Quantum Matter

Magnetic skyrmions are topological solitons which occur in a large class of ferromagnetic materials and which  are currently attracting much attention, not least because of their potential use for  low-energy magnetic information storage and manipulation. The talk is about an integrable model for magnetic skyrmions, introduced in a recent paper (arxiv:1812.07268) and generalised in  arxiv:1905.06285. The model is based on a geometrical interpretation of the Dzyaloshinskii-Moriya interaction in terms of a non-abelian gauge field.  In the talk will explain the model and the geometry behind its solution, and discuss solutions and their applications.

Monolayer WTe2, an inversion-symmetric transition metal dichalcogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we show that generally a superconducting inversion-symmetric quantum spin Hall material whose normal state is ``effectively gapped", such as gated monolayer WTe2, can be an inversion-protected topological crystalline superconductor featuring ``higher-order topology" if the superconductivity is parity-odd. We explicitly demonstrate how the bulk-boundary correspondence naturally emerges in such type of superconductors within a two-dimensional minimal model. We then study the pairing symmetry of superconducting WTe2 with a microscopic model, and find two types of self-consistently obtained exotic pairings. First is an odd-parity pairing that possesses a nontrivial bulk symmetry indicator and hosts two Majorana Kramers pairs localizing at opposite corners. Even when the conventional pairing is energetically favored, we find that an intermediate in-plane field exceeding the Pauli limit stabilizes an equal-spin pairing aligning with the field. Our findings suggest gated monolayer WTe2 is a playground for exotic odd-parity superconductivity, and possibly the first material realization for inversion-protected Majorana corner modes without utilizing proximity effect. 

Topological crystalline states are short-range entangled states jointly protected by onsite and crystalline symmetries. While the non-interacting limit of these states, e.g., the topological crystalline insulators, have been intensively studied in band theory and have been experimentally discovered, the classification and diagnosis of their strongly interacting counterparts are relatively less well understood. Here we present a unified scheme for constructing all topological crystalline states, bosonic and fermionic, free and interacting, from real-space "building blocks" and "connectors". Building blocks are finite-size pieces of lower dimensional topological states protected by onsite symmetries alone, and connectors are "glue" that complete the open edges shared by two or multiple pieces of building blocks. The resulted assemblies are selected against two physical criteria we call the "no-open-edge condition" and the "bubble equivalence", which, respectively, ensure that each selected assembly is gapped in the bulk and cannot be deformed to a product state. The scheme is then applied to obtaining the full classification of bosonic topological crystalline states protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our real-space recipes give the complete set of topological crystalline states for bosons and fermions, and prove the boson case analytically using a spectral sequence expansion of group cohomology.

We present a paradigm for effective descriptions of quantum magnets. Typically, a magnet has many classical ground states — configurations of spins (as classical vectors) that have the least energy. The set of all such ground states forms an abstract space. Remarkably, the low energy physics of the quantum magnet maps to that of a single particle moving in this space.

This presents an elegant route to simulate simple quantum mechanical models using molecular magnets. For instance, a dimer coupled by an XY bond maps to a particle moving on a ring. An XY triangular magnet maps to a particle moving on two disjoint rings. We can even simulate Berry phases; when the spin has half-integer values, the particle sees a pi-flux  threaded through the rings.

A particularly interesting example is the XY tetrahedral magnet. Here, the ground state space is a 'non-manifold' due to singularities. These singularities behave like strong impurities to create bound states. The entire low energy physics of the magnet is dominated by these bound states. We call this phenomenon 'order by singularity’. This leads to a preference for certain classical ground states purely due to topology, rather than due to thermal or quantum fluctuations. Unlike order-by-disorder, this effect persists even in the classical limit.

The Wilson formulation of lattice gauge theories provides a first-principles study of many properties of strongly interacting theories, such as quantum chromodynamics (QCD). Certain other properties, such as real-time dynamics, pose insurmountable challenges in this paradigm. Quantum Link Models are generalized lattice gauge theories, which not only offer novel approaches to study dynamics of gauge theories with quantum simulators, but also connect to
gauge theory descriptions of certain condensed matter systems (such as frustrated magnetism). In this talk, we explore some novel properties of gauge theories using quantum link models on classical computers, and explain how to realize them on quantum simulators.  

In this talk, I will focus on topological aspects and edge states of a spin system on a Kagome lattice. with the anisotropic XXZ and Dzyaloshinskii-Moriya interaction (DMI). I will begin with the rich phase diagram in the classical limit arising as a result of the interplay of the two interaction strengths, followed by a spin-wave analysis in some of these phases. These spin-waves (or magnons) are studied using the Holstein-Primakoff transformations.   In the  ferromagnetic phase in which all the spins point along the +z or -z direction the bulk bands are separated from each other by finite energy gaps. Finding the Chern numbers here one finds that, there are four topologically distinct phases sharing the same ground state spin-configuration.  Hence an infinite strip of the system hosts robust edge states which are directly related to the Chern number of the bands. The other phases also are found to have edge modes. However, these are not topologically protected because of the gapless nature of the energy dispersion. 

We uncover a rich phenomenology of the self-organized honeycomb network superstructure of one-dimensional metals in a nearly-commensurate charge-density wave 1T-TaS${}_2$, which may play a significant role in understanding global topology of phase diagrams and superconductivity. The key observation is that the emergent honeycomb network magically supports a cascade of flat bands, whose unusual stability we thoroughly investigate. Furthermore, by combining the weak-coupling mean-field and strong-coupling approaches, we argue that the superconductivity will be strongly enhanced in the network. This provides a natural cooperative mechanism of the charge order and superconductivity, which coexist side-by-side in the 1T-TaS${}_2$. Not only explaining the superconductivity, we show that abundant topological band structures including several symmetry-protected band crossings and corner states, which are closely related to that of the higher-order topology, appear. The results reported here can be generically applicable to various other systems with similar network superstructures.

Self-learning Monte Carlo (SLMC) method is a general-purpose numerical method to simulate many-body systems. SLMC can efficiently cure the critical slowing down in both bosonic and fermionic systems. Moreover, for fermionic systems, SLMC can generally reduce the computational complexity and speed up simulations even away from the critical points. For example, SLMC is more than 1000 times faster than the conventional method for the double exchange model in 8*8*8 cubic lattice. In addition, SLMC also provides a general framework to naturally integrate the advanced machine learning techniques into Monte Carlo. In this talk, I will give an introduction about the background, basic idea and the design principle of SLMC. Later, I will explicitly show how to use SLMC and its great accelerations in classic systems, free fermions coupled with classical spins systems, and interacting fermion systems. At the end, I will talk our recent developed a new type of neural network, which is motivated by the mean field theory and could be used to represent many physical Hamiltonians. We believe the new structural network would be highly efficient to identify different phases and accelerate the numerical simulations for even complex models.

The appearance of scale invariance and diverging response functions in many-body systems is inseparably linked to the presence of a critical point and spontaneous symmetry breaking. In thermal equilibrium critical points mostly correspond to isolated spots in parameter space, which require rather strong fine tuning of e.g., the temperature of magnetic fields, in order to be reached. Pushing systems away from thermal equilibrium, e.g., by exposing them to external drive fields or dissipation, can give rise to more unconventional forms of criticality. External driving and dissipation may even turn critical points robust and attractive, imposing scale invariance for a large variety of external parameters, a phenomenon known as self-organized criticality (SOC).

I will discuss the manifestation of SOC in driven dissipative cold atom ensembles, for which a direct mapping of their microscopic parameters to an SOC field theory exists. This yields an opportunity to study SOC, typically associated with hard-to-control, large scale non-equilibrium setups, such as earthquakes, solar flares, or neural networks, under very controlled conditions. In addition, I will briefly explain the theoretical challenges for a quantum theory of self-organized criticality and point out current steps towards their realization.

Fracton order is a new kind of phase of matter which is similar to topological order, except its excitations have mobility constraints. The excitations are bound to various n-dimensional surfaces with exotic fusion rules that determine how excitations on intersecting surfaces can combine.

This talk will focus on one of the simplest fracton models: the X-cube model. I will explain how this fracton model can be thought of as starting from a 3D toric code and stacks of 2D toric code layers, and then condensing certain excitations on the 2D layers. This is the first TQFT picture of the X-cube model as a 3D order with non-invertible 2D "defects". When we zoom out so that the layers appear close together, I'll show that the model can be described by what I call a foliated field theory. I define a foliated field theory to be a field theory that couples to a foliation field, which describes the geometry of the foliating layers (in analogy to a metic which describes the geometry of spacetime). In this case, the foliated field theory takes the form of a 3+1D BF theory which is coupled to stacks of 2+1 BF theories, with stacking structure described by the foliation field.

This talk is based on arXiv:1812.01613 and another forthcoming work.

Using Monte Carlo methods we explore how well does the recent proposal for computing conformal dimensions, using a large charge expansion, work. We focus on the O(2) and the O(4) Wilson-Fisher fixed points as test cases. Since the traditional Monte Carlo approach suffers from a severe signal-to-noise ratio problem in the large charge sectors, we use worldline formulations that eliminate such problems. In particular we argue that the O(4) model can be simplified drastically by studying what we refer to as a "qubit" formulation. Such simpler  formulations of quantum field theories have become interesting recently from the perspective of quantum computing. Using our studies we confirm that the conformal dimensions of both conformal field theories with O(2) and O(4) symmetries obey a simple formula predicted by the large charge expansion. We also compute the two leading universal low energy constants in both cases , that play an important role in the large charge expansion.

We discover an infinite hierarchical web of the products of supersymmetric generators sustained by the superconformal Virasoro algebra. This hierarchy structure forms the mathematical foundation underpinning the explicit derivation of the character for the self-symmetric Ramond highest weight $c/24$. To consistently fit these exact results into the modular-invariant torus partition function, we advocate a necessary augmentation of the representation theory in the original Friedan--Qiu--Shenker construction via symmetrizing the ground-state manifold associated with the $c/24$ Verma module. Under the newly-proposed scheme, we invoke a quantum-interference mechanism between the two independent Ishibashi states to construct the boundary Cardy states for the whole family of the $\mathcal{N}=1$ superconformal minimal series, based on which the extra fusion channels are unveiled through the obtained Verlinde formula. Our work thus provides the first complete solution to this thirty-year-old question.