Foliated-Exotic Duality in Fractonic BFTheories Kantaro Ohmori and Shutaro Shimamura Department of Physics The University of Tokyo Bunkyo-ku Tokyo

2025-05-06 0 0 634.77KB 46 页 10玖币

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Foliated-Exotic Duality in Fractonic BF Theories

Kantaro Ohmori and Shutaro Shimamura

Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo

113-0033, Japan

Abstract

There has been proposed two continuum descriptions of fracton systems: foliated

quantum ﬁeld theories (FQFTs) and exotic quantum ﬁeld theories. Certain fracton

systems are believed to admit descriptions by both, and hence a duality is expected

between such a class of FQFTs and exotic QFTs. In this paper we study this duality in

detail for concrete examples in 2+1 and 3+1 dimensions. In the examples, both sides

of the continuum theories are of BF -type, and we ﬁnd the explicit correspondences

of gauge-invariant operators, gauge ﬁelds, parameters, and allowed singularities and

discontinuities. This deepens the understanding of dualities in fractonic quantum ﬁeld

theories.

arXiv:2210.11001v3 [hep-th] 6 Feb 2023

Contents

1 Introduction 2

2BF -type Theory in 2+1 Dimensions 8

2.1 2+1d Foliated BF Theory ............................ 8

2.1.1 Foliation and Foliated Gauge Fields ................... 8

2.1.2 2+1d Foliated BF Lagrangian ...................... 9

2.1.3 Gauge-Invariant Operators ........................ 13

2.2 2+1d Exotic BF Theory ............................. 15

2.2.1 Tensor Gauge Fields ........................... 15

2.2.2 2+1d Exotic BF Lagrangian ....................... 16

2.2.3 Gauge-Invariant Operators ........................ 18

2.3 Correspondences in 2+1 Dimensions ...................... 19

3BF -type Theory in 3+1 Dimensions 22

3.1 3+1d Foliated BF Theory ............................ 22

3.1.1 Foliated Gauge Fields .......................... 23

3.1.2 3+1d Foliated BF Lagrangian ...................... 23

3.1.3 Gauge-Invariant Operators ........................ 25

3.2 3+1d Exotic BF Theory ............................. 28

3.2.1 Tensor Gauge Fields ........................... 28

3.2.2 3+1d Exotic BF Lagrangian ....................... 31

3.2.3 Gauge-Invariant Operators ........................ 32

3.3 Correspondences in 3+1 Dimensions ...................... 34

4 Conclusion 38

A Electric-Magnetic Dual Description in 2+1 Dimensions 39

A.1 Foliated Gauge Theory .............................. 39

A.2 Exotic Gauge Theory ............................... 40

A.3 Correspondences ................................. 41

1 Introduction

A Fracton phase is a new kind of phase of matter that exhibits excitations with restricted

mobility, which can only move in certain dimensional submanifolds (see [1,2] for reviews).

The characteristic excitations are called fractons, lineons, and planons, depending on the

spatial dimension of the excitation. Such fracton models, studied as lattice models in con-

densed matter physics [3–6], have various novel properties: a new type of symmetry and the

exponential growth of ground state degeneracy in terms of the linear sizes of the system.

The fracton systems are not only theoretically interesting in its own right, but expected to

be applied to quantum information [3,7,8] and gravity [9].

While fracton phases ﬁrst appeared in lattice systems, one would also expect a contin-

uum description in the low-energy limit of a lattice system. There have been proposed such

descriptions by continuum quantum ﬁeld theories (QFTs) in various situations [10–26]. The

QFTs do not have the Lorentz invariance or even the full rotational invariance, and can have

the discontinuous ﬁeld conﬁgurations. In the low-energy descriptions, the gapped excitations

are not dynamical and arise as the gauge-invariant defects. The identiﬁcation and construc-

tion of these QFTs are based on the subsystem symmetry, which is one of the generalizations

of symmetry. A subsystem symmetry is a symmetry that acts on a spatial submanifold, e.g.

a plane along a particular directions, and can have diﬀerent values on each submanifold [12].1

For lattice models, some fracton models can be written as foliated fracton phases [6,28–30].

A foliation is a decomposition of a manifold and regarding it as a stack of an inﬁnite number

of submanifolds. For example, the X-cube model [5], which is a gapped fracton lattice model

in 3+1 dimensions, can be written as a stack of the (2 + 1)-dimensional toric codes [31]

by using foliations [6]. For QFTs, there are fractonic QFTs coupled to foliations, which are

called foliated quantum ﬁled theories (FQFTs) [20–22]. On the other hand, some fractonic

QFTs can be written as tensor gauge theories [10,13–16,32] respecting the lattice rotational

symmetries, which we call the exotic QFTs [19]. The continuum QFT description of the

X-cube model can be written as BF -type theories in terms of both a foliated QFT in the

ﬂat foliations2and an exotic QFT [11,15]. The foliated and exotic descriptions are believed

to represent the same physics, but the duality between them has not been made clear.

In this paper, we will consider the foliated and exotic BF -type theories in 2 + 1 and 3 + 1

dimensions. In 2+1 dimensions, the BF -type theories are the continuum description of the

ZNplaquette Ising model (see [33] for a review) and the ZNlattice tensor gauge theory [13].

In 3+1 dimensions, the BF -type theories are the continuum description of the X-cube model

and the ZNlattice tensor gauge theory [15].

The goal of this paper is to show the explicit correspondences of the gauge ﬁelds and

1While a subsystem symmetry is similar to a higher form symmetry [27] as its corresponding symme-

try operator has codimension higher than one, the operator is not topological in the directions out of the

submanifold.

2The foliation is characterized by a foliation ﬁled e. The foliation is ﬂat when eis ﬂat, i.e., de = 0. See

Section 2.1.1.

parameters between the foliated BF theory and the exotic BF theory, completing the pre-

vious observation made in [22]. We will see that both foliated and exotic BF theories have

the same type of gauge-invariant operators and subsystem symmetries, and by matching the

operators, we will derive the correspondences of the ﬁelds and parameters. It is novel to

exhibit the explicit correspondences between the foliated ﬁelds, including the bulk ﬁelds, and

the exotic tensor gauge ﬁelds. This establishes the duality between the foliated and exotic

BF theories, which we call the foliated-exotic duality.

The organization of the rest of the paper is as follows. In Section 2, we will discuss the

BF -type theories in 2+1 dimensions. In Section 2.1, we will consider the foliated BF theory

with attention to singularities and discontinuities. The foliated BF Lagrangian in the ﬂat

foliations3is

Lf=

k=1

2π(dBk+b)∧Ak∧dxk+iN

2πb∧da . (1.1)

In Section 2.2, we will review the exotic BF theory [13]. The exotic BF Lagrangian4is

Le=iN

2πφ12(∂0A12 −∂1∂2A0).(1.2)

In Section 2.3, we show the explicit correspondences between them by matching the gauge-

invariant operators. In order to match the gauge-invariant operators, we need to modify

the strip operators in the foliated BF theory. The modiﬁcation turns out to be only by an

operator that is not remotely detectable [34,35]. The correspondences of the gauge ﬁelds and

parameters are shown in Table 1. In the correspondences, the singularities and discontinuities

are also matched. In Section 3, we will discuss BF -type theories in 3+1dimensions as in

the case of 2+1dimensions. In Section 3.1, we will review the foliated BF theory [20–22].

The foliated BF Lagrangian in the ﬂat foliations is

Lf=

k=1

2π(dBk+b)∧Ak∧dxk+iN

2πb∧da . (1.3)

In Section 3.2, we will review the exotic BF theory [11,15]. The exotic BF Lagrangian is

Le=iN

2πX

i,j 1

2Aij(∂0ˆ

Aij −∂kˆ

Ak(ij)

0) + 1

2A0∂i∂jˆ

Aij.(1.4)

In Section 3.3, we show the explicit correspondences between them. The correspondences of

the gauge ﬁelds and parameters are shown in Table 2,3. In Appendix A, we will consider

the electric-magnetic dual descriptions of the BF -type theories in 2+1dimensions.

Along the way, we ﬁnd that there are gauge-invariant operators that cannot be remotely

3The superscripts kindex the directions of the foliations. The subscripts in Table 1,2and 3are the

spatial indices.

4In the exotic theories, the superscripts and subscripts are the spacetime indices. As the metric is ﬂat,

we do not need to distinguish them.

detected by other spatially placed operators, but represents a time-like symmetry [36]. This

makes a contrast to the case of ordinary topological order or topological ﬁeld theory, where

every operator is remotely detectable.

The establishment of the foliated-exotic duality deepens the understanding of both of the

continuum descriptions of the fractonic systems. In general it is not known when a fractonic

system admits a description by a foliated or an exotic QFT, and this result will be a clue in

this interesting question. It would also serve as a starting point of exploring more general

dualities in quantum ﬁeld theories without Lorentz invariance.

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Foliated-ExoticDualityinFractonicBFTheoriesKantaroOhmoriandShutaroShimamuraDepartmentofPhysics,TheUniversityofTokyo,Bunkyo-ku,Tokyo113-0033,JapanAbstractTherehasbeenproposedtwocontinuumdescriptionsoffractonsystems:foliatedquantumeldtheories(FQFTs)andexoticquantumeldtheories.Certainfractonsystemsar...

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Foliated-Exotic Duality in Fractonic BFTheories Kantaro Ohmori and Shutaro Shimamura Department of Physics The University of Tokyo Bunkyo-ku Tokyo

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