Quantum Matter

The first part of this talk will introduce generalized Jordan–Wigner
transformation on arbitrary triangulation of any simply connected
manifold in 2d, 3d and general dimensions. This gives a duality
between all fermionic systems and a new class of Z2 lattice gauge
theories. This map preserves the locality and has an explicit
dependence on the second Stiefel–Whitney class and a choice of spin
structure on the manifold. In the Euclidean picture, this mapping is
exactly equivalent to introducing topological terms (Chern-Simon term
in 2d or the Steenrod square term in general) to the Euclidean action.
We can increase the code distance of this mapping, such that this
mapping can correct all 1-qubit and 2-qubits errors and is useful for
the simulation of fermions on the quantum computer. The second part of
my talk is about SPT phases. By the boson-fermion duality, we are able
to show the equivalent between any supercohomology fermionic SPT and
some higher-group bosonic SPT phases. Particularly in (3+1)D, we have
constructed a unitary quantum circuit for any supercohomology
fermionic SPT state with gapped boundary construction. This fermionic
SPT state is derived by gauging higher-form Z2 symmetry in the
higher-group bosonic SPT and apply the boson-fermion duality.

I will briefly review the pseudogap phenomenology in high Tc cuprate superconductor, especially the recent experiments, and propose a unified picture of the phenomenology under only one assumption: the fluctuating pair density wave. By quantum disordering a pair density wave, we found a state composed of insulating antinodal pairs and nodal electron pocket. We compare the theoretical predictions with ARPES results, optical conductivity, quantum oscillation and other experiments.

We derive general results relating revivals in the dynamics of quantum
many-body systems to the entanglement properties of energy eigenstates.
For a D-dimensional lattice system of N sites initialized in a
low-entangled and short-range correlated state, our results show that a
perfect revival of the state after a time at most poly(N) implies the
existence of "quantum many-body scars", whose number grows at least as
the square root of N up to poly-logarithmic factors. These are energy
eigenstates with energies placed in an equally-spaced ladder and with
Rényi entanglement entropy scaling as log(N) plus an area law term for
any region of the lattice. This shows that quantum many-body scars are a
necessary condition for revivals, independent of particularities of the
Hamiltonian leading to them. We also present results for approximate
revivals, for revivals of expectation values of observables and prove
that the duration of revivals of states has to become vanishingly short
with increasing system size.

Work in collaboration with Henrik Wilming (ETH)

It has recently been shown that quenched randomness, via the phenomenon of many-body localization, can stabilize dynamical phases of matter in periodically driven (Floquet) systems, with one example being discrete time crystals. This raises the question: what is the nature of the transitions between these Floquet many-body-localized phases, and how do they differ from equilibrium? We argue that such transitions are generically controlled by infinite randomness fixed points. By introducing a real-space renormalization group procedure for Floquet systems, asymptotically exact in the strong-disorder limit, we characterize the criticality of the periodically driven interacting quantum Ising model, finding forms of (multi-)criticality novel to the Floquet setting. We validate our analysis via numerical simulations of free-fermion models sufficient to capture the critical physics.