String-net Research Articles

Emergent gauge symmetries: making symmetry as well as breaking it.

Gauge symmetries play an essential role in determining the interactions of particle physics. Where do they come from? Might the gauge symmetries of the Standard Model unify in the ultraviolet or might they be emergent in the infrared, below some large scale close to the Planck scale? Emergent gauge symmetries are important in quantum many-body systems in quantum phases associated with long range entanglement and topological order, e.g. they arise in high temperature superconductors, with string-net condensation and in the A-phase of superfluid 3He. String-nets and superfluid 3He exhibit emergent properties similar to the building blocks of particle physics. Emergent gauge symmetries also play an important role in simulations of quantum field theories. This article discusses recent thinking on possible emergent gauge symmetries in particle physics, commenting also on Higgs phenomena and the vacuum energy or cosmological constant puzzle in emergent gauge systems. This article is part of the theme issue 'Quantum technologies in particle physics'.

Characterizing Topological Order with Matrix Product Operators

One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries of the local tensors that encode the wavefunction. In their most general form, these symmetries are represented by matrix product operators acting on the virtual level, which leads to a set of algebraic rules characterizing states with topological quantum order. This construction generalizes the concepts of {mathsf {G}}- and twisted injectivity; the corresponding matrix product operators encode all topological features of the theory and provide a complete picture of the ground state manifold on the torus. We show how the string-net models of Levin and Wen fit within this formalism and in doing so provide a particularly intuitive interpretation of the pentagon equation for F-symbols as the pulling of matrix product operators through the string-net tensor network. Our approach paves the way to finding novel topological phases beyond string nets and elucidates the description of topological phases in terms of entanglement Hamiltonians and edge theories.

Generalized string-net model for unitary fusion categories without tetrahedral symmetry

The Levin-Wen model of string-net condensation explains how topological phases emerge from the microscopic degrees of freedom of a physical system. However, the original construction is not applicable to all unitary fusion category since some additional symmetries for the $F$-symbols are imposed. In particular, the so-called tetrahedral symmetry is not fulfilled by many interesting unitary fusion categories. In this paper, we present a generalized construction of the Levin-Wen model for arbitrary multiplicity-free unitary fusion categories that works without requiring these additional symmetries. We explicitly calculate the matrix elements of the Hamiltonian and, furthermore, show that it has the same properties as the original one.

Tensor-network approach to phase transitions in string-net models

We use a recently proposed class of tensor-network states to study phase transitions in string-net models. These states encode the genuine features of the string-net condensate such as, e.g., a nontrivial perimeter law for Wilson loops expectation values, and a natural order parameter detecting the breakdown of the topological phase. In the presence of a string tension, a quantum phase transition occurs between the topological phase and a trivial phase. We benchmark our approach for $\mathbb{Z}_2$ string nets and capture the second-order phase transition which is well known from the exact mapping onto the transverse-field Ising model. More interestingly, for Fibonacci string nets, we obtain first-order transitions in contrast with previous studies but in qualitative agreement with mean-field results.

Neutrophil Extracellular Traps Are an Intrinsic Feature in Sickle Cell Disease

Polymerisation of deoxygenated-HbS in sickle cell disease (SCD) leads to alteration of the shape and function of the red blood cells (RBC), resulting in hemolytic anemia and recurrent episodes of micro-vascular occlusion, triggering a cascade of downstream events and inflammation that underlies the progressive multi-systemic organ damage. Neutrophils have a critical role in promoting sickle pathophysiology; patients with higher leukocyte count, particularly neutrophils, tend to have a higher risk for more severe manifestations as well as earlier mortality. SCD neutrophils display a specific overreactive phenotype, including increased adhesive properties and production of reactive oxygen species. Activated neutrophils can also produce extracellular traps (NETs) in response to pathogens or other inflammatory stimuli. Studies have shown that prolonged exposure of healthy neutrophils to SCD plasma leads to formation of mature string-like NETs. These strings of decondensed chromatin covered with elastase (NE) or myeloperoxidase (MPO) might contribute to SCD pathophysiology. Here, we investigated if SCD neutrophils had enhanced intrinsic potential for NETs formation (i.e. could form NETs in the absence of a stimulus).We evaluated NETosis both indirectly, in which neutrophils isolated from healthy subjects were incubated with plasma from HbSS patients and directly, in which neutrophils isolated from HbSS patients were compared with those isolated from healthy donors. 17 HbSS patients in steady-state (all on hydroxyurea) and 11 race and sex matched healthy donors were included in the direct assays; the NETosis-related events were observed over a period of 7 minutes to 4 hours. For the indirect assay, we used plasma from 19 HbSS patients (14 on hydroxyurea, 5 off hydroxyurea and/or transfused), all of whom provided samples in steady-state and acute pain crisis (paired samples); the healthy neutrophils were treated with SCD plasma for up to 7 hours. A pain crisis was defined as an episode of acute pain that has no evident cause other than SCD, resulting in hospitalization and treatment with parenteral opioids. Steady state was defined as the period from at any time 8 weeks prior to or after a crisis. Patients were excluded when they were <18 or >80 years of age, pregnant or had a history of blood transfusion in the previous 8 weeks. The IRB approved the studies under NIH protocol numbers NCT00081523 (SCD subjects) and 03-H-0015 (healthy controls and SCD subjects). The NETosis features (histone citrullination H4cit3, nuclear decondensation, DNA-NE-MPO strings) were assessed by imaging flow cytometry and fluorescence microscopy.In the indirect assays, healthy neutrophils made more string NETs when treated with the SCD plasma (from both steady state and crisis) compared to plasma from healthy donors (Figure 1A), and NETs response was further increased with plasma from acute compared to their paired steady state sample (N=19, p=0.0114) (Figure 1B). Plasma from patients on or off hydroxyurea induced comparable NETosis responses. This confirmed a higher ability for the crisis milieu to modulate NETosis.Directly, neutrophils isolated from SCD patients as compared to those from healthy controls, showed increased number of decondensed H4cit3 positive nuclei, after only 7 min of incubation in RPMI (N=7, p=0.0688). By comparison the response to high concentration (20 uM) of Hemin - a regular product of hemolysis, was similar in the SCD and the healthy neutrophils (p=0.6871) (Figure 1C). This data suggested that the SCD environment might activate neutrophils to produce NETs in the absence of a specific stimulus. Fluorescence microscopy confirmed that neutrophils isolated from the patients did produce significantly more DNA-NE strings without any stimulation compared to healthy neutrophils (N=9, p=0.0079). After Hemin treatment, NETs response in SCD neutrophils was also higher than that of the healthy controls, but not statistically significant (Figure 1D).Our data provide the first direct in vitro evidence that neutrophils in SCD are innately prone to NETosis but whether and how these might contribute to the initiation or progress of SCD vaso-occlusive crises is yet to be determined. [Display omitted] DisclosuresNo relevant conflicts of interest to declare.

Neutrophil extracellular traps are a heterogeneous feature in sickle cell disease

Abstract Sickle cell disease (SCD) is an inherited hemoglobin disorder caused by a single base change (A-T) in the beta-globin gene of hemoglobin with ensuing recurring inflammation that results in progressive end-organ damage and early mortality. Production of neutrophil extracellular traps (NETs) have been shown to be a key component of this inflammatory process in SCD pathophysiology. We sought to determine the relevance of NETosis versus a different type of programmed cell death (apoptosis) in SCD. We used an indirect assay using healthy neutrophils exposed to paired plasma samples (n=16) from SCD patients in steady-state and crisis for 30 min to 7 hrs, and a direct approach of observing neutrophils (3 healthy and 3 SCD) themselves up to 4 hrs. Features of NETosis (histone citrullination, nuclear decondensation, string nets) and apoptosis (piknotic nuclei, c-Parp, membrane damage) were assessed by imaging flow cytometry and fluorescence microscopy. Additionally, plasma titers for 20 inflammatory analytes were determined for the 16 paired SCD samples and 4 healthy controls. High circulating IL-6 levels at crisis found in 9/16 of the paired samples correlated with an increased number of NETs at crisis compared to steady-state. NETs were also observed directly in SCD neutrophils at steady state but not consistently. When NETs or histone citrullination were absent or low, c-Parp was typically detected. Additionally, membrane damage was observed in neutrophils from all SCD donors. The data suggest that the SCD environment disrupts neutrophil homeostasis and activate multiple cell death programs. SCD NETosis appears to be heterogeneous, its contribution to the disease pathophysiology remains to be determined.

Condensation-driven phase transitions in perturbed string nets

We develop methods to probe the excitation spectrum of topological phases of matter in two spatial dimensions. Applying these to the Fibonacci string nets perturbed away from exact solvability, we analyze a topological phase transition driven by the condensation of non-Abelian anyons. Our numerical results illustrate how such phase transitions involve the spontaneous breaking of a topological symmetry, generalizing the traditional Landau paradigm. The main technical tool is the characterization of the ground states using tensor networks and the topological properties using matrix-product-operator symmetries. The topological phase transition manifests itself by symmetry breaking in the entanglement degrees of freedom of the quantum transfer matrix.

Symmetry-enriched string nets: Exactly solvable models for SET phases

We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group $G$ and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group $G$, the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a $\mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.

Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models

We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows to carry over to these string net models the representation-theoretic classification of the excitations in quantum double models, as well as define them in arbitrary lattices, and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalising the map to broader classes of string nets is considered.

Topology-driven quantum phase transitions in time-reversal-invariant anyonic quantum liquids

Indistinguishable particles in two dimensions can be characterized by anyonic quantum statistics, which is more general than that of bosons or fermions. Anyons emerge as quasiparticles in fractional quantum Hall states and in certain frustrated quantum magnets. Quantum liquids of anyons show degenerate ground states, where the degeneracy depends on the topology of the underlying surface. Here, we present a new type of continuous quantum phase transition in such anyonic quantum liquids, which is driven by quantum fluctuations of the topology. The critical state connecting two anyonic liquids on surfaces with different topologies is reminiscent of the notion of a ‘quantum foam’ with fluctuations on all length scales. This exotic quantum phase transition arises in a microscopic model of interacting anyons for which we present an exact solution in a linear geometry. We introduce an intuitive physical picture of this model that unifies string nets and loop gases, and provide a simple description of topological quantum phases and their phase transitions. Quantum many-body systems can show an elusive form of order known as topological order. Theoretical work now unifies several microscopic models whereby topological phases have been found, and predicts quantum phase transitions that are driven by quantum fluctuations of the topology.

From String Nets to Nonabelions

We discuss Hilbert spaces spanned by the set of string nets, i.e. trivalent graphs, on a lattice. We suggest some routes by which such a Hilbert space could be the low-energy subspace of a model of quantum spins on a lattice with short-ranged interactions. We then explain conditions which a Hamiltonian acting on this string net Hilbert space must satisfy in order for the system to be in the DFib (Doubled Fibonacci) topological phase, that is, be described at low energy by an SO(3)3 × SO(3)3 doubled Chern-Simons theory, with the appropriate non-abelian statistics governing the braiding of the low-lying quasiparticle excitations (nonabelions). Using the string net wavefunction, we describe the properties of this phase. Our discussion is informed by mappings of string net wavefunctions to the chromatic polynomial and the Potts model.

Quantum graphity: A model of emergent locality

Quantum graphity is a background-independent model for emergent macroscopic locality, spatial geometry and matter. The states of the system correspond to dynamical graphs on $N$ vertices. At high energy, the graph describing the system is highly connected and the physics is invariant under the full symmetric group acting on the vertices. We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low dimensional and local. Consideration of the free energy associated with the dominant terms in the dynamics shows that this low-energy state is thermodynamically stable under local perturbations. The model can also give rise to an emergent U(1) gauge theory in the ground state by the string-net condensation mechanism of Levin and Wen. We also reformulate the model in graph-theoretic terms and compare its dynamics to some common graph processes.

Detecting Topological Order in a Ground State Wave Function

A large class of topological orders can be understood and classified using the string-net condensation picture. These topological orders can be characterized by a set of data (N, di, F(lmn)(ijk), delta(ijk). We describe a way to detect this kind of topological order using only the ground state wave function. The method involves computing a quantity called the "topological entropy" which directly measures the total quantum dimension D= Sum(id2i).

Quantum ether: Photons and electrons from a rotor model

We give an example of a purely bosonic model -- a rotor model on the 3D cubic lattice -- whose low energy excitations behave like massless U(1) gauge bosons and massless Dirac fermions. This model can be viewed as a ``quantum ether'': a medium that gives rise to both photons and electrons. It illustrates a general mechanism for the emergence of gauge bosons and fermions known as ``string-net condensation.'' Other, more complex, string-net condensed models can have excitations that behave like gluons, quarks and other particles in the standard model. This suggests that photons, electrons and other elementary particles may have a unified origin: string-net condensation in our vacuum.

Emergent U(1) gauge theory with fractionalized boson/fermion from the Bose condensation of excitons in a multiband insulator

Fractionalized phases are studied in a low energy theory of exciton bose condensate in a multi-band insulator. It is shown that U(1) gauge theory with either fractionalized boson or fermion can emerge out of a single model depending on the coupling constants. Both the statistics and spin of the fractionalized particles are dynamically determined, satisfying the spin-statistics theorem in the continuum limit. We present two mutually consistent descriptions for the fractionalization. In the first approach, it is shown that fractionalized degree of freedom emerges from reduced phase space constrained by strong interaction and that the U(1) gauge field arises as a collective excitation of the low energy modes. In the second approach, complimentary descriptions are provided for the fractionalization based on world line picture of the original excitons. The emergent gauge structure is identified from the fluctuating web of exciton world lines which, in turn, realizes the string net condensation in a space-time picture.

Colloquium: Photons and electrons as emergent phenomena

Recent advances in condensed-matter theory have revealed that new and exotic phases of matter can exist in spin models (or more precisely, local bosonic models) via a simple physical mechanism, known as ``string-net condensation.'' These new phases of matter have the unusual property that their collective excitations are gauge bosons and fermions. In some cases, the collective excitations can behave just like the photons, electrons, gluons, and quarks in our vacuum. This suggests that photons, electrons, and other elementary particles may have a unified origin---string-net condensation in our vacuum. In addition, the string-net picture indicates how to make artificial photons, artificial electrons, and artificial quarks and gluons in condensed-matter systems.

An introduction to quantum order, string-net condensation, and emergence of light and fermions

We review some recent work on new states of matter. Those states cannot be described by symmetry breaking and hence contain a new kind of order—quantum order. Some quantum orders are shown to be closely related to string-net condensations. Those quantum orders lead to an emergence of gauge bosons and fermions from pure bosonic models.

String-net condensation: A physical mechanism for topological phases

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases - namely topological phases. These phases occur when the extended objects, called ``string-nets'', become highly fluctuating and condense. We derive exactly soluble Hamiltonians for 2D local bosonic models whose ground states are string-net condensed states. Those ground states correspond to 2D parity invariant topological phases. These models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians - a spin-1/2 system on the honeycomb lattice - is a simple theoretical realization of a fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

From New States of Matter to a Unification of Light and Electrons

For a long time, people believe that all possible states of matter are described by Landau symmetry-breaking theory. Recently we find that string-net condensation provide a mechanism to produce states of matter beyond the symmetry-breaking description. The collective excitations of the string-net condensed states turn out to be our old friends, photons and electrons (and other gauge bosons and fermions). This suggests that our vacuum is a string-net condensed state. Light and electrons in our vacuum have a unified origin -- string-net condensation.

Artificial light and quantum order in systems of screened dipoles

The origin of light is an unsolved mystery in nature. Recently, it was suggested that light may originate from a new kind of order, quantum order. To test this idea in experiments, we study systems of screened magnetic/electric dipoles in two-dimensional (2D) and 3D lattices. We show that our models contain an artificial light--a photonlike collective excitation. We discuss how to design realistic devices that realize our models. We show that the ``speed of light'' and the ``fine-structure constant'' of the artificial light can be tuned in our models. The properties of artificial atoms (bound states of pairs of artificial charges) are also discussed. The existence of artificial light (as well as artificial electron) in condensed-matter systems suggests that elementary particles, such as light and electron, may not be elementary. They may be collective excitations of quantum order in our vacuum. In our model, light is realized as a fluctuation of string-nets and charges as the ends of open strings (or nodes of string nets).