size $n_\text{rare} = 8$ (gray shading), with $H_\text{eff}$ formed at various offsets $\ell$ from the center. We contrast the case of $W=4$ with $W=12$, which are expected to be unstable and stable to avalanches, respectively. (c) Same as (b) but for the level spacing ratio $\langle r \rangle$. The Poisson (GOE) value is marked with a dashed (dotted) line. (d)-(e) The maximum of $\langle S_E \rangle$ with respect to position as a function of $L_\text{eff}$, for various $W$ and $n_\text{rare}$, for (d) the XXZ model $\Delta = 1$, and (e) an XY model with $\Delta = 0$. An increase in this quantity suggests susceptibility to an avalanche. [In all cases we use at least 500 disorder realizations.]
• Figure S1: (a)-(b) Bipartite entanglement entropy $S_E$ against energy $E$ for eigenstates of $H_\text{eff}$, for a single disorder realization at two values of $W$: (a) $W = 0.5$, and (b) $W = 5$. The data are colored according to the energy variance $\sigma^2_H$, with darker colors indicating a lower variance. We also show the marginal distributions of $S_E$ and $E$. In both cases, we pick site $\ell = 25$ and form $H_\text{eff}$ with $M = 1$, then restrict to the $m_z = 0$ sector. (c) $h_\ell$ for this disorder realization; see also Eq. \ref{['eq:single-realisation']}.