String-like Excitations Research Articles

Phonons in stringlet-land and the boson peak

Solid materials that deviate from the harmonic crystal paradigm exhibit characteristic anomalies in the specific heat and vibrational density of states (VDOS) with respect to Debye's theory predictions. The boson peak (BP), a low-frequency excess in the VDOS over Debye lawg(ω)∝ω2, is certainly the most famous among them; nevertheless, its origin is still subject of fierce debate. Recent simulation works provided strong evidence that localized one-dimensional string-like excitations (stringlets) might be the microscopic origin of the BP. In this work, we study the dynamics of acoustic phonons interacting with a bath of vibrating 1D stringlets with exponentially distributed size, as observed in simulations. We show that stringlets strongly renormalize the phonon propagator and naturally induce a BP anomaly in the VDOS, corresponding to the emergence of a dispersionless BP flat mode. Additionally, phonon-stringlet interactions produce a strong enhancement of sound attenuation and a dip in the speed of sound near the BP frequency, consistent with experimental and simulation data. The qualitative trends of the BP frequency and intensity are predicted within the model and shown to be in good agreement with previous observations. In summary, our results substantiate with a simple theoretical model the recent simulation results by Hu and Tanaka claiming the origin of the BP from stringlet dynamics.

Stringlet excitation model of the boson peak.

The boson peak (BP), a low-energy excess in the vibrational density of states over the Debye contribution, is often identified as a characteristic of amorphous solid materials. Despite decades of efforts, its microscopic origin still remains a mystery. Recently, it has been proposed, and corroborated with simulations, that the BP might stem from intrinsic localized modes involving one-dimensional (1D) string-like excitations ("stringlets"). We build on a theory originally proposed by Lund that describes the localized modes as 1D vibrating strings, but we specify the stringlet size distribution to be exponential, as observed in simulations. We provide an analytical prediction for the BP frequency ωBP in the temperature regime well below the observed glass transition temperature Tg. The prediction involves no free parameters and accords quantitatively with prior simulation observations in 2D and 3D model glasses based on inverse power law potentials. The comparison of the string model to observations is more uncertain when compared to simulations of an Al-Sm metallic glass material at temperatures well above Tg. Nonetheless, our stringlet model of the BP naturally reproduces the softening of the BP frequency upon heating and offers an analytical explanation for the experimentally observed scaling with the shear modulus in the glass state and changes in this scaling in simulations of glass-forming liquids. Finally, the theoretical analysis highlights the existence of a strong damping for the stringlet modes above Tg, which leads to a large low-frequency contribution to the 3D vibrational density of states, observed in both experiments and simulations.

Gapped boundaries of (3+1) -dimensional topological order

Given a gapped boundary of a $(3+1)$-dimensional [$(3+1)\mathrm{D}$] topological order (TO), one can stack on it a decoupled $(2+1)\mathrm{D}$ TO to get another boundary theory. Should one view these two boundaries as ``different''? A natural choice would be no. Different classes of gapped boundaries of $(3+1)\mathrm{D}$ TO should be defined modulo the decoupled $(2+1)\mathrm{D}$ TOs. But is this enough? We examine the possibility of coupling the boundary of a $(3+1)\mathrm{D}$ TO to additional $(2+1)\mathrm{D}$ TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk pointlike excitations, when touching the boundary, become excitations in the added $(2+1)\mathrm{D}$ phase, while the stringlike excitations in the bulk may end on the boundary but with end points dressed by some other excitations in the $(2+1)\mathrm{D}$ phase. For a good definition of ``class'' for gapped boundaries of $(3+1)\mathrm{D}$ TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the stringlike excitations that can end on the boundary, whatever their end points are. A concrete example is the $(3+1)\mathrm{D}$ bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary, and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the $m$-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a $(3+1)\mathrm{D}$ TO characterized by a finite group $G$, different classes correspond to different normal subgroups of $G$. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model, and field theory.

Optimally rejuvenated model binary glasses

Using the creation relaxation algorithm developed for the atomistic modeling of the high-dose irradiation limit of crystalline systems, we explore the limits of the structural rejuvenation of a highly excited model binary glass. This high-energy athermal amorphous structure exhibits a direct transition to homogeneous plastic flow and a microstructure that is largely insensitive to this flow, being characterized by a porous system-spanning network of minimally frustrated structural motifs. The observed homogeneous plasticity is mediated by the same string-like structural excitations, which mediate structural relaxation and microplasticity at finite temperature in more relaxed structures. This highly rejuvenated structural asymptote is not far from the structural state of regions, which have experienced athermal shear localization in more relaxed samples, suggesting an optimally rejuvenated glassy structure will always be limited by that produced by shear localization.

Crossing with the circle in Dijkgraaf–Witten theory and applications to topological phases of matter

Given a fully extended topological quantum field theory, the “crossing with the circle” conditions establish that the dimension, or categorification thereof, of the quantum invariant assigned to a closed k-manifold Σ is equivalent to that assigned to the (k + 1)-manifold Σ×S1. We compute in this paper these conditions for the 4-3-2-1 Dijkgraaf–Witten theory. In the context of the lattice Hamiltonian realization of the theory, the quantum invariants assigned to the circle and the torus encode the defect open string-like and bulk loop-like excitations, respectively. The corresponding “crossing with the circle” condition, thus, formalizes the process by which loop-like excitations are formed out of string-like ones. Exploiting this result, we revisit the statement that loop-like excitations define representations of the linear necklace group as well as the loop braid group.

Vortex structure of excitation fields in a supercooled glass-forming liquid and its relationship with relaxations

In the framework of the dynamic facilitation model, a glass former is structured with local cooperative excitations in space and time when approaching the glass transition, which is closely related to dynamic relaxations. We investigate the structure of excitation fields in supercooled ${\mathrm{Cu}}_{50}{\mathrm{Zr}}_{50}$ glass former, and reveal that the excitation fields feature the vortexlike structure beyond the well-known stringlike excitations. The vortex is developed from the stringlike excitations and frequently annihilates via the interactions between vortex and antivortex. The region favored by vortices exhibits percolationlike morphology. We find that the probability distributions of vortex core number are characterized by the discrete multiple mode, which is very analogous to the quantized mode of the slow-\ensuremath{\beta} type relaxation. Furthermore, the relaxation time derived from the formation rate of vortices as well as the corresponding activation energy coincide with the characteristic values of the slow-\ensuremath{\beta} relaxation in metallic glasses. Therefore, it is concluded that the vortex excitation is the origin of the slow-\ensuremath{\beta} relaxation in dynamic space. We further find that the activation energy and relaxation time associated with individual excitations agree well with the Johari-Goldstein \ensuremath{\beta} relaxation. It manifests a more intrinsic structural sign of the Johari-Goldstein \ensuremath{\beta} relaxation than the interpretation based on stringlike excitations.

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic string-like excitations that terminate at gapped boundaries. Using a tube algebra approach, we classify such excitations and derive the corresponding representation theory. Via a dimensional reduction argument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations at interfaces between two gapped boundaries. Such point-like excitations are well known to be encoded into a bicategory of module categories over the input fusion category. Exploiting this correspondence, we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showing that it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. In the process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebra objects over this input bicategory.

Excitations of elementary fermions in gauge Higgs theories

Static quark-antiquark states in QCD, at finite quark separation, have a spectrum of metastable states corresponding to string-like excitations of the gauge field. In this article I suggest that there may also exist an excitation spectrum of heavy fermions in some gauge Higgs theories deep in the Higgs phase. In this situation there are no color electric flux tubes connecting quarks with antiquarks. There may, nonetheless, exist stable excitations of the bosonic fields surrounding an isolated fermion, below the particle production threshold. I present numerical evidence indicating the existence of such excitations in an SU(3) gauge Higgs theory, with the scalar field in the fundamental representation of the gauge group.

Fractons from Vector Gauge Theory.

Motivated by the prediction of fractonic topological defects in a quantum crystal, we utilize a reformulated elasticity duality to derive a description of a fracton phase in terms of coupled vector U(1) gauge theories. The fracton order and restricted mobility emerge as a result of an unusual Gauss law where electric field lines of one gauge field act as sources of charge for others. At low energies this vector gauge theory reduces to the previously studied fractonic symmetric tensor gauge theory. We construct the corresponding lattice model and a number of generalizations, which realize fracton phases via a condensation of stringlike excitations built out of charged particles, analogous to the p-string condensation mechanism of the gapped X-cube fracton phase.

Polarons leave a trace.

Spin and charge interplay leads to stringlike excitations in the 2D Hubbard model

Classification of 3+1D Bosonic Topological Orders (II): The Case When Some Pointlike Excitations Are Fermions

In this paper, we classify EF topological orders for 3+1D bosonic systems where some emergent pointlike excitations are fermions. (1) We argue that all 3+1D bosonic topological orders have gappable boundary. (2) All the pointlike excitations in EF topological orders are described by the representations of $G_f=Z_2^f\leftthreetimes_{e_2} G_b$ -- a $Z_2^f$ central extension of a finite group $G_b$ characterized by $e_2\in H^2(G_b,Z_2)$. (3) We find that the EF topological orders are classified by 2+1D anomalous topological orders $\mathcal{A}_b^3$ on their unique canonical boundary. Here $\mathcal{A}_b^3$ is a unitary fusion 2-category with simple objects labeled by $\hat G_b=Z_2^m\leftthreetimes G_b$. $\mathcal{A}_b^3$ also has one invertible fermionic 1-morphism for each object as well as quantum-dimension-$\sqrt 2$ 1-morphisms that connect two objects $g$ and $gm$, where $g\in \hat G_b$ and $m$ is the generator of $Z_2^m$. (4) When $\hat G_b$ is the trivial $Z_2^m$ extension, the EF topological orders are called EF1 topological orders, which is classified by simple data $(G_b,e_2,n_3,\nu_4)$. (5) When $\hat G_b$ is a non-trivial $Z_2^m$ extension, the EF topological orders are called EF2 topological orders, where some intersections of three stringlike excitations must carry Majorana zero modes. (6) Every EF2 topological order with $G_f=Z_2^f\leftthreetimes G_b$ can be associated with a EF1 topological order with $G_f=Z_2^f\leftthreetimes \hat G_b$. (7) We find that all EF topological orders correspond to gauged 3+1D fermionic symmetry protected topological (SPT) orders with a finite unitary symmetry group. (8) We further propose that the general classification of 3+1D topological orders with finite unitary symmetries for bosonic and fermionic systems can be obtained by gauging or partially gauging the finite symmetry group of 3+1D SPT phases of bosonic and fermionic systems.

Stable quantum memories with limited measurement

We demonstrate the existence of a finite temperature threshold for a 1D stabilizer code under an error correcting protocol that requires only a fraction of the syndrome measurements. Below the threshold temperature, encoded states have exponentially long lifetimes, as demonstrated by numerical and analytical arguments. We sketch how this algorithm generalizes to higher dimensional stabilizer codes with string-like excitations, like the toric code.

Entanglement entropy for (3+1)-dimensional topological order with excitations

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $\ln d_i$ to the entanglement entropy, where $d_i$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.

Exactly soluble local bosonic cocycle models, statistical transmutation, and simplest time-reversal symmetric topological orders in 3+1 dimensions

We propose a generic construction of exactly soluble \emph{local bosonic models} that realize various topological orders with gappable boundaries. In particular, we construct an exactly soluble bosonic model that realizes a 3+1D $Z_2$ gauge theory with emergent fermionic Kramer doublet. We show that the emergence of such a fermion will cause the nucleation of certain topological excitations in space-time without pin$^+$ structure. The exactly soluble model also leads to a statistical transmutation in 3+1D. In addition, we construct exactly soluble bosonic models that realize 2 types of time-reversal symmetry enriched $Z_2$-topological orders in 2+1D, and 20 types of simplest time-reversal symmetry enriched topological (SET) orders which have only one non-trivial point-like and string-like topological excitations. Many physical properties of those topological states are calculated using the exactly soluble models. We find that some time-reversal SET orders have point-like excitations that carry Kramer doublet -- a fractionalized time-reversal symmetry. We also find that some $Z_2$ SET orders have string-like excitations that carry anomalous (non-on-site) $Z_2$ symmetry, which can be viewed as a fractionalization of $Z_2$ symmetry on strings. Our construction is based on cochains and cocycles in algebraic topology, which is very versatile. In principle, it can also realize emergent topological field theory beyond the twisted gauge theory.

On the Defect Group of a 6D SCFT

We use the F-theory realization of 6D superconformal field theories (SCFTs) to study the corresponding spectrum of stringlike, i.e. surface defects. On the tensor branch, all of the stringlike excitations pick up a finite tension, and there is a corresponding lattice of string charges, as well as a dual lattice of charges for the surface defects. The defect group is data intrinsic to the SCFT and measures the surface defect charges which are not screened by dynamical strings. When non-trivial, it indicates that the associated theory has a partition vector rather than a partition function. We compute the defect group for all known 6D SCFTs, and find that it is just the abelianization of the discrete subgroup of U(2) which appears in the classification of 6D SCFTs realized in F-theory. We also explain how the defect group specifies defining data in the compactification of a (1,0) SCFT.

Dynamics of Dirac strings and monopolelike excitations in chiral magnets under a current drive

Skyrmion lines in metallic chiral magnets carry an emergent magnetic field experienced by the conduction electrons. The inflow and outflow of this field across a closed surface is not necessarily equal, thus it allows for the existence of emergent monopoles. One example is a segment of skyrmion line inside a crystal, where a monopole and antimonopole pair is connected by the emergent magnetic flux line. This is a realization of Dirac string-like excitations. Here we study the dynamics of monopoles in chiral magnets under an electric current. We show that in the process of creation of skyrmion lines, skyrmion line segments are first created via the proliferation of monopoles and antimonopoles. Then these line segments join and span the whole system through the annihilation of monopoles. The skyrmion lines are destroyed via the proliferation of monopoles and antimonopoles at high currents, resulting in a chiral liquid phase. We also propose to create the monopoles in a controlled way by applying an inhomogenous current to a crystal. Remarkably, an electric field component in the magnetic field direction proportional to the current squared in the low current region is induced by the motion of distorted skyrmion lines, in addition to the Hall and longitudinal voltage. The existence of monopoles can be inferred from transport or imaging measurements.

Some recent progress on magnetic effects for non-Abelian vortices in color-flavor-locked quark matter

It is recently known that strong magnetic fields may lead to highly nontrivial effects on strongly interacting matter,including the color-flavor-locked phase of dense quark matter. Dense quark matter provides various kind of topological solitons such as vortices,domain walls,monopoles,kinks,boojums etc.,it is hence natural to expect that the magnetic field can also affect these topological solitons. In this mini-review,we update our recent progress on the studies of magnetic field effects in the color-flavor-locked quark matter. The particular emphasis is given on the non-Abelian vortices since they are the most fundamental string-like topological excitations under certain circumstance. Their relevant properties( e. g.,the profile function and the tension energy) and the magnetic-field dependence are investigated with the Ginzburg-Landau framework.

Low-energy electrodynamics of novel spin excitations in the quantum spin ice Yb2Ti2O7

In condensed matter systems, formation of long-range order (LRO) is often accompanied by new excitations. However, in many geometrically frustrated magnetic systems, conventional LRO is suppressed, while non-trivial spin correlations are still observed. A natural question to ask is then what is the nature of the excitations in this highly correlated state without broken symmetry? Frequently, applying a symmetry breaking field stabilizes excitations whose properties reflect certain aspects of the anomalous state without LRO. Here we report a THz spectroscopy study of novel excitations in quantum spin ice Yb2Ti2O7 under a <001> directed magnetic field. At large positive fields, both right- and left-handed magnon and two-magnon-like excitations are observed. The g-factors of these excitations are dramatically enhanced in the low-field limit, showing a crossover of these states into features consistent with the quantum string-like excitations proposed to exist in quantum spin ice in small <001> fields.

Covariant calculus for effective string theories

A covariant calculus for the construction of effective string theories is developed. Effective string theory, describing quantum string-like excitations in arbitrary dimension, has in the past been constructed using the principles of conformal field theory, but not in a systematic way. Using the freedom of choice of field definition, a particular field definition is made in a systematic way to allow an explicit construction of effective string theories with manifest exact conformal symmetry. The impossibility of a manifestly invariant description of the Polchinski–Strominger Lagrangian is demonstrated and its meaning is explained.

Vortices and other topological solitons in dense quark matter

Abstract Dense quantum chromodynamic matter accommodates various kind of topological solitons such as vortices, domain walls, monopoles, kinks, boojums, and so on. In this review, we discuss various properties of topological solitons in dense quantum chromodynamics (QCD) and their phenomenological implications. Particular emphasis is placed on the topological solitons in the color–flavor-locked (CFL) phase, which exhibits both superfluidity and superconductivity. The properties of topological solitons are discussed in terms of effective field theories such as the Ginzburg–Landau theory, the chiral Lagrangian, or the Bogoliubov–de Gennes equation. The most fundamental string-like topological excitations in the CFL phase are non-Abelian vortices, which are 1/3 quantized superfluid vortices and color magnetic flux tubes. These vortices are created at a phase transition by the Kibble–Zurek mechanism or when the CFL phase is realized in compact stars, which rotate rapidly. The interaction between vortices is found to be repulsive and consequently a vortex lattice is formed in rotating CFL matter. Bosonic and fermionic zero-energy modes are trapped in the core of a non-Abelian vortex and propagate along it as gapless excitations. The former consists of translational zero modes (a Kelvin mode) with a quadratic dispersion and ${\mathbb {C}}P^2$ Nambu–Goldstone gapless modes with a linear dispersion, associated with the CFL symmetry spontaneously broken in the core of a vortex, while the latter is Majorana fermion zero modes belonging to the triplet of the symmetry remaining in the core of a vortex. The low-energy effective theory of the bosonic zero modes is constructed as a non-relativistic free complex scalar field and a relativistic ${\mathbb {C}}P^2$ model in 1+1 dimensions. The effects of strange quark mass, electromagnetic interactions, and non-perturbative quantum corrections are taken into account in the ${\mathbb {C}}P^2$ effective theory. Various topological objects associated with non-Abelian vortices are studied; colorful boojums at the CFL interface, the quantum color magnetic monopole confined by vortices, which supports the notion of quark–hadron duality, and Yang–Mills instantons inside a non-Abelian vortex as lumps are discussed. The interactions between a non-Abelian vortex and quasiparticles such as phonons, gluons, mesons, and photons are studied. As a consequence of the interaction with photons, a vortex lattice behaves as a cosmic polarizer. As a remarkable consequence of Majorana fermion zero modes, non-Abelian vortices are shown to behave as a novel kind of non-Abelian anyon. In the order parameters of chiral symmetry breaking, we discuss fractional and integer axial domain walls, Abelian and non-Abelian axial vortices, axial wall–vortex composites, and Skyrmions.