Nonthermal States Arising from Confinement in One and Two Dimensions

We show that confinement in the quantum Ising model leads to nonthermal eigenstates, in both continuum and lattice theories, in both one (1D) and two dimensions (2D). In the ordered phase, the presence of a confining longitudinal field leads to a profound restructuring of the excitation spectrum, with the low-energy two-particle continuum being replaced by discrete “meson” modes (linearly confined pairs of domain walls). These modes exist far into the spectrum and are atypical, in the sense that expectation values in the state with energy $E$ do not agree with the microcanonical (thermal) ensemble prediction. Single meson states persist above the two-meson threshold due to a surprising lack of hybridization with the ($n\ge 4$)-domain wall continuum, a result that survives into the thermodynamic limit and that can be understood from analytical calculations. The presence of such states is revealed in anomalous postquench dynamics, such as the lack of a light cone, the suppression of the growth of entanglement entropy, and the absence of thermalization for some initial states. The nonthermal states are confined to the ordered phase—the disordered (paramagnetic) phase exhibits typical thermalization patterns in both 1D and 2D in the absence of integrability.

Figure 1

(Upper panel) EEV spectrum of the spin operator $\sigma (0)$ as a function of energy $E$ for the 1D Ising field theory (2) with $m=1$, $g=0.1$, $R=35$. Arrows show the energies of the first 40 meson states [83]. The MCE result is shown within the continuum, with error bars denoting the standard deviation of results averaged over. (Inset) The average magnetization for the $n=11–15$ meson states compared to the MCE at the same average energy for a number of volumes $R$. (Lower panel) EEV spectrum of $\sum _j{\sigma }_j^z/N$ as a function of energy $E$ in the 1D lattice model (1) with $J=-1$, $h_x=-0.5$, $h_z=0.05$ for $N=40$ sites, computed with DMRG for open boundary conditions. The nonthermal states are mesonlike and confined to the vicinity of a boundary.

Figure 2

We plot the relative second order correction to the energy of the first 19 zero momentum mesons ($\delta E_2/E$) coming from mixing with zero and four domain wall states in the 1D Ising field theory (2) with $m=1$ and $g=0.1$. For comparison, we also plot the energy correction to the zero momentum spin flip excitation in the disordered phase. Note that the correction for the mesons is from ${10}^{-2}$ to ${10}^{-4}$ that of spin flip excitation.

Figure 3

(Upper panels) The time evolution of the connected correlation function, $|⟨{\sigma }_{i+y}(x,t){\sigma }_i(x,t)⟩-⟨{\sigma }_{i+y}(x,t)⟩⟨{\sigma }_i(x,t)⟩|$, following quenches in the 2D quantum Ising model (4) for $R=8$, $J_{\perp }=0$ to $-0.15$, and (left panels) ordered chains $m=1$ and (right panels) disordered chains $m=-1$. Both cases start from a $J_{\perp }=0$ ground state. Dynamics are computed via ChainAMPS [59]. (Lower panels) The time evolution of the entanglement entropy $S_E$ following the same quenches. Note the $y$ axis of the lower left panel has been increased by a factor of 100. A detailed discussion of the simulations is provided in the Supplemental Material [74].