35 research outputs found
Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
In topologically ordered quantum states of matter in 2+1D (space-time
dimensions), the braiding statistics of anyonic quasiparticle excitations is a
fundamental characterizing property which is directly related to global
transformations of the ground-state wavefunctions on a torus (the modular
transformations). On the other hand, there are theoretical descriptions of
various topologically ordered states in 3+1D, which exhibit both point-like and
loop-like excitations, but systematic understanding of the fundamental physical
distinctions between phases, and how these distinctions are connected to
quantum statistics of excitations, is still lacking. One main result of this
work is that the three-dimensional generalization of modular transformations,
when applied to topologically ordered ground states, is directly related to a
certain braiding process of loop-like excitations. This specific braiding
surprisingly involves three loops simultaneously, and can distinguish different
topologically ordered states. Our second main result is the identification of
the three-loop braiding as a process in which the worldsheets of the three
loops have a non-trivial triple linking number, which is a topological
invariant characterizing closed two-dimensional surfaces in four dimensions. In
this work we consider realizations of topological order in 3+1D using
cohomological gauge theory in which the loops have Abelian statistics, and
explicitly demonstrate our results on examples with topological
order
Chiral spin density wave, spin-charge-Chern liquid and d+id superconductivity in 1/4-doped correlated electronic systems on the honeycomb lattice
Recently two interesting candidate quantum phases --- the chiral spin density
wave state featuring anomalous quantum Hall effect and the d+id superconductor
--- were proposed for the Hubbard model on the honeycomb lattice at 1/4 doping.
Using a combination of exact diagonalization, density matrix renormalization
group, the variational Monte Carlo method and quantum field theories, we study
the quantum phase diagrams of both the Hubbard model and t-J model on the
honeycomb lattice at 1/4-doping. The main advantage of our approach is the use
of symmetry quantum numbers of ground state wavefunctions on finite size
systems (up to 32 sites) to sharply distinguish different quantum phases. Our
results show that for in the Hubbard model and for in the t-J model, the quantum ground state is either a chiral spin
density wave state or a spin-charge-Chern liquid, but not a d+id
superconductor. However, in the t-J model, upon increasing the system goes
through a first-order phase transition at into the d+id
superconductor. Here the spin-charge-Chern liquid state is a new type of
topologically ordered quantum phase with Abelian anyons and fractionalized
excitations. Experimental signatures of these quantum phases, such as tunneling
conductance, are calculated. These results are discussed in the context of
1/4-doped graphene systems and other correlated electronic materials on the
honeycomb lattice.Comment: Some parts of text revised for clarity of presentatio
Deconfined quantum critical point in one dimension
We perform a numerical study of a spin-1/2 model with symmetry in one dimension which demonstrates an interesting
similarity to the physics of two-dimensional deconfined quantum critical points
(DQCP). Specifically, we investigate the quantum phase transition between Ising
ferromagnetic and valence bond solid (VBS) symmetry-breaking phases. Working
directly in the thermodynamic limit using uniform matrix product states, we
find evidence for a direct continuous phase transition that lies outside of the
Landau-Ginzburg-Wilson paradigm. In our model, the continuous transition is
found everywhere on the phase boundary. We find that the magnetic and VBS
correlations show very close power law exponents, which is expected from the
self-duality of the parton description of this DQCP. Critical exponents vary
continuously along the phase boundary in a manner consistent with the
predictions of the field theory for this transition. We also find a regime
where the phase boundary splits, as suggested by the theory, introducing an
intermediate phase of coexisting ferromagnetic and VBS order parameters.
Interestingly, we discover a transition involving this coexistence phase which
is similar to the DQCP, being also disallowed by Landau-Ginzburg-Wilson
symmetry-breaking theory.Comment: 20 pages, 18 figure
Ising ferromagnet to valence bond solid transition in a one-dimensional spin chain: Analogies to deconfined quantum critical points
We study a one-dimensional (1d) system that shows many analogies to proposed
two-dimensional (2d) deconfined quantum critical points (DQCP). Our system is a
translationally invariant spin-1/2 chain with on-site symmetry
and time reversal symmetry. It undergoes a direct continuous transition from a
ferromagnet (FM), where one of the symmetries and the time reversal are
broken, to a valence bond solid (VBS), where all on-site symmetries are
restored while the translation symmetry is broken. The other symmetry
remains unbroken throughout, but its presence is crucial for both the direct
transition (via specific Berry phase effect on topological defects, also
related to a Lieb-Schultz-Mattis-like theorem) and the precise characterization
of the VBS phase (which has crystalline-SPT-like property). The transition has
a description in terms of either two domain wall species that "fractionalize"
the VBS order parameter or in terms of partons that "fractionalize" the FM
order parameter, with each picture having its own gauge structure. The
two descriptions are dual to each other and, at long wavelengths, take the form
of a self-dual \emph{gauged} Ashkin-Teller model, reminiscent of the self-dual
easy-plane non-compact CP model that arises in the description of the 2d
easy-plane DQCP. We also find an exact reformulation of the transition that
leads to a simple field theory description that explicitly unifies the FM and
VBS order parameters; this reformulation can be interpreted as a new parton
approach that does not attempt to fractionalize either of the two order
parameters but instead encodes them in instantons. Besides providing explicit
realizations of many ideas proposed in the context of the 2d DQCP, here in the
simpler and fully tractable 1d setting with continuous transition, our study
also suggests possible new line of approach to the 2d DQCP.Comment: 36 pages, 1 figure; published versio
Symmetric topological phases and tensor network states:
Thesis advisor: Ying RanClassification and simulation of quantum phases are one of main themes in condensed matter physics. Quantum phases can be distinguished by their symmetrical and topological properties. The interplay between symmetry and topology in condensed matter physics often leads to exotic quantum phases and rich phase diagrams. Famous examples include quantum Hall phases, spin liquids and topological insulators. In this thesis, I present our works toward a more systematically understanding of symmetric topological quantum phases in bosonic systems. In the absence of global symmetries, gapped quantum phases are characterized by topological orders. Topological orders in 2+1D are well studied, while a systematically understanding of topological orders in 3+1D is still lacking. By studying a family of exact solvable models, we find at least some topological orders in 3+1D can be distinguished by braiding phases of loop excitations. In the presence of both global symmetries and topological orders, the interplay between them leads to new phases termed as symmetry enriched topological (SET) phases. We develop a framework to classify a large class of SET phases using tensor networks. For each tensor class, we can write down generic variational wavefunctions. We apply our method to study gapped spin liquids on the kagome lattice, which can be viewed as SET phases of on-site symmetries as well as lattice symmetries. In the absence of topological order, symmetry could protect different topological phases, which are often referred to as symmetry protected topological (SPT) phases. We present systematic constructions of tensor network wavefunctions for bosonic symmetry protected topological (SPT) phases respecting both onsite and spatial symmetries
Variational Tensor Wavefunctions for the Interacting Quantum Spin Hall Phase
The quantum spin hall (QSH) phase, also known as the 2D topological
insulator, is characterized by protected helical edge modes arising from time
reversal symmetry. While initially proposed for band insulators, this phase can
also manifest in strongly-correlated systems where conventional band theory
fails. To overcome the challenge of simulating this phase in realistic
correlated models, we propose a novel framework utilizing fermionic tensor
network states. Our approach involves constructing a tensor representation of
the fixed-point wavefunction based on an exact solvable model, enabling us to
derive a set of tensor equations governing the transformation rules of local
tensors under symmetry operations. These tensor equations lead to the anomalous
edge theory, which provides a comprehensive description of the QSH phase. By
solving these tensor equations, we obtain variational ansatz for the QSH phase,
which we subsequently verify through numerical calculations. This method serves
as an initial step towards employing tensor algorithms to simulate the QSH
phase in strongly-correlated systems, opening new avenues for investigating and
understanding topological phenomena in complex materials.Comment: 6+15 pages,12 figures. Numerical calculations are adde
Unveiling Correlated Topological Insulators through Fermionic Tensor Network States -- Classification, Edge Theories and Variational Wavefunctions
The study of topological band insulators has revealed fascinating phases
characterized by band topology indices, harboring extraordinary boundary modes
protected by anomalous symmetry actions. In strongly correlated systems, where
the traditional notion of electronic bands becomes obsolete, it has been
established that topological insulator phases persist as stable phases,
separate from trivial insulators. However, due to the inability to express the
ground states of such systems as Slater determinants, the formulation of
generic variational wavefunctions for numerical simulations is highly
desirable. In this paper, we tackle this challenge by developing a
comprehensive framework for fermionic tensor network states. Starting from
simple assumptions, we obtain possible sets of tensor equations for any given
symmetry group, capturing consistent relations governing symmetry
transformation rules on tensor legs. We then examine the connections between
these tensor equations and topological insulators by construing edge theories
and extracting quantum anomaly data from each set of tensor equations. By
exhaustively exploring all possible sets of equations, we achieve a systematic
classification of topological insulator phases. Imposing the solutions of a
given set of equations onto local tensors, we obtain generic variational
wavefunctions for corresponding topological insulator phases. Our methodology
provides a crucial first step towards simulating topological insulators in
strongly correlated systems. We discuss the limitations and potential
generalizations of our results, paving the way for further advancements in this
field.Comment: 32+20 pages, 11 figure