Research Overview
Quantum information creates quantum matter I: How to observe Quasiparticles from entanglement?
The understanding of quantum many-body theory has been deeply affected by the quantum information revolution' which has, among other ideas, led to the systematic study of the entanglement present in quantum states. The quantum entanglement of a wave function builds the information entropy carried by each Qubit of the many-body system and correspondingly creates unique quantum matters.
Much is explored for the entanglement properties in the ground state or highly excited states with finite energy density, then what about low energy quasiparticles? Can we visualize distinct quasiparticles and collective motions from an entanglement perspective?
We present a quantum information framework to examine the nature of the entanglement in the quasiparticle excitations. We argue that the salient features of the quasiparticles, including their quantum numbers, locality, and fractionalization are reflected in the entanglement spectrum and in the mutual information. Much of the exotic properties, including hidden orders and QP coherence, can be tackled from their entanglement properties.
Higher-order entanglement!
We developed a new entanglement property-dubbed 'higher-order entanglement' which can potentially reveal many salient features and properties of a many-body system!
For many interacting HOTI, the entanglement spectrum under any spatial cut could be gapped. In addition, the 'gap closing' in the entanglement Hamiltonian does not guarantee a phase transition in the physical system. Then can one still use `entanglement spectroscopy' to test different phases?
To solve these enigmas, we propose a new concept -- `higher-order entanglement', which refers to a hierarchy sequence of the entanglement spectrum instead of a single reduced density matrix.
Review on Fracton phase of matter with Mike Pretko and Xie Chen
We made it! A state-of-the-art review on Fracton phase of matter from various aspects.
In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets.
Q: Is Higher-order topological insulator a Fracton?
A: Conditional yes, from the viewpoint of topological multipolar field theory.
Is higher-order TI related to Fracton phase of matter? I had been questioned on this frequently since 2018. At that time, the relation between HOTI and Fracton is still superficial, or even abstract nonsense to me. I did not take this issue seriously until recently, my collaborator and I developed a topological multipolar theory description for HOTI in various dimensions. Such field theory share many similarities with the Fractonic tensor gauge theory we pursued a while ago.
Remarkably, we find that despite their distinct symmetry structure, some classes of fractons and HOTIs can be connected through their essentially identical topological response theories. More precisely, we propose a topological quadrupole response theory that describes both a 2D symmetry enriched fracton phase and a related bosonic quadrupolar HOTI with strong interactions. Such a topological quadrupole term encapsulates the protected corner charge modes and, for the HOTI, predicts an anomalous edge with fractional dipole moment. In 3D we propose a dipolar Chern-Simons theory with a quantized coefficient as a description of the response of both second order HOTIs harboring chiral hinge currents, and of a related fracton phase. This theory correctly predicts chiral currents on the hinges and anomalous dipole currents on the surfaces. We generalize these results to higher dimensions to reveal a family of multipolar Chern-Simons terms and related theta-term actions that can be reached via dimensional reduction or extension from the Chern-Simons theories.
How Fracton becomes real in frustrated magnetism?
An algebraic quantum liquid from plaquette melting transitions will tell!
Paramagnetic spin systems with spontaneously broken spatial symmetries, such as valence bond solid (VBS) phases, can host topological defects carrying non-trivial quantum numbers, which enables the paradigm of deconfined quantum criticality.
Recently, we study the properties of topological defects in valence plaquette solid (VPS) phases. The defects of the VPS order parameter, in addition to possessing non-trivial quantum numbers, have fracton mobility constraints deep in the VPS phase, which has been overlooked previously.
These mobility constraints, while they persist, can potentially inhibit the condensation of vortices and preclude a continuous transition from the VPS to the Neel antiferromagnet. Instead, the VPS melting transition can be driven by proliferation of spinon dipoles. A 2d VPS can melt into a stable gapless phase in the form of an algebraic bond liquid with algebraic correlations and long-range entanglement. Such a bond liquid phase yields a concrete example of the elusive 2d Bose metal with symmetry fractionalization!
The beauty is in the decoration:
Higher-order topological phase without crystalline symmetry
1）Is there a higher-order topological phase with gapless hinge/corner which does not require spatial symmetry?
Yes. A subsystem symmetric HOTI does the job. While most previous HOTIs are descendants of topological crystalline phases, our result probably escapes from this circle. Such subsystem symmetric HOTI only exists in strongly interacting systems. We propose several strong interacting models with gapless hinge or corner based on a `decorated hinge-wall condensate' picture.
2) In particular, we establish a no-go theorem to demonstrate the ungappable nature of the hinge by making a connection between generalized Lieb-Schultz-Mattis theorem and the boundary anomaly of HOSPT state.
What is the relation between fracton and higher-order TI? Another forthcoming collaboration will tell.
Fractonic Chern-Simons and BF theory in 3+1d
1) Fracton phase of matter had been extensively studied in terms of discrete tensor gauge theory.
However, once we get into the U(1) limit, most of these phases are unstable due to instanton confinement. Is there a universal way to construct the deconfined U(1) Fracton phase?
2) It is obvious that the CSS type Fracton stabilizer codes can be interpreted as zoology of symmetric tensor Electromagnetism.
Then what about the non-CSS Fracton code, e.g. the Chamon code?
In our recent work, we introduce a Fracton Chern-Simons coupling in addition to the tensor Maxwell theory. Such theory supports a gapped deconfined phase and the quasiparticle excitations display fractional statistics with restrict motion. The fractonic Chern-Simons type coupling reproduces the salient features of the ordinary 2D Chern-Simons theory including the transverse electromagnetic response, the Wilson-Algebra and the current anomaly on the boundary. By studying the resulting quantum theory on the lattice, we show that it describes a Z_n generalization of the Chamon code.
Non-Abelian defects in Fracton phase of matter
A wide variety of fracton models with abelian excitations had been proposed and extensively studied while the candidates for non-Abelian fracton phases are less explored.
In this work, we investigate the effect of twisted defect in abelian fracton models. The twisted defect is launched by introducing a branch cut line hosting anyon condensate. In particular, these twisted defects, which alter different types of quasiparticles, carry projective non-Abelian zero modes. En route, such defects can be engineered via strong onsite hybridization along a branch cut which provides wide tunability and flexibility in experiment platforms. The braiding of twisted defects with projective non-Abelian Berry phase renders a new avenue toward fault-tolerant quantum computation.
Majorana Quantum Lego, a Route Towards Fracton Matter
arXiv:1812.06901
Fracton topological phases host fractionalized topological quasiparticles with restricted mobility, with promising applications to fault-tolerant quantum computation. While a variety of exactly solvable fracton models have been proposed, there is a need for platforms to realize them experimentally.
We show that a rich set of fracton phases emerges in interacting Majorana band models whose building blocks are within experimental reach. Specifically, our Majorana constructions overcome a principal obstacle, namely the implementation of the complicated spin cluster interactions underlying fracton stabilizer codes. The basic building blocks of the proposed constructions include Coulomb blockaded Majorana islands and weak inter-island Majorana hybridizations. This setting produces a wide variety of fracton states and promises numerous opportunities for probing and controlling fracton phases experimentally. Our approach also reveals the relation between fracton phases and Majorana fermion codes and further generates a hierarchy of fracton spin liquids.
Higher order topological superconductors, a generator for long-range entangled phases
arXiv:1810.10556
While some early literature shows that interactions can both trivialize and enrich these phases, most efforts to date have focused on mathematical classification rather than microscopic Hamiltonian.
Recent, we investigate the interaction effect in some Higher-order TSC models. We demonstrate that strong local interactions can induce more exotic topologically ordered phases in a class of higher order topological superconductors. In two dimensions, interacting HOTSC realizes various topologically ordered surface and color codes. In three dimensions, interactions can drive HOTSC into recently discovered fracton phases. We explicitly relate fermion parity operators underlying the gapless excitations of the HOTSC to the Wilson algebra of symmetry enriched quantum codes.
Such interaction induced topological phase can be implemented in Majorana network with tunable interaction. This tunability allows for experimental access to such topological phase transition, and thus provides a new route towards the realization of fracton phases and color codes.
Higher order SPT states in interacting systems
Higher-order topological insulators have a modified bulk-boundary correspondence compared to other topological phases: instead of gapless edge or surface states, they have gapped edges and surfaces, but protected modes at corners or hinges. Here, we explore symmetry protected topological phases in strongly interacting many-body systems with this generalized bulk-boundary correspondence. We introduce several exactly solvable bosonic lattice models as candidates for interacting higher order symmetry protected topological (HOSPT) phases protected by spatial symmetries, and develop a topological field theory that captures the non-trivial nature of the gapless corner and hinge modes. We show how, for rotational symmetry, this field theory leads to a natural relationship between HOSPT phases and conventional SPT phases with an enlarged internal symmetry group. We also explore the connection between bosonic and fermionic HOSPT phases in the presence of strong interactions, and comment on the implications of this connection for the classification of interacting fermionic HOSPT phases. Finally, we explore how gauging internal symmetries of these phases leads to topological orders characterized by nontrivial braiding statistics between topological vortex excitations and geometrical defects related to the spatial symmetry.
Fractonic matters with a twist!
Fractonic matter is a strongly interacting many-body system with sub-dimensional excitation, where the quasiparticle has restricted mobility and dynamics. For conventional interacting boson/fermion theory with additional symmetry, there exists a class of symmetry protected short-range entangled states with quantized topological response. Then what happens for fractonic matters? Do they support symmetry protected fracton phase?
In this paper (https://arxiv.org/abs/1805.09800), we explore the interplay between symmetry and fracton order, motivated by the analogous close relationship for topologically ordered systems. Specifically, we consider models with 3D planar subsystem symmetry, and show that these can realize subsystem symmetry protected topological phases with gapless boundary modes. Gauging the planar subsystem symmetry leads to a fracton order in which particles restricted to move along lines exhibit a new type of statistical interaction that is specific to the lattice geometry. We show that both the gapless boundary modes of the ungauged theory, and the statistical interactions after gauging, are naturally captured by a higher-rank version of Chern-Simons theory. We also show that gauging only part of the subsystem symmetry can lead to symmetry-enriched fracton orders, with quasiparticles carrying fractional symmetry charge.
Subsystem symmetry protected topological order
Strongly interacting SPT phases could be `topological' under global symmetry protection. But what could happen when the symmetry only acts on a subset of the many-body system, e.g. along lines or planes?
In this work, we are looking for subsystem SPT states where symmetry acts on a subset of the many-body system. In principle, such SSPT order cannot appear in non-interacting(weak interacting) system. We develop some exactly solvable models as candidates for SSPT states, investigate their properties and topological response.
What do you expect for that vortex metal?
reference: arxiv.1711.03969
Interacting 2d electrons in the presence of strong magnetic field exhibit rich phase diagram and exotic phenomenon. Beyond the incompressible quantum Hall insulator, there also exists a class of compressible metallic state when electrons are at even filling factor $\nu=\frac{1}{2n}$. Motivated by the idea of composite Fermion(CF) with flux attachment, Halperin, Lee, and Read(HLR) initially developed a theoretical framework for partial filled Landau level with metallic behavior. When each fermion is attached with $2n$ flux, the composite fermion forms a Fermi surface with strong and nonlocal interaction mediated by dynamical gauge fields. The gauge boson is overdamped by the gapless Fermi surface, and the fermion encounters inelastic scattering mediate by gauge boson. Hence, the composite Fermi surface becomes chaotic and decoherent at low T and the quasiparticles acquire finite lifetime.
The HLR theory provides a simple but primitive framework to clarify the existence of a stable Fermi surface in $\nu=\frac{1}{2n}$ filled Landau Levels(LL). Such composite Fermi surface with non-Fermi liquid nature contains unconventional transport properties. The existence of `composite Fermi surface' with finite wave vector could be probed in experiment by measuring the Weiss oscillation and static dielectric response. One can also subtract the Fermi wave vector of the composite Fermi surface in numerical simulations by scaling the entanglement entropy with a leading order of $L\ln(L)$ or by figuring out the singularity of Lindhard function, which mostly agrees with the HLR predication.
In the large magnetic field limit where the inter LL gap is much larger than any other interaction scale, one could ignore the effect from LL mixing and project the Hilbert space to the lowest LL in IR theory. In such limit, the half-filled Landau level contains an exact Particle-Hole(PH) symmetry. In addition, for LL bilayer with filling $\nu_1=\frac{1}{2n}$ and $\nu_2=1-\frac{1}{2n}$ at each layer, the system also contains an explicit PH symmetry(up to layer switching). However, based on the previous flux attachment picture in HLR theory, all these phases contain a composite Fermi surface with finite Fermi wave vector. While a Fermi surface indicates non-zero chemical potential, it is paradoxical to get a `survival Fermi surface' in the PH symmetric limit
To solve this enigma, we develop a vortex metal theory for partial filled Landau Level at $\nu=\frac{1}{2n}$, whose ground state contains a composite Fermi surface(FS) formed by the vortex of electrons. In the projected Landau Level limit, the composite Fermi surface contains $\frac{-\pi}{n}$ Berry phase. Such fractional Berry phase is a consequence of LL projection which produces the GMP guiding center algebra and embellishes an anomalous velocity to the equation of motion for the vortex metal. The PH symmetry acts on the composite Fermi surface in the form of time-reversal, and this makes the composite Fermi surface survive in the lowest Landau Level limit. Further, we investigate a particle-hole symmetric bilayer system with $\nu_1=\frac{1}{2n}$ and $\nu_2=1-\frac{1}{2n}$ at each layer, and demonstrate that the $\frac{-\pi}{n}$ Berry phase on the composite Fermi surface leads to the suppression of $2k_f$ back-scattering between the PH partner bilayer, which could be a smoking gun to detect the fractional Berry phase. We also mention various instabilities and competing orders in such bilayer system including a $Z_{4n}$ topological order phase driven by quantum criticality.