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arXiv:2012.11592 · Physical review. B./Physical review. B

Fractons with twisted boundary conditions and their symmetries

Tom Rudelius, Nathan Seiberg B, Shu-Heng Shao

Published May 7, 2021• 42 citations
arXivHTMLPDF

Abstract

We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the $xy$ plane. The ground-state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground-state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in $2+1$ and $3+1$ dimensional systems. It originates from the underlying subsystem symmetry but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus $\ensuremath{\tau}={\ensuremath{\tau}}_{1}+i{\ensuremath{\tau}}_{2}$ has rational ${\ensuremath{\tau}}_{1}=k/m$. Then, in systems based on $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized ${\mathbb{Z}}_{m}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{m}$, which leads to an $m$-fold ground state degeneracy. In systems based on ${\mathbb{Z}}_{N}$ symmetries, like the X-cube model, each of these two ${\mathbb{Z}}_{m}$ factors is replaced by ${\mathbb{Z}}_{gcd(N,m)}$.

DOI
10.1103/physrevb.103.195113

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We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the xyxyxy plane. The ground-state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground-state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in 2+12+12+1 and 3+13+13+1 dimensional systems. It originates from the underlying subsystem symmetry but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus \ensuremathτ=\ensuremathτ1+i\ensuremathτ2\ensuremath{\tau}={\ensuremath{\tau}}_{1}+i{\ensuremath{\tau}}_{2}\ensuremathτ=\ensuremathτ1​+i\ensuremathτ2​ has rational \ensuremathτ1=k/m{\ensuremath{\tau}}_{1}=k/m\ensuremathτ1​=k/m. Then, in systems based on U(1)\ifmmode×\else\texttimes\fiU(1)\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)U(1)\ifmmode×\else\texttimes\fiU(1) subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized Zm\ifmmode×\else\texttimes\fiZm{\mathbb{Z}}_{m}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{m}Zm​\ifmmode×\else\texttimes\fiZm​, which leads to an mmm-fold ground state degeneracy. In systems based on ZN{\mathbb{Z}}_{N}ZN​ symmetries, like the X-cube model, each of these two Zm{\mathbb{Z}}_{m}Zm​ factors is replaced by Zgcd(N,m){\mathbb{Z}}_{gcd(N,m)}Zgcd(N,m)​.
We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile ``fracton'' excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub ``twisted fracton models.'' A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a 3-torus and computing their ground-state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility.
In the study of three-dimensional gapped models, two-dimensional gapped states should be considered as a free resource. This is the basic idea underlying the notion of ``foliated fracton order'' proposed in Shirley et al. [Phys. Rev. X 8, 031051 (2018)]. We have found that many of the known type-I fracton models, although they appear very different, have the same foliated fracton order, known as ``X-cube'' order. In this paper, we identify three-dimensional fracton models with different kinds of foliated fracton order. Whereas the X-cube order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry, the different orders correspond to twisted versions of the gauge theory for which the system prior to gauging has nontrivial order protected by the planar subsystem symmetry. We present constructions of the twisted models and demonstrate that they possess nontrivial order by studying their fractional excitation contents.
We investigate two-dimensional compactifications of three-dimensional fractonic stabilizer models. We find the two-dimensional topological phases produced as a function of compactification radius for the XXX-cube model and Haah's cubic code. Furthermore, we uncover translation symmetry enrichment in the compactified cubic code that leads to twisted boundary conditions.
Subsystem symmetries are intermediate between global and gauge symmetries. One can treat these symmetries either like global symmetries that act on subregions of a system, or gauge symmetries that act on the regions transverse to the regions acted upon by the symmetry. We show that this latter interpretation can lead to an understanding of global, topology-dependent features in systems with subsystem symmetries. We demonstrate this with an exactly solvable lattice model constructed from a two-dimensional system of bosons coupled to a vector field with a one-dimensional subsystem symmetry. The model is shown to host a robust ground-state degeneracy that depends on the spatial topology of the underlying manifold, and localized zero-energy modes on corners of the system. A continuum field theory description of these phenomena is derived in terms of an anisotropic, modified version of the Abelian KKK-matrix Chern-Simons field theory. We show that this continuum description can lead to geometric-type effects such as corner states and edge states whose character depends on the orientation of the edge.
This paper reports an extensive study of three-dimensional topological ordered phases that in one of the directions behave like usual topological order concerning mobility of excitations, but in the perpendicular plane manifest type-II fracton physics dictated by a fractal subsystem symmetry. We obtain an expression for the ground state degeneracy, which depends intricately on the sizes of the plane, signaling a strong manifestation of UV/IR mixing. The ground state degeneracy can be interpreted in terms of spontaneous/explicit breaking of fractal subsystem symmetries. We also study the boundary physics, which in turn is useful to understand the connection with certain two-dimensional phases. Finally, we derive a low-energy but not long-distance effective field theory, by Higgsing a fractal U(1)U(1)U(1) symmetry and taking the deep IR limit. This description embodies in a natural way several aspects of the phases, such as the content of generalized global symmetries, the role of fractal symmetries on the mobility of excitations, the anomalies, and the boundary physics.
We study the boundary theory of the ZN{\mathbb{Z}}_{N}ZN​ X-cube model using a continuum perspective, from which the exchange statistics of a subset of bulk excitations can be recovered. We discuss various gapped boundary conditions that either preserve or break the translation and rotation symmetries on the boundary, and further present the corresponding ground-state degeneracies on ${T}^{2}\ifmmode\times\else\texttimes\fi{}I$. The low-energy physics is highly sensitive to the boundary conditions: even the extensive part of the ground-state degeneracy can vary when different sets of boundary conditions are chosen on the two boundaries. We also examine the anomaly inflow of the boundary theory and find that the X-cube model is not the unique (3+1)(3+1)(3+1)-dimensional theory that cancels the 't Hooft anomaly of the boundary.