Scarred ferromagnetic phase in the long-range transverse-field Ising model

Abstract

We report the existence of a large set of ferromagnetic scarred states in the one-dimensional transverse-field Ising model with long-range interactions, in a regime with no ferromagnetic phase at finite temperature. These scarred states are distributed over different spectral regions, surrounded by paramagnetic states. We show that simple initial conditions, consisting in a few small magnetic domains, selectively populate these scarred states. This leads to the appearance of a special dynamical phase, which we call scarred ferromagnetic phase. As a consequence, initial states with a small number of small magnetic domains evolve towards ferromagnetic equilibrium states, whereas initial states with larger domains or no magnetic structure relax to the expected thermal paramagnetic equilibrium state.

Figures (5)

• Figure 1: (a)-(f) Eigenvalues of the normalized total magnetization in $\mathcal{E}_n$ subspaces for different values of the power-law interaction, $\alpha$, $J=1$ and $h=0.1$, as a function of the energy of the subspace. System size is $N=22$. (g) Scaling of the number of scarred states with absolute values of the eigenvalues of the total magnetization greater than $0.2$ for $\alpha=0.6$ (red), $1.4$ (blue), $2.2$ (orange), $3$ (green), $5$ (purple) and $10$ (pink).
• Figure 2: (a) Local density of states for two initial states populating different regions of the energy spectrum. The initial states considered are $\ket{\psi_{1}}=\ket{\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow}_{x}$ (blue) and $\ket{\psi_{2}}=\ket{\uparrow\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\uparrow\uparrow\uparrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow}_{x}$ (red). (b) Distribution of the eigenvalues of the reduced total magnetization in each $\mathcal{E}_n$ subspace. (c) Time evolution of the normalized total magnetization for these two initial states. System size is $N=20$, and $\alpha=2.2$, $J=1$, $h=0.1$.
• Figure 3: Equilibrium value of the total magnetization for initial states with one ferromagnetic domain (top row) and two domains (bottom row), for different values of the magnetic field $h$, $\alpha=2.2$ and $J=1$. System size is $N=12,16,20$ (blue, orange, green). Errorbars represent the standard deviation with respect to the long-time average. Dashed horizontal lines indicate zero magnetization.
• Figure 4: Absolute expectation value of the intensive magnetization, $\hat{m}=2\hat{M}/N$, over the Hamiltonian eigenstates of opposite parity, $\bra{E_{n,-}}\hat{m}\ket{E_{n,+}}$, as a function of the gap $\Delta E_{nm}=|E_{n,+}-E_{m,-}|$. Results are averages over all possible combinations of Hamiltonian eigenstates. Red points correspond to expectation values in the case of scarred states, and red triangles correspond to normal states (see text). System parameters are $J=1$, $h=0.1$, $\alpha=2.2$ and $N=20$.
• Figure 5: (a)-(f) Eigenvalues of the normalized total magnetization in individual energy eigenspaces for different values of the power-law interaction, $\alpha$, $h$, and $J=1$ as a function of eigenspace energy. System size is $N=22$.