arXiv subjects

Liam Watson

Publications and source records attributed to Liam Watson.

At least 19 recordsLinked to original sources

arXiv2026

Excitonic order in quantum materials: fingerprints, platforms and opportunities

The exciton insulator (EI) is a unique many-body ground state of condensed, spontaneously formed excitons (electron-hole pairs) in equilibrium, distinct from conventional band or Mott insulators. Originally proposed over half a century ago, the concept has recently gained renewed experimental traction thanks to advances in spectroscopic resolution, ultrafast probes, and materials synthesis. In this Review, we outline the essential theoretical ingredients underpinning excitonic order and discuss how dimensionality, disorder and screening affect stability. We then examine the diverse experimental fingerprints of the excitonic state, with central focus on strategies to disentangle excitonic order from competing phases such as charge density waves, Mott insulating states, and hybridization-driven insulators, particularly in systems where non-trivial band topology plays a role. We survey the rapidly expanding family of candidate materials, from layered chalcogenides and correlated rare-earth compounds to artificial excitonic platforms and optically driven non-equilibrium condensates. Finally, we discuss the key challenges and emerging opportunities in the field, identifying the theoretical and experimental frontiers that promise to shape the next decade of research.

cond-mat.str-el↗

arXiv2026

On mutation invariance in Khovanov homology

We show that reduced Khovanov homology over any field is invariant under component-preserving Conway mutation. Our proof relies on strong geography restrictions for a certain Khovanov multicurve invariant associated with Conway tangles that we introduced in previous work [arXiv:1910.14584]. Applying ideas from homological mirror symmetry, we give a full classification of the components of this invariant.

math.GT↗

arXiv2025

Observation of charge density wave excitonic order parameter in topological insulator monolayer WTe2

Strong electron-hole interactions in a semimetal or narrow-gap semiconductor may drive a ground state of condensed excitons. Monolayer WTe2 has been proposed as a host material for such an exciton condensate, but the order parameter - the key signature of a macroscopic quantum-coherent condensate - has not been observed. Here we use Fourier-transform scanning tunnelling spectroscopy (FT-STS) to study quasi-particle interference (QPI) and periodic modulations of the local density of states (LDOS) in monolayer WTe2. In WTe2 on graphene, in which the carrier density can be varied via back-gating, FT-STS shows QPI features in the 2D bulk bands, confirming the interacting nature of the bandgap in neutral WTe2 and the semi-metallic nature of highly n- and p-doped WTe2. We observe additional non-dispersive spatial modulations in the LDOS imprinted on the topological edge mode of neutral WTe2 on metallic substrates (graphene and graphite), which we interpret as the interaction of the topological edge mode with the expected charge density wave order parameter of the excitonic condensate in WTe2 at low interaction strength due to screening by the metallic substrates.

cond-mat.str-el↗

arXiv2024

Reversal of charge transfer doping on the negative electronic compressibility surface of MoS2

The strong electron-electron interaction in transition metal dichalcogenides (TMDs) gives rise to phenomena such as strong exciton and trion binding and excitonic condensation, as well as large negative exchange and correlation contributions to the electron energies, resulting in negative electronic compressibility. Here we use angle-resolved photoemission spectroscopy to demonstrate a striking effect of negative electronic compressibility in semiconducting TMD MoS2 on the charge transfer to and from a partial overlayer of monolayer semimetallic WTe2. We track the changes in binding energy of the valence bands of both WTe2 and MoS2 as a function of surface transfer doping with donor (K) and acceptor (F4-TCNQ) species. Donor doping increases the binding energy of the MoS2 valence band, as expected, while counterintuitively reducing the binding energy of the WTe2 valence bands and core levels. The inverse effect is observed for acceptor doping, where a typical reduction in the MoS2 binding energies is accopanied by an unexpected increase in those of WTe2. The observations imply a reversal of the expected charge transfer; donor (acceptor) deposition decreases (increases) the carrier density in the WTe2 adlayer. The charge transfer reversal is a direct consequence of the negative electronic compressibility of the MoS2 surface layer, for which addition (subtraction) of charge leads to attraction (repulsion) of further charge from neighbouring layers. These findings highlight the importance of many-body interactions for the electrons in transition metal dichalcogenides and underscore the potential for exploring strongly correlated quantum states in two-dimensional semiconductors.

cond-mat.str-el↗

arXiv2023

Defects, band bending and ionization rings in MoS2

Chalcogen vacancies in transition metal dichalcogenides are widely acknowledged as both donor dopants and as a source of disorder. The electronic structure of sulphur vacancies in MoS2 however is still controversial, with discrepancies in the literature pertaining to the origin of the in-gap features observed via scanning tunneling spectroscopy (STS) on single sulphur vacancies. Here we use a combination of scanning tunnelling microscopy (STM) and STS to study embedded sulphur vacancies in bulk MoS2 crystals. We observe spectroscopic features dispersing in real space and in energy, which we interpret as tip position- and bias-dependent ionization of the sulphur vacancy donor due to tip induced band bending (TIBB). The observations indicate that care must be taken in interpreting defect spectra as reflecting in-gap density of states, and may explain discrepancies in the literature.

cond-mat.mtrl-sci↗

arXiv2023

Imaging the breakdown and restoration of topological protection in magnetic topological insulator MnBi$_2$Te$_4$

Quantum anomalous Hall (QAH) insulators transport charge without resistance along topologically protected chiral one-dimensional edge states. Yet, in magnetic topological insulators (MTI) to date, topological protection is far from robust, with the zero-magnetic field QAH effect only realised at temperatures an order of magnitude below the N\'eel temperature TN, though small magnetic fields can stabilize QAH effect. Understanding why topological protection breaks down is therefore essential to realising QAH effect at higher temperatures. Here we use a scanning tunnelling microscope to directly map the size of the exchange gap (Eg,ex) and its spatial fluctuation in the QAH insulator 5-layer MnBi$_2$Te$_4$. We observe long-range fluctuations of Eg,ex with values ranging between 0 (gapless) and 70 meV, uncorrelated to individual point defects. We directly image the breakdown of topological protection, showing that the chiral edge state, the hallmark signature of a QAH insulator, hybridizes with extended gapless metallic regions in the bulk. Finally, we unambiguously demonstrate that the gapless regions originate in magnetic disorder, by demonstrating that a small magnetic field restores Eg,ex in these regions, explaining the recovery of topological protection in magnetic fields. Our results indicate that overcoming magnetic disorder is key to exploiting the unique properties of QAH insulators.

cond-mat.mtrl-sci↗

arXiv2022

Khovanov multicurves are linear

In previous work we introduced a Khovanov multicurve invariant $\operatorname{\widetilde{Kh}}$ associated with Conway tangles. Applying ideas from homological mirror symmetry we show that $\operatorname{\widetilde{Kh}}$ is subject to strong geography restrictions: Every component of the invariant is linear, in the sense that it admits a lift to a curve homotopic to a straight line in an appropriate planar cover of the tangle boundary.

math.GT↗

arXiv2021

Cosmetic operations and Khovanov multicurves

We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$ detect if a Conway tangle is split.

math.GT↗

arXiv2021

Thin links and Conway spheres

When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $\delta$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname{HFT}$ and the Khovanov invariant $\operatorname{\widetilde{Kh}}$ that were developed by the authors in previous works.

math.GT↗

arXiv2021

Khovanov homology and strong inversions

There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author. In particular, we provide a counterexample to [Conjecture 29, arXiv:1311.1085] as well as a refinement of and additional evidence for [Conjecture 28, arXiv:1311.1085].

math.GT↗

arXiv2020

A mnemonic for the Lipshitz-Ozsv\'ath-Thurston correspondence

When $\mathbf{k}$ is a field, type D structures over the algebra $\mathbf{k}[u,v]/(uv)$ are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure over $\mathbf{k}[u,v]/(uv)$, can be viewed as a set of immersed curves. With this observation as a starting point, given a knot $K$ in $S^3$, we realize the immersed curve invariant $\widehat{\mathit{HF}}(S^3 \smallsetminus \mathring{\nu}(K))$ [arXiv:1604.03466] by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsv\'ath, and Thurston [arXiv:0810.0687] calculating the bordered invariant of $S^3 \smallsetminus \mathring{\nu}(K)$ in terms of the knot Floer homology of $K$.

math.GT↗

arXiv2020

Khovanov invariants via Fukaya categories: the tangle invariants agree

Given a pointed 4-ended tangle $T \subset D^3$, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere $(S^2,4\text{pt})=\partial (D^3, T)$. We prove that these two invariants are the same.

math.GT↗

arXiv2019

Immersed curves in Khovanov homology

We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}(T)$ to define two immersed curve invariants $\widetilde{\operatorname{Kh}}(T)$ and $\operatorname{Kh}(T)$, which satisfy similar gluing theorems that recover reduced and unreduced Khovanov homology of links, respectively. As a first application, we prove that Conway mutation preserves reduced Bar-Natan homology over the field with two elements and Rasmussen's $s$-invariant over any field. As a second application, we give a geometric interpretation of Rozansky's categorification of the two-stranded Jones-Wenzl projector. This allows us to define a module structure on reduced Bar-Natan and Khovanov homologies of infinitely twisted knots, generalizing a result by Benheddi.

math.GT↗

arXiv2019

L-spaces, taut foliations, and graph manifolds

If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $π_1(Y)$ is not left-orderable.

math.GT↗

arXiv2019

Cabling in terms of immersed curves

In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formula for the behaviour of these immersed curves under cabling.

math.GT↗

arXiv2019

A refinement of Khovanov homology

We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of filtrations. We focus primarily on strongly invertible knots and show, for instance, that this refinement is able to detect mutation.

math.GT↗

arXiv2018

Heegaard Floer homology for manifolds with torus boundary: properties and examples

This is a companion paper to earlier work of the authors, which interprets the Heegaard Floer homology for a manifold with torus boundary in terms of immersed curves in a punctured torus. We prove a variety of properties of this invariant, paying particular attention to its relation to knot Floer homology, the Thurston norm, and the Turaev torsion. We also give a geometric description of the gradings package from bordered Heegaard Floer homology and establish a symmetry under spin$^c$ conjugation; this symmetry gives rise to genus one mutation invariance in Heegaard Floer homology for closed three-manifolds. Finally, we include more speculative discussions on relationships with Seiberg-Witten theory, Khovanov homology, and $HF^\pm$. Many examples are included.

math.GT↗

arXiv2017

On the geography and botany of knot Floer homology

This note explores two questions: (1) Which bigraded groups arise as the knot Floer homology of a knot in the three-sphere? (2) Given a knot, how many distinct knots share its Floer homology? Regarding the first, we show there exist bigraded groups satisfying all previously known constraints of knot Floer homology which do not arise as the invariant of a knot. This leads to a new constraint for knots admitting lens space surgeries, as well as a proof that the rank of knot Floer homology detects the trefoil knot. For the second, we show that any non-trivial band sum of two unknots gives rise to an infinite family of distinct knots with isomorphic knot Floer homology. We also prove that the fibered knot with identity monodromy is strongly detected by its knot Floer homology, implying that Floer homology solves the word problem for mapping class groups of surfaces with non-empty boundary. Finally, we survey some conjectures and questions and, based on the results described above, formulate some new ones.

math.GT↗