Theory of metastable states in many-body quantum systems

TL;DR

The paper develops a rigorous theory of metastable pure states in finite-dimensional quantum many-body systems by defining a local-energy-barrier notion that remains robust under nearby perturbations. It shows that short-range-entangled metastable states are essentially scars of a perturbed Hamiltonian and that local observables decay only on nonperturbatively long timescales via a local-diagonalization scheme built around Schrieffer–Wolff transformations, yielding lifetimes t* that scale at least as exp(R^α). The authors prove a sharp lower bound on false-vacuum lifetimes in d-dimensional Ising-like models, tFVD ≥ exp[(Δ'/ε')^d], match semiclassical predictions, and present explicit metastable constructions in the PXP model, generalized helical chains, and the 2D Ising model, illustrating broad connections to prethermalization, scars, and quantum nucleation. This framework provides a rigorous foundation for understanding slow dynamics, metastable basins, and robust nonthermal behavior in a wide class of quantum many-body systems, with potential implications for quantum simulation and controlled slow thermalization.

Abstract

We present a mathematical theory of metastable pure states in closed many-body quantum systems with finite-dimensional Hilbert space. Given a Hamiltonian, a pure state is defined to be metastable when all sufficiently local operators either stabilize the state, or raise its average energy. We prove that short-range entangled metastable states are necessarily eigenstates (scars) of a perturbatively close Hamiltonian. Given any metastable eigenstate of a Hamiltonian, in the presence of perturbations, we prove the presence of prethermal behavior: local correlation functions decay at a rate bounded by a time scale nonperturbatively long in the inverse metastability radius, rather than Fermi's Golden Rule. Inspired by this general theory, we prove that the lifetime of the false vacuum in certain $d$-dimensional quantum models grows at least as fast as $\exp(ε^{-d} )$, where $ε\rightarrow 0$ is the relative energy density of the false vacuum; this lower bound matches, for the first time, explicit calculations using quantum field theory. We identify metastable states at finite energy density in the PXP model, along with exponentially many metastable states in "helical" spin chains and the two-dimensional Ising model. Our inherently quantum formalism reveals precise connections between many problems, including prethermalization, robust quantum scars, and quantum nucleation theory, and applies to systems without known semiclassical and/or field theoretic limits.

Figures (9)

• Figure 1: A summary of our results. (a) Metastable states have a "local gap" -- all sufficiently small operators acting on a set of diameter $\le R$ raise the average energy by $\ge \Delta$. While our exposition mostly focuses on the all-zero metastable state (or, more generally, short-range entangled states) as depicted here, the metastability concept generalizes to any state. (b) Theorem \ref{['thm:H=H0+V']} proves that any short-range entangled metastable state of Hamiltonian $H$ is an eigenstate of a "nearby" Hamiltonian $H_0 = H-V$, where (heuristically) $V$ is perturbatively small in $R^{-1}$. (c) Theorem \ref{['thm:main']} proves that given the decomposition $H=H_0+V$, all metastable states exhibit prethermal slow dynamics: local correlation functions decay on the nonperturbatively slow time scale depicted. (d) Our metastability theory leads us to new insight into a broad range of many-body models, giving us rigorous guarantees on the lifetime of false vacua, as well as uncovering slow thermalization in helical spin chains. We also provide a new perspective on slow thermalization in the PXP model, and the two-dimensional Ising model.
• Figure 2: Proof ideas for (a) Theorem \ref{['thm:main']} and (b) Theorem \ref{['thm:Ising_t_d']}. (a) We perform Schrieffer-Wolff transformations to locally diagonalize the Hamiltonian as if $H_0$ was gapped, because metastability implies local operators feel an effective gap. (b) By keeping track of the operator volume instead of diameter, resonance happens at a much higher order $k\sim R^d$ instead of $k\sim R$.
• Figure 3: (a) Energy spectrum vs perturbation strength $\epsilon$ in the zero momentum, inversion even sector. The color indicates, for each energy eigenstate, the expectation value of the number of motifs $n_m=\sum_i \lvert012210\rangle\langle012210\rvert_{i,\dots, i+5}$. The arrow points at the metastable state $\lvert0122101221\rangle$. (b) Expectation value of the number of $0$s $n_0=\sum_i \lvert0\rangle\langle0\rvert_i$ as a function of time for different values of the perturbation strength $\epsilon$. The initial state is the metastable state $\lvert0122101221\rangle$. (c) Same as (b), but for the non-metastable initial state $\lvert0000000012\rangle$. All the results were obtained for $L=10$, $\mu_1=0.2$, $\mu_2=0.12$.
• Figure 4: Time evolution of the local observable $n_j=\lvert1\rangle\langle1\rvert_j+2\lvert2\rangle\langle2\rvert_j$ for two initial states with the same energy $E=3.2$: (a) a metastable state $\lvert0122100122100...\rangle$ and (b) a non-metastable state $\lvert00200200...00220020...\rangle$. Entanglement entropy $S_{j,j+1}$ at a cut between site $j$ and $j+1$ in the evolution from the same (c) metastable and (d) non-metastable states as in (a), (b) respectively. All the results were obtained for $L=30$, $\mu_1=0.2$, $\mu_2=0.12$, and maximal bond dimension $\chi=40$. (e) Half chain entanglement entropy for different values of the bond dimension, which demonstrates that the slow dynamics of the metastable state is not limited by the finite bond dimension.
• Figure 5: Metastability of state $\lvert\psi_0\rangle=\lvert0-\rangle$ in the PXP model with periodic boundaries from ED. (a) For a given $R$, the energy difference \ref{['eq:H0>Delta1']} is minimized to obtain $\Delta$, which shows the metastability range $R=7$. To this end, an $(R+1)$-site metastable Hamiltonian with suitable boundary conditions is diagonalized. The data is compared with more usual boundary conditions, where the periodic one only applies to odd $R$. (b) Overlap of the $\lvert0-\rangle$ state with the ground state and first excited state. Both overlaps decay exponentially with the system size $N$. (c) Low energy spectrum (dots) $E$ of $H$ in the constrained subspace $\mathcal{H}_{\rm PXP}$, with the ground state energy set to $0$. For each $N$, the energy levels in the $\mathcal{I} =+1, k=0$ sector ($\mathcal{I} =-1, k=\pi$ sector) are plotted on the left (right) of the dashed line. The energy of $\lvert0-\rangle$ (pink solid line) is deep in a continuum of excited states at large $N$. The color of each eigenstate represents its overlap with $\lvert0-\rangle$, showing that $\lvert0-\rangle$ is approximately supported on a minority of eigenstates. (d) An alternative view of the eigenstate overlap of $|0-\rangle$, showing that the state is predominantly supported on a handful of eigenstates of approximately equal energy spacing (circle markers for $\mathcal{I} =+1, k=0$ sector, square markers for $\mathcal{I} =-1, k=\pi$, no line). For comparison, we have shown (solid line) what the spectrum would look like if the state was made of a Poisson-distributed collection of non-interacting "quasiparticles" $\sum \alpha^{n/2} \mathrm{e}^{-\alpha n/2}(n!)^{-1/2} \cdot |n\rangle$, where $|n\rangle$ denotes the state with $n$ quasiparticles at momentum $k\approx \pi$. The similarity in the spectra is consistent with a quasiparticle interpretation for the metastable state. (e) We observe slow thermalization and persistent oscillations in a quench from the metastable state, as evidenced by $Z_e = \langle 2N^{-1}\sum_{i \; \text{even}} Z_i\rangle$, for example. Data for $N=24,26,28,30$ are plotted (they almost exactly overlap), with color intensity increasing with $N$.

Theorems & Definitions (45)

• Theorem 1: Lieb-Robinson Theorem
• Definition 2
• Proposition 3
• proof
• Proposition 4
• proof
• Theorem 5
• Proposition 6
• proof
• Definition 7