31 research outputs found
Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems
Topological phases protected by symmetry can occur in gapped
and---surprisingly---in critical systems. We consider non-interacting fermions
in one dimension with spinless time-reversal symmetry. It is known that the
phases are classified by a topological invariant and a central charge
. We investigate the correlations of string operators, giving insight into
the interplay between topology and criticality. In the gapped phases, these
non-local string order parameters allow us to extract . Remarkably,
ratios of correlation lengths are universal. In the critical phases, the
scaling dimensions of these operators serve as an order parameter, encoding
and . We derive exact asymptotics of these correlation functions
using Toeplitz determinant theory. We include physical discussion, e.g.,
relating lattice operators to the conformal field theory. Moreover, we discuss
the dual spin chains. Using the aforementioned universality, the topological
invariant of the spin chain can be obtained from correlations of local
observables.Comment: 35 pages, 5 page appendi
One-Dimensional Symmetry Protected Topological Phases and their Transitions
We present a unified perspective on symmetry protected topological (SPT)
phases in one dimension and address the open question of what characterizes
their phase transitions. In the first part of this work we use symmetry as a
guide to map various well-known fermionic and spin SPTs to a Kitaev chain with
coupling of range . This unified picture uncovers new
properties of old models --such as how the cluster state is the fixed point
limit of the Affleck-Kennedy-Lieb-Tasaki state in disguise-- and elucidates the
connection between fermionic and bosonic phases --with the Hubbard chain
interpolating between four Kitaev chains and a spin chain in the Haldane phase.
In the second part, we study the topological phase transitions between these
models in the presence of interactions. This leads us to conjecture that the
critical point between any SPT with -dimensional edge modes and the trivial
phase has a central charge . We analytically verify this for
many known transitions. This agrees with the intuitive notion that the phase
transition is described by a delocalized edge mode, and that the central charge
of a conformal field theory is a measure of the gapless degrees of freedom.Comment: 18 pages, 9 figures, 3 page appendi
Strong quantum interactions prevent quasiparticle decay
Quantum states of matter---such as solids, magnets and topological
phases---typically exhibit collective excitations---phonons, magnons, anyons.
These involve the motion of many particles in the system, yet, remarkably, act
like a single emergent entity---a quasiparticle. Known to be long-lived at the
lowest energies, common wisdom says that quasiparticles become unstable when
they encounter the inevitable continuum of many-particle excited states at high
energies. Whilst correct for weak interactions, we show that this is far from
the whole story: strong interactions generically stabilise quasiparticles by
pushing them out of the continuum. This general mechanism is straightforwardly
illustrated in an exactly solvable model. Using state-of-the-art numerics, we
find it at work also in the spin- triangular lattice Heisenberg
antiferromagnet (TLHAF) near the isotropic point---this is surprising given the
common expectation of magnon decay in this paradigmatic frustrated magnet.
Turning to existing experimental data, we identify the detailed phenomenology
of avoided decay in the TLHAF material BaCoSbO, and even in liquid
helium---one of the earliest instances of quasiparticle decay. Our work unifies
various phenomena above the universal low-energy regime in a comprehensive
description. This broadens our window of understanding of many-body
excitations, and provides a new perspective for controlling and stabilising
quantum matter in the strongly-interacting regime.Comment: 4 pages, appendix (5 pages
Dynamics of the Kitaev-Heisenberg Model
We introduce a matrix-product state based method to efficiently obtain
dynamical response functions for two-dimensional microscopic Hamiltonians,
which we apply to different phases of the Kitaev-Heisenberg model. We find
significant broad high energy features beyond spin-wave theory even in the
ordered phases proximate to spin liquids. This includes the phase with zig-zag
order of the type observed in -RuCl, where we find high energy
features like those seen in inelastic neutron scattering experiments. Our
results provide an example of a natural path for proximate spin liquid features
to arise at high energies above a conventionally ordered state, as the diffuse
remnants of spin-wave bands intersect to yield a broad peak at the Brillouin
zone center.Comment: 7 pages, 8 figure
Gapless topological phases and symmetry-enriched quantum criticality
We introduce topological invariants for critical bosonic and fermionic
chains. More generally, the symmetry properties of operators in the low-energy
conformal field theory (CFT) provide discrete invariants, establishing the
notion of symmetry-enriched quantum criticality. For nonlocal operators, these
invariants are topological and imply the presence of localized edge modes.
Depending on the symmetry, the finite-size splitting of this topological
degeneracy can be exponential or algebraic in system size. An example of the
former is given by tuning the spin-1 Heisenberg chain to an Ising phase. An
example of the latter arises between the gapped Ising and cluster phases: this
symmetry-enriched Ising CFT has an edge mode with finite-size splitting . More generally, our formalism unifies various examples previously
studied in the literature. Similar to gapped symmetry-protected topological
phases, a given CFT can split into several distinct symmetry-enriched CFTs.
This raises the question of classification, to which we give a partial
answer---including a complete characterization of symmetry-enriched Ising CFTs.Comment: 18 pages + appendi
Topology and Excitations in Low-Dimensional Quantum Matter
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations.
Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing.
In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension.
In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies.
Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples.
Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium.
In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime
Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians
We show that the combination of charge and dipole
conservation---characteristic of fracton systems---leads to an extensive
fragmentation of the Hilbert space, which in turn can lead to a breakdown of
thermalization. As a concrete example, we investigate the out-of-equilibrium
dynamics of one-dimensional spin-1 models that conserve charge (total )
and its associated dipole moment. First, we consider a minimal model including
only three-site terms and find that the infinite temperature auto-correlation
saturates to a finite value---showcasing non-thermal behavior. The absence of
thermalization is identified as a consequence of the strong fragmentation of
the Hilbert space into exponentially many invariant subspaces in the local
basis, arising from the interplay of dipole conservation and local
interactions. Second, we extend the model by including four-site terms and find
that this perturbation leads to a weak fragmentation: the system still has
exponentially many invariant subspaces, but they are no longer sufficient to
avoid thermalization for typical initial states. More generally, for any finite
range of interactions, the system still exhibits non-thermal eigenstates
appearing throughout the entire spectrum. We compare our results to charge and
dipole moment conserving random unitary circuit models for which we reach
identical conclusions.Comment: close to published version: 10 pages + Appendices. Updated
discussions and conten
Quantum Dynamics of the Square-Lattice Heisenberg Model
Despite nearly a century of study of the Heisenberg model on the
square lattice, there is still disagreement on the nature of its high-energy
excitations. By tuning toward the Heisenberg model from the exactly soluble
Ising limit, we find that the strongly attractive magnon interactions of the
latter naturally account for a number of spectral features of the Heisenberg
model. This claim is backed up both numerically and analytically. Using the
density matrix renormalization group method, we obtain the dynamical structure
factor for a cylindrical geometry, allowing us to continuously connect both
limits. Remarkably, a semi-quantitative description of certain observed
features arises already at the lowest non-trivial order in perturbation theory
around the Ising limit. Moreover, our analysis uncovers that high-energy
magnons are localized on a single sublattice, which is related to the
entanglement properties of the ground state.Comment: 11 pages, 10 figures, appendi