Localization and Symmetry Breaking in the Quantum Quasiperiodic Ising Glass

Quasiperiodic modulation can prevent isolated quantum systems from equilibrating by localizing their degrees of freedom. In this article, we show that such systems can exhibit dynamically stable long-range orders forbidden in equilibrium. Specifically, we show that the interplay of symmetry breaking and localization in the quasiperiodic quantum Ising chain produces a quasiperiodic Ising glass stable at all energy densities. The glass order parameter vanishes with an essential singularity at the melting transition with no signatures in the equilibrium properties. The zero-temperature phase diagram is also surprisingly rich, consisting of paramagnetic, ferromagnetic, and quasiperiodically alternating ground-state phases with extended, localized, and critically delocalized low-energy excitations. The system exhibits an unusual quantum Ising transition whose properties are intermediate between those of the clean and infinite randomness Ising transitions. Many of these results follow from a geometric generalization of the Aubry-André duality that we develop. The quasiperiodic Ising glass may be realized in near-term quantum optical experiments.

Popular Summary

Most phases of matter—such as solids, liquids, and gases—are classified according to how the constituent particles (e.g., atoms and molecules) are arranged relative to each other. In everyday magnets (or ferromagnets), the intrinsic magnetism of the atoms all point in the same direction throughout the sample. Typically, at sufficiently high energy, the sample splits into a number of magnetic domains, each with its own magnetic orientation. The motion of the domain walls destroys the long-range correlation of the magnetization. In isolated disordered quantum systems, however, the domain walls might be localized. This could permit the long-range order in an initial state to remain frozen indefinitely. Very little is known about this “localization-protected order.” Here, we show that disorder is unnecessary to protect such order, and we present an analytic theory of how it can melt.

By introducing spatial quasiperiodic modulation to the canonical transverse-field Ising chain—a simple model of ferromagnetism based on a string of interacting atomic spins—we uncover a quasiperiodic Ising glass in which the domain walls are localized at all energies. This opens a route to near-term quantum optical experiments, as quasiperiodic modulation is much easier to produce than disorder. Using quasiperiodicity rather than disorder also permits the study of the melting transition, as the domain walls can delocalize at small modulation.

In passing, we uncover a new class of quantum phase transitions where the localization properties of the excitations and the long-range order in the ground state change simultaneously, and we develop a new technique for studying quasiperiodic chains.

Figure 1

Combined symmetry breaking and localization phase diagram of the quasiperiodic Ising model at $A_h=0$. The system has quasiperiodic Ising glass excited-state order at all energy densities in the red region, where all single-particle excitations are localized. The ground state is paramagnetic (PM) in the striped region; it breaks Ising symmetry ferromagnetically (FM) above the diagonal (dashed line) and with quasiperiodically alternating modulation (QPFM) below. The blue and purple shading indicates whether the low-energy spectrum is fully extended (blue) or critically delocalized (purple).

Figure 2

(a) The transverse-field Ising chain with spatially varying fields $h_i$ (blue) and bonds $J_{i+\frac{1}{2}}$ (orange). (b) The Jordan-Wigner transformation maps the spins ${\sigma }_i$ to Majorana fermions ${\gamma }_{2i}$, ${\gamma }_{2i+1}$ arranged in a “hopping” chain. (c) The two-dimensional hopping model Eq. (15) obtained by treating the phase ${\varphi }_h$ as quasimomentum $k_y$ and inverting the Fourier transformation. Each unit cell (of two sites) is pierced by uniform flux $Q$. The solid and dashed blue bonds have hopping strength $h$ and $A_h$ respectively, while the solid and dashed orange bonds correspond to $J$ and $A_J$ respectively.

Figure 3

The Ising glass order parameter $\mathcal{O}$ along three representative cuts in the phase diagram (inset). The definitions of ${\mathcal{O}}_{\mathrm{gs}}$ and ${\mathcal{O}}_{\mathrm{exc}}$ are given by Eqs. (24) and (35), respectively. The spatial averaging window is of size $w=5$, and we sample 50 eigenstates in the excited-state average. In the first panel, both the ground- and infinite-temperature order turn on at the same coupling as the excitations localize across the transition. In the second panel, the ground-state order is unchanged even as the excited-state order turns on because of the localization of the domain walls. In the third panel, the ground-state order turns on, but the excitations are extended so that the excited-state order remains zero.

Figure 4

The 2D model associated with the incommensurate TFIM at $A_h=0$, $J=0$ decouples into two interpenetrating layers (blue and orange) of a honeycomb lattice. Each hexagonal plaquette is pierced by flux $2Q$. We have adjusted the geometric embedding of the lattice points relative to Fig. 2 in order to emphasize the symmetry associated with rotation by $2\pi /3$. All parallel bonds carry the same coupling magnitude (though the hopping phase depends on the choice of gauge). The incommensurate TFIM at $J=0$ is embedded into a larger AAA family of models with couplings $A_1=A_J/2$, $A_2=A_J/2$, $A_3=h$ on bonds as shown.

Figure 5

Fermionic excitation spectra (upper row) and scaling exponent of low-energy IPR (lower row) on representative cuts in the phase diagram, indicated by black lines on the inset phase diagram. On each panel, vertical lines indicate the ground-state symmetry-breaking phase boundary and the diagonal ($A_J=J$) critically delocalized to extended boundary. (Upper row) Excitation spectra at size $L=1000$. The color value of each level indicates the extension of the corresponding state. The color value is given by $-\log \mathrm{IPR}/\log L$, which varies between 0 (localized, black) and 1 (extended, copper). (Lower row) The IPR exponent $\alpha$ is defined by the scaling $[\log \mathrm{IPR}]\sim -\alpha \log L$, where the mean $[·]$ is taken over the lowest twentieth of the excitations. Here, $\alpha$ varies between 0 (localized) and 1 (extended). To indicate the finite-size trends approaching the thermodynamic limit, we fit data from both $L=100$, 200, 350, 500, 750, 1000 (blue) and $L=350$, 500, 750, 1000 (green).

Figure 6

Spin-spin correlation function $⟨{\sigma }_{L/2}^x{\sigma }_{L/2+d}^x⟩$ as a function of the distance $d$ from the center of the chain in an $L=100$ site chain with open boundary conditions. The blue line is the ground state, while the green line shows a random excited state from the infinite-temperature ensemble. From left to right, the parameters are chosen so that the system is critically delocalized paramagnetic, extended ferromagnetic, and localized ferromagnetic, as indicated in the inset.