Condensed Matter

In this mini course, I shall introduce the basic concepts in 2D topological orders by studying simple models of topological orders and then introduce topological quantum computing based on Fibonacci anyons. Here is the (not perfectly ordered) syllabus.                      

•        Overview of topological phases of matter
•        Z2 toric code model: the simplest model of 2D topological orders
•        Quick generalization to the quantum double model
•        Anyons, topological entanglement entropy, S and T matrices
•        Fusion and braiding of anyons: quantum dimensions, pentagon and hexagon identities
•        Fibonacci anyons
•        Topological quantum computing.

In an isolated quantum many-body system undergoing unitary evolution, the entropy of a subsystem (smaller than half the system size) thermalizes if at long times, it is to leading order equal to the thermodynamic entropy of the subsystem at the same energy. We prove entropy thermalization for a nearly integrable Sachdev-Ye-Kitaev model initialized in a pure product state. The model is obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. In this model, there is a regime of “thermalization without eigenstate thermalization.” Thus, the eigenstate thermalization hypothesis is not a necessary condition for thermalization. Joint work with Aram W. Harrow

Zoom Link: https://pitp.zoom.us/j/91710478120?pwd=OVRDOStOSkdIVG9mcGJqMWJlU1FRdz09

Isolated quantum systems, investigated a century ago, exhibit coherent, unitary dynamics. When such a system is coupled to an environment, the resulting loss of coherence is modeled by completely positive, trace preserving (CPTP) quantum maps for the density matrix. A lossy atom, when it has not decayed, exhibits a coherent dynamics that is in a distinct, new class. Non-Hermitian Hamiltonians with parity-time symmetry govern this class and exhibit exceptional-point (EP) degeneracies with topological features. After a historical introduction to PT symmetry, I will present examples of coherent, quantum dynamics in the static and Floquet regimes for such systems with a superconducting transmon (Nature Phys. 15, 1232 (2019)), ultracold atoms (Nature Comm. 10, 855 (2019)), and integrated quantum photonics (Phys. Rev. Res. 4, 013051 (2022); Nature 557, 660 (2018)) as platforms. These include topological quantum state transfer, entanglement/coherence control, and super-quantum correlations. I will conclude with speculations on applicability of these ideas to quantum matter, particle physics, and strong gravity. 

(* with Anthony Laing group, Kater Murch group, Le Luo group, Sourin Das group).

Zoom link:  https://pitp.zoom.us/j/92391441075?pwd=QmRYSnYveUZCci9QZFcwUHBFS29QZz09

This talk is motivated by the question: why do we put so much effort and investment into quantum computing? A short answer is that we expect quantum advantages for practical problems. To achieve this goal, it is essential to reexamine existing experiments and propose new protocols for future quantum advantage experiments. In 2019, Google published a paper in Nature claiming to have achieved quantum computational advantage, also known as quantum supremacy. In this talk, I will explain how they arrived at their claim and its implications. I will also discuss recent theoretical and numerical developments that challenge this claim and reveal fundamental limitations in their approach. Due to these new developments, it is imperative to design the next generation of experiments. I will briefly mention three potential approaches: efficient verifiable quantum advantage, hardware-efficient fault-tolerance, and quantum algorithms on analog devices, including machine learning and combinatorial optimization.

Zoom Link: https://pitp.zoom.us/j/96945612624?pwd=ckRKMFJqZ0Q0dGtFOU91c1hnMzIzZz09

Recent advances in programmable quantum devices brought to the fore the intriguing possibility of using them to realize and investigate topological quantum spin liquids (QSLs) phase. This new and exciting direction brings about important research questions on how to probe and determine the presence of such exotic, highly entangled phases. In this talk, I will discuss how to construct Z2 QSLs states as the ground state of a static Hamiltonian with only local two-qubit interactions and a transverse field, and demonstrate its realization in the classical limit at the endpoint of quantum annealing protocol, using D-Wave DW-2000Q machine . I will also demonstrate how to probe signatures of Z2 QSLs fractional statistics in quantum simulators via quasiparticle interferometry. At the end, I will show the robustness of this probe against disorders and dephasing -- effects that are generally pervasive in quantum devices nowadays.

Zoom Link: https://pitp.zoom.us/j/99207971920?pwd=N3R3YUJtWXJWVjFQWlRoZ3hYSDlMZz09

In this talk I will describe a topological response theory that predicts the physical manifestation of a class of topological invariants in systems with crystalline symmetry. I focus on two such invariants, the 'discrete shift' and a quantized charge polarization. Guided by theory, I discuss how these invariants can be extracted from lattice models by measuring the fractional charge at lattice disclinations and dislocations, as well as from the angular and linear momentum of magnetic flux. These methods are illustrated using the Hofstadter model of spinless fermions in a background magnetic field; they give new topological invariants in this model for the first time since the quantized Hall conductance was computed by TKNN in 1982.

Zoom link:  https://pitp.zoom.us/j/93633131128?pwd=d2h4U1l0ZVU5aE1ORURkdFNSanB4dz09

Periodic driving is a ubiquitous tool for controlling experimental quantum systems. When the drive fields are of comparable, incommensurate frequencies, new theoretical tools are required to treat the resulting quasiperiodic time dependence. Similarly, new and surprising phenomena of topological origin may emerge in this regime, including the quantized pumping of energy from one drive field to another. I will describe how to exploit this energy pumping to coherently translate––or boost––quantum states of a cavity in the Fock basis. This protocol enables the preparation of highly excited Fock states for use in quantum metrology––one need only boost low occupation Fock states. Energy pumping, and hence boosting, may be achieved nonadiabatically as a robust edge effect associated to a topological phase. I will present a simple coupled-layer model for the phase, and briefly describe the topological classification which characterizes its robust properties.

Zoom link:  https://pitp.zoom.us/j/94040881668?pwd=THh1WlIxZmZnYlp6QVRKRDhMWnk1UT09

It has long been known that the quantum ground state of a metal is characterized by an abstract manifold in the momentum space called the Fermi sea. Fermi sea can be distinguished topologically in much the same way that a ball can be distinguished from a donut by counting the number of holes. The associated topological invariant, i.e. the Euler characteristic (χ_F), serves to classify metals. Here I will survey two recent proposals relating χ_F  to experimental observables, namely: (i) equal-time density/number correlations, and (ii) Andreev state transport along a planar Josephson junction. Moreover, from the perspective of quantum information, I will explain how multipartite entanglement in real space probes the Fermi sea topology in momentum space. Our works not only suggest a new connection between topology and entanglement in gapless quantum matters, but also suggest accessible experimental platforms to extract the topology in metals. 

Zoom link:  https://pitp.zoom.us/j/98944473905?pwd=ak5nVmd4N0pSdXpjOFM0YnFJdnJ4dz09

Symmetry-protected topological (SPT) phases are short-range entanglement (SRE) quantum states which cannot be adiabatically connected to trivial product states in the presence of symmetries. Recently, it is shown that symmetry-protected short-range entanglement can still prevail even if part of the protecting symmetry is broken by quenched disorder locally but restored upon disorder averaging, dubbed as the average symmetry-protected topological (ASPT) phases. In this talk, I will systematically construct the ASPT phases as a mixed ensemble or density matrix, which may not be realized in a clean system without any disorder. I will also design the strange correlator of the ASPT phases via a strange density matrix to detect the nontrivial ASPT state. Moreover, it is amazing that the strange correlator of ASPT can be precisely mapped to the loop correlation functions of some proper statistical loop models, with power-law behaviors.

Zoom link:  https://pitp.zoom.us/j/91672345456?pwd=N0dNQXNmVVoybnNxYXJuWnVRME8rUT09

I will talk about the generalization of Berry classes for quantum lattice spin systems. It defines invariants of topologically ordered states or families thereof. In particular, its equivariant version for 2d gapped states gives the zero-temperature Hall conductance and its various generalizations. I will also discuss the construction of chiral states realizing the topological order associated with a unitary rational vertex operator algebra for which these invariants are non-trivial

Zoom link:  https://pitp.zoom.us/j/99910103969?pwd=VlVHVGRiV29iVEFyTXZyR3ovMkRaQT09

The multipartite entanglement structure for the ground states of two dimensional topological phases is an interesting albeit not well understood question. Utilizing the bulk-boundary correspondence, the tripartite entanglement calculation of 2d topological phases can be reduced to that on the vertex state, defined by the boundary conditions at the interfaces between spatial regions. In this work, we use the conformal interface technique to calculate the entanglement measures of the vertex state, which include the area-law, geometrical and topological pieces, and the possible extra order one contribution. This explains our previous observation of Markov gap h=\frac{c}{3}\ln 2 in the 3-vertex state, and generalizes it to the p-vertex state as well as rational conformal field theory, and more general choices of subsystem. Finally, we support our prediction by numerical evidence. 

Zoom link:  https://pitp.zoom.us/j/93914854044?pwd=eWl3eGVLU25XUGhKbnFRSm5ab0JuUT09

We construct an exactly solvable lattice model for a deconfined quantum critical point (DQCP) in (1+1) dimensions. This DQCP occurs in an unusual setting, namely at the edge of a (2+1) dimensional bosonic symmetry protected topological phase (SPT) with ℤ2×ℤ2 symmetry. The DQCP describes a transition between two gapped edges that break different ℤ2 subgroups of the full ℤ2×ℤ2 symmetry. Our construction is based on an exact mapping between the SPT edge theory and a ℤ4 spin chain. This mapping reveals that DQCPs in this system are directly related to ordinary ℤ4 symmetry breaking critical points. Based on arXiv:2206.01222.

Zoom link:  https://pitp.zoom.us/j/93794543360?pwd=Y3lidGZhRFNrUjEyMVpaNEgwTnE3QT09