Fragmentation and Emergent Integrable Transport in the Weakly Tilted Ising Chain

We investigate emergent quantum dynamics of the tilted Ising chain in the regime of a weak transverse field. Within the leading order perturbation theory, the Hilbert space is fragmented into exponentially many decoupled sectors. We find that the sector made of isolated magnons is integrable with dynamics being governed by a constrained version of the $XXZ$ spin Hamiltonian. As a consequence, when initiated in this sector, the Ising chain exhibits ballistic transport on unexpectedly long timescales. We quantitatively describe its rich phenomenology employing exact integrable techniques such as generalized hydrodynamics. Finally, we initiate studies of integrability-breaking magnon clusters whose leading-order transport is activated by scattering with surrounding isolated magnons.

Figure 1

Magnetization profiles $⟨S_j^z⟩$ in the Ising chain at $J=1$ and $h_{\perp }=0.2$ initialized by joining the ferromagnetic and Neel states. (a) For $h_{\parallel }=-0.7$ we observe ballistic transport with a characteristic light cone. (b) For $h_{\parallel }=0.7$ we find strong suppression of spin transport. TEBD simulations are done for a chain of length $L=80$. The peculiar transport is captured by the integrable dynamics governed by the Hamiltonian [Eq. (3)] which emerges for a weak transverse field $h_{\perp }\ll h_{\parallel },J$. The validity of the phase diagram is within this limit; see main text for discussion.

Figure 2

(a) The magnetization profile of a chain of length $L=80$ evolved with TEBD from $|\text{Neel}⟩\otimes |\text{ferro}⟩$ at large time (measured in the AB units $[{\mathcal{J}}^{-1}]$) $t_{\mathrm{AB}}=20$ approaches the GHD prediction. For the Ising model we choose parameters $h_{\perp }=0.5$, $h_{\parallel }=6$, and $J=1$, corresponding to $\mathrm{\Delta }=0.5$ in the AB model. In the inset, we show the collapse of the AB simulations on the GHD analytical prediction. (b) To highlight magnetization jumps in $|\mathrm{\Delta }|>1$ (precisely, $\mathrm{\Delta }=1.5,\mathcal{J}=-1$), we consider the partitioning from $|{\mathrm{GS}}_{⟨Z⟩}⟩\otimes |\text{ferro}⟩$ with $|{\mathrm{GS}}_{⟨Z⟩}⟩$ the ground state of the AB model in the sector at fixed magnetization $⟨Z⟩$ for a chain of length $L=120$. For the left-side magnetization being below (top) and above (bottom) the half filling dotted line, the profile exhibits qualitatively different behavior.

Figure 3

(a) The level statistics analysis shows compatibility with the Gaussian orthongonal ensemble [49], suggesting that the sector with a two-magnon cluster is not integrable. The distribution function $P(r)$ is defined in the Supplemental Material [61] and computed with the exact diagonalization package quspin [83, 84]. (b) A two-magnon cluster, initially at the center of a chain of length $L=80$ bipartitioned into antiferromagnetic and ferromagnetic halves, can move to the left by virtue of the magnon-assisted hopping. We track its position by measuring the projector on two consecutive flipped spins ${\mathcal{P}}_{i,i+1}^{\downarrow \downarrow }$. (c) At large times, the position $⟨x⟩$ and variance $⟨x^2⟩-⟨x{⟩}^2$ of the cluster evolve linearly in time, as described in the main text. The TEBD simulations for (b) and (c) are done with the Ising Hamiltonian with parameters corresponding to $\mathrm{\Delta }=0.5$ and $\mathcal{J}=-1$.