Unidirectional subsystem symmetry in a hole-doped honeycomb-lattice Ising magnet Sambuddha Sanyal1Alexander Wietek2 3yand John Sous4z 1Department of Physics Indian Institute of Science Education and Research IISER Tirupati Tirupati 517507 India

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Unidirectional subsystem symmetry in a hole-doped honeycomb-lattice Ising magnet

Sambuddha Sanyal,1, ∗Alexander Wietek,2, 3, †and John Sous4, ‡

1Department of Physics, Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati 517507, India

2Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA

3Max Planck Institute for the Physics of Complex Systems,

N¨othnitzer Strasse 38, Dresden 01187, Germany

4Department of Physics, Stanford University, Stanford, CA 93405, USA

(Dated: October 4, 2022)

We study a model of a hole-doped collinear Ising antiferromagnet on the honeycomb lattice as a

route toward realization of subsystem symmetry. We ﬁnd nearly exact conservation of dipole sym-

metry veriﬁed both numerically with exact diagonalization (ED) on ﬁnite clusters and analytically

with perturbation theory. The emergent symmetry forbids the motion of single holes – or fractons

– but allows hole pairs – or dipoles – to move freely along a one-dimensional line, the antiferromag-

netic direction, of the system; in the transverse direction both fractons and dipoles are completely

localized. This presents a realization of a ‘unidirectional’ subsystem symmetry. By studying inter-

actions between dipoles, we argue that the subsystem symmetry is likely to continue to persist up

to ﬁnite (but probably small) hole concentrations.

Introduction. Fractons are the newest addition in the

line of exotic quasiparticles in condensed matter physics.

Recent years have witnessed tremendous progress in un-

derstanding fractons [1–40], which unmasked connec-

tions with other areas of physics including topological

order [2,4,8,9], gauge theory [12,13], quantum comput-

ing [4], glasses and soft matter [21,22], These exotic prop-

erties result from unusual mobility constraints whereby

a fracton is a charge-like excitation which has restricted

mobility when in isolation, but which, nonetheless, can

easily move in a subdimension of space when bound to an

oppositely charged fracton in a dipolar bound state [41].

The realization of these aberrant mobility constraints

and subsystem symmetry in physical systems is, however,

not naturally available, stimulating many interesting the-

oretical proposals [42–44]. One particularly appealing

approach is based on the idea that defect motion-induced

frustration in an ordered background leads to emergent

immobility constraints on the motion of defects them-

selves which in turn gives rise to fractonic quasiparticles,

with hole-doped Ising antiferromagnets [45–54] being a

prime example of this mechanism [27,28]. Unlike other

constructions proposed to realize fracton topological or-

der (e.g. [55]), this approach does not require an exten-

sive number of locally conserved quantities (which results

in an extensively degenerate ground state), making the

physical realization of fracton conservation laws (with-

out topological order) in quantum magnets signiﬁcantly

more accessible. However, the emergence of fracton-like

quasiparticles in fully two-dimensional (2D) hole-doped

Ising antiferromagnets has been demonstrated only in the

asymptotic limit of tJ(where tis the hole hopping

and Jis the Ising coupling) [27], where already at the

leading order in t2/J, the direction of the dipole moment

is not conserved even though its magnitude is because

hole pairs can rotate as they move in the 2D plane.

In this letter we overcome this limitation and pro-

FIG. 1. Ground state of the Hamiltonian HIsing (Eq. (2))

with one electron per site. The three bonds of the honey-

comb lattice are denoted as x0, y0and z0respectively (xand

yare the Cartesian directions). The Hamiltonian describes

an Ising magnet on a honeycomb lattice with antiferromag-

netic exchange along the x0and y0bonds and ferromagnetic

exchange along the z0bonds. The vectors δx0,δy0and δz0

connect near-neighbor sites in the x0,y0and z0directions, re-

spectively. The unit cell of the honeycomb lattice with A and

B sublattices is shown in the dotted region, and a1and a2are

the primitive lattice vectors.

pose an essentially exact physical realization of fractonic

quasiparticles with subsystem dipolar symmetry in a 2D

non-degenerate, ordered spin system with local two-spin

interactions. The key ingredient in our proposal is the

collinear antiferromagnetic order in which defect motion-

induced frustration completely prevents hole pairs or

dipoles from moving in the perpendicular direction, re-

sulting in exact conservation of both the magnitude and

direction of dipole moment in the asymptotic limit tJ.

This further endows the system with a subsystem symme-

try since dipole motion is restricted to a one-dimensional

(1D) submanifold of the system. This symmetry mani-

fests only along the antiferromagnetic direction, thus we

arXiv:2210.00012v1 [cond-mat.str-el] 30 Sep 2022

FIG. 2. (a) An isolated hole can move by one site only in the ferromagnetic direction but cannot move in the antiferromagnetic

direction without frustrating the antiferromagnetic bonds. (b) A pair of holes on neighboring sites can move only along the

antiferromagnetic x−ydirection. Note that a pair of holes on neighboring sites connected by a bond in the z0direction cannot

move without frustrating the background and as such will be localized.

denote it as ‘unidirectional’ subsystem symmetry. Impor-

tantly, we ﬁnd via exact diagonalization (ED) that this

symmetry continues to hold quantitatively away from the

perturbative limit when t&J. We further suggest that

interaction between hole pairs is weak implying continu-

ity of these immobility constraints to small, ﬁnite con-

centrations.

Model. Using a coupled spin chain construction to fer-

romagnetically couple antiferromagnetic Ising chains we

construct a 2D ordered Ising magnet, in which we study

doped holes. Speciﬁcally, we consider a model of holes

doped into an Ising collinear antiferromagnet on the hon-

eycomb lattice given by

H=−tX

ri,δj,σ c†

ri+δj,σcri,σ + h.c.+HIsing,(1)

where tdenotes the hopping amplitude, c†(c) are

fermionic creation (annihilation) operators, and σ∈ {↑,↓

}is the fermion spin. The coordinates riare deﬁned by

the sites of a Bravais lattice given by rj=mja1+nja2

where a1,a2denote the primitive lattice vectors of the

honeycomb lattice. The basis has two sites referred as A

and B. The vectors δx0, δy0, δz0connect spins on nearest-

neighbor sites in the three directions of the honeycomb

lattice, see Fig. 1. A no double occupancy constraint,

nri=Pσc†

ri,σcri,σ ≤1, ∀ri∈A, B is enforced on

the Hilbert space. The Ising spin Hamiltonian HIsing

is constructed by ferromagnetically coupling alternating

sites on antiferromagnetic chains (this can be viewed as

a model of a striped antiferromagnet in a brick-wall lat-

tice). The ferromagnetic spin couplings are taken to be

along the δz0direction and antiferromagnetic spin cou-

plings along the δx0and δy0directions:

HIsing =JX

riSz

ri+δx0Sz

ri+Sz

ri+δy0Sz

ri−Sz

ri+δz0Sz

ri,

(2)

where Sz

ridenotes the spin-zoperator. One

ground state of HIsing is given by |ψGSi=

Qri.(a1+a2)∈2Zc†

ri,↓c†

ri+δx0,↓c†

ri+a1,↑c†

ri+a1+δz0,↑|0i.

A hole can be created (annihilated) on a given site by

annihilating (creating) an electron on that site. For any

given hole density, the total magnetization is conserved.

Thus, we associate the removal (addition) of a fermion

with spin σwith the creation (annihilation) of a hole with

spin −σ, as either amounts to a total net change of the

magnetization of the entire system by −σ. Therefore the

hole creation operator is given by f†

ri,σ =cri,−σ. A hole

can move to a neighboring site along the antiferromag-

netic direction if the electron with antialigned spin on

that site moves to the hole’s original site. One can view

this as a spin ﬂip operation at the original hole site ac-

companied by hopping of the hole to the concerned neigh-

bor site. The original hole site with a ﬂipped spin is now

in a “wrong” orientation with respect to its two remain-

ing neighbors and thus we view this as defect creation. A

hole dressed by such bosonic (spin wave) defects forms a

magnetic polaron. We can represent a misaligned spin as

a bosonic magnon defect for the sites ri.(a1+a2)∈2Z

as b†

ri=σ−

ri,b†

ri+δx0=σ+

ri+δx0,b†

ri+δy0=σ+

ri+δy0, and

b†

ri+δz0=σ−

ri+δz0, and for the sites ri.(a1+a2)∈2Z+ 1

as bri=σ+

ri,bri+δx0=σ−

ri+δx0,bri+δy0=σ−

ri+δy0, and

bri+δz0=σ+

ri+δz0, where σ±

riare the Pauli Ladder oper-

ators. In contrast to motion along the antiferromagnetic

direction, a single hole can move – only by only one site –

in the ferromagnetic z0direction since there no wrongly

aligned spin.

A unique characteristic of this model on the honey-

comb lattice is that each site has two antiferromagnetic

neighbors in the x0and y0directions and one ferromag-

netic neighbor in the z0direction. This means that in

order for a hole to move from a site to another in the x0

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Unidirectionalsubsystemsymmetryinahole-dopedhoneycomb-latticeIsingmagnetSambuddhaSanyal,1,AlexanderWietek,2,3,yandJohnSous4,z1DepartmentofPhysics,IndianInstituteofScienceEducationandResearch(IISER)Tirupati,Tirupati517507,India2CenterforComputationalQuantumPhysics,FlatironInstitute,162FifthAvenue,Ne...

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Unidirectional subsystem symmetry in a hole-doped honeycomb-lattice Ising magnet Sambuddha Sanyal1Alexander Wietek2 3yand John Sous4z 1Department of Physics Indian Institute of Science Education and Research IISER Tirupati Tirupati 517507 India

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