Biao Lian

We study the phase diagram of a one-dimensional spin quantum breakdown model, which has an exponential U ( 1 ) symmetry with charge unit decaying as 2 − j with site position j . By exact diagonalization and density matrix renormalization group, we show that the model with spin S ≥ 2 exhibits an exponential U ( 1 ) spontaneous symmetry-breaking (SSB) phase dubbed a quantum breakdown condensate. It exhibits a bulk gap violating the Goldstone theorem and an edge mode only on the left edge if in open boundary condition. In a length L lattice, the condensate has O ( 2 L ) number of SSB ground states originating from the O ( 2 L ) number of exponential U ( 1 ) charge sectors, leading to a finite entropy density ln 2 . This enforces a first-order SSB phase transition into this phase, as observed numerically and verified in the large S limit on an exactly solvable Rokhsar-Kivelson line. The condensate has an SSB order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli (dyadic) map and thus is effectively random. Moreover, we show the condensate exhibits nondecaying local autocorrelations and does not have an off-diagonal long-range order. The quantum breakdown condensate thus behaves as a disorder-free quantum glass and is beyond the existing classifications of phases of matter.

Quantum Hall edge states in proximity to a superconductor (SC) usually acquire a nonquantized electron-to-hole conversion probability in transport, due to nonuniversal SC couplings and disorders. With counterpropagating modes, we show that the situation can be the opposite in the ν=2/3 fractional quantum Hall (FQH) edge states with SC proximity, where disordered SC couplings can reconstruct the edge states into an infinite set of stable phases with quantized electron-to-hole conversion probability along a long edge. Each phase is dominated by a disordered SC coupling that tunnels ±|q_{N}| Cooper pairs, which can take values |q_{N}|=1, 4, 15, etc. We predict that this gives rise to a quantized downstream resistance R_{d}=h/(2q_{N}^{2}e^{2}) in an FQH-SC junction, serving as a quantized electrical transport signature beyond the Hall conductance. Higher-order nonlinear transport due to irrelevant Cooper pair tunneling or vortex dissipation is further studied, which becomes dominant when the edge is in a normal phase. Our results apply to both the single-layer state (as a particle-hole conjugate of ν=1/3) and the bilayer Halperin-(112) state, revealing a rich landscape of disorder-stabilized phases in FQH edge states with SC proximity, and may as well apply to fractional Chern insulators recently observed at the same filling.

We introduce a dissipative lattice gauge model that exhibits the many-body version of the non-Hermitian skin effect. The dissipative couplings between dynamical gauge fields on the lattice links and the surrounding environment generate chiral motions of particles residing on lattice sites. Despite the complexity arising from many-body interactions, the local gauge symmetry enables the exact construction of a steady state that displays the many-body non-Hermitian skin effect. Furthermore, our approach can be generalized to realize a new type of many-body non-Hermitian skin effect, dubbed the hierarchical skin effect, where different subsystem degrees of freedom exhibit boundary accumulation of multiple moments at different orders. Our findings can be readily observed by engineering dissipation in state-of-the-art lattice gauge simulators.

Relaxed quantum systems with conservation laws are believed to be approximated by the generalized Gibbs ensemble (GGE), which incorporates the constraints of certain conserved quantities serving as integrals of motion. By drawing an analogy between eigenstate reduced density matrix and GGE, we conjecture that a natural set of conserved quantities for GGE can emerge from the reduced density matrices of properly chosen eigenstates by the entanglement Hamiltonian superdensity matrix (EHSM) framework, and we demonstrate this explicitly for models mappable to free fermions. The framework proposes that such conserved quantities are linear superpositions of eigenstate entanglement Hamiltonians of a larger auxiliary system, where the eigenstates are Fock states occupying what we call the common eigenmodes, which remain eigenmodes when truncated within the physical subsystem. For 1D homogeneous free fermions with (anti)periodic boundary conditions, which maps to 1D hardcore bosons with nearest neighbor hoppings, these conserved quantities lead to a non-Abelian GGE, which predicts the relaxation of both fermion and boson bilinears more accurately than the conventional Abelian GGE. Generalization of this framework may provide novel numerical insights for quantum integrability.

Hofstadter’s butterfly, the predicted energy spectrum for non-interacting electrons confined to a two-dimensional lattice in a magnetic field, is one of the most remarkable fractal structures in nature¹. At rational ratios of magnetic flux quanta per lattice unit cell, this spectrum shows self-similar distributions of energy levels that reflect its recursive construction. For most materials, Hofstadter’s butterfly is predicted under experimental conditions that are unachievable using laboratory-scale magnetic fields^1,2,3. More recently, electrical transport studies have provided evidence for Hofstadter’s butterfly in materials engineered to have artificially large lattice constants^4,5,6, such as those with moiré superlattices^7,8,9,10. Yet, so far, direct spectroscopy of the fractal energy spectrum predicted by Hofstadter nearly 50 years ago has remained out of reach. Here we use high-resolution scanning tunnelling microscopy/spectroscopy (STM/STS) to investigate the flat electronic bands in twisted bilayer graphene (TBG) near the predicted second magic angle^11,12, an ideal setting for spectroscopic studies of Hofstadter’s spectrum. Our study shows the fractionalization of flat moiré bands into discrete Hofstadter subbands and discerns experimental signatures of self-similarity of this spectrum. Moreover, our measurements uncover a spectrum that evolves dynamically with electron density, showing phenomena beyond that of Hofstadter’s original model owing to the combined effects of strong correlations, Coulomb interactions and the quantum degeneracy of electrons in TBG.

The one-dimensional quantum breakdown model, which features spatially asymmetric fermionic interactions simulating the electrical breakdown phenomenon, exhibits an exponential U(1) symmetry and a variety of dynamical phases including many-body localization and quantum chaos with quantum scar states. We investigate the minimal quantum breakdown model with the minimal number of on-site fermion orbitals required for the interaction and identify a large number of local conserved charges in the model. We then reveal a mapping between the minimal quantum breakdown model in certain charge sectors and a quantum link model which simulates the U(1) lattice gauge theory and show that the local conserved charges map to the gauge symmetry generators. A special charge sector of the model further maps to the PXP model, which shows quantum many-body scars. This mapping unveils the rich dynamics in different Krylov subspaces characterized by different gauge configurations in the quantum breakdown model.