X-cube model on generic lattices: Fracton phases and geometric order

Fracton order is a new kind of quantum order characterized by topological excitations that exhibit remarkable mobility restrictions and a robust ground-state degeneracy (GSD) which can increase exponentially with system size. In this paper, we present a generic lattice construction (in three dimensions) for a generalized X-cube model of fracton order, where the mobility restrictions of the subdimensional particles inherit the geometry of the lattice. This helps explain a previous result that lattice curvature can produce a robust GSD, even on a manifold with trivial topology. We provide explicit examples to show that the (zero-temperature) phase of matter is sensitive to the lattice geometry. In one example, the lattice geometry confines the dimension-1 particles to small loops, which allows the fractons to be fully mobile charges, and the resulting phase is equivalent to (3+1)-dimensional toric code. However, the phase is sensitive to more than just lattice curvature; different lattices without curvature (e.g., cubic or stacked kagome lattices) also result in different phases of matter, which are separated by phase transitions. Unintuitively, however, according to a previous definition of phase [X. Chen , Phys. Rev. B 82, 155138 (2010)PRBMDO1098-012110.1103/PhysRevB.82.155138], even just a rotated or rescaled cubic results in different phases of matter, which motivates us to propose a coarser definition of phase for gapped ground states and fracton order. This equivalence relation between ground states is given by the composition of a local unitary transformation and a quasi-isometry (which can rotate and rescale the lattice); equivalently, ground states are in the same phase if they can be adiabatically connected by varying both the Hamiltonian and the positions of the degrees of freedom (via a quasi-isometry). In light of the importance of geometry, we further propose that fracton orders should be regarded as a geometric order.

Note:

9+3 pages, 11+2 figures; published version

References(65)

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[1]

Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity

- Phys.Rev.B 91 (2015) 12, 125121
•
e-Print:
- 1406.5090
•
DOI:
- 10.1103/physrevb.91.125121

[2]

Anyons in an exactly solved model and beyond

Alexei Kitaev
(
- Caltech, Pasadena (main)
)

- Annals Phys. 321 (2006) 1, 2-111
•
e-Print:
- cond-mat/0506438
•
DOI:
- 10.1016/j.aop.2005.10.005

[3]

A theory of 2+1D bosonic topological orders

Xiao-Gang Wen

- Natl.Sci.Rev. 3 (2016) 1, 68-106
•
e-Print:
- 1506.05768
•
DOI:
- 10.1093/nsr/nwv077

[4]

T. Lan
,
L. Kong
,
X.-G. Wen

e-Print:
- 1704.04221

[5]

Fault tolerant quantum computation by anyons

A.Yu. Kitaev
(
- Landau Inst.
)

- Annals Phys. 303 (2003) 2-30
•
e-Print:
- quant-ph/9707021
•
DOI:
- 10.1016/S0003-4916(02)00018-0

[6]

Topological Gauge Theories of Antisymmetric Tensor Fields

Matthias Blau
(
- SISSA, Trieste
)
,
George Thompson
(
- Paris U., VI-VII
)

- Annals Phys. 205 (1991) 130-172
•
DOI:
- 10.1016/0003-4916(91)90240-9

[7]

Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions

Pavel Putrov
(
- Princeton, Inst. Advanced Study
)
,
Juven Wang
(
)
,
Shing-Tung Yau
(
- Harvard U., Dept. Math. and
- Harvard U., Phys. Dept.
)

- Annals Phys. 384 (2017) 254-287
•
e-Print:
- 1612.09298
•
DOI:
- 10.1016/j.aop.2017.06.019

[8]

A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations

- Phys.Rev.B 92 (2015) 23, 235136
•
e-Print:
- 1505.02576
•
DOI:
- 10.1103/PhysRevB.92.235136

[9]

Local stabilizer codes in three dimensions without string logical operators

Jeongwan Haah
(
- Caltech
)

- Phys.Rev.A 83 (2011) 4, 042330
•
e-Print:
- 1101.1962
•
DOI:
- 10.1103/physreva.83.042330

[10]

Exotic topological order in fractal spin liquids

Beni Yoshida

- Phys.Rev.B 88 (2013) 12, 125122
•
e-Print:
- 1302.6248
•
DOI:
- 10.1103/PhysRevB.88.125122

[11]

Quantum Glassiness

Claudio Chamon

- Phys.Rev.Lett. 94 (2005) 4, 040402
•
e-Print:
- cond-mat/0404182
•
DOI:
- 10.1103/physrevlett.94.040402

[12]

Topological order in an exactly solvable 3D spin model

Sergey Bravyi
(
- IBM Watson Res. Ctr.
)
,
Bernhard Leemhuis
(
- Amsterdam U.
)
,
Barbara M. Terhal
(
- IBM Watson Res. Ctr.
)

- Annals Phys. 326 (2011) 4, 839-866
•
DOI:
- 10.1016/j.aop.2010.11.002

[13]

Fracton Topological Order, Generalized Lattice Gauge Theory and Duality

- Phys.Rev.B 94 (2016) 23, 235157
•
e-Print:
- 1603.04442
•
DOI:
- 10.1103/PhysRevB.94.235157

[14]

A Generalization of Non-Abelian Anyons in Three Dimensions

e-Print:
- 1706.07070

[15]

Fracton topological order via coupled layers

- Phys.Rev.B 95 (2017) 24, 245126
•
e-Print:
- 1701.00747
•
DOI:
- 10.1103/PhysRevB.95.245126

[16]

Fractons from partons

- Phys.Rev.B 96 (2017) 16, 165105
•
e-Print:
- 1703.02973
•
DOI:
- 10.1103/PhysRevB.96.165105

[17]

Fracton Topological Phases from Strongly Coupled Spin Chains

- Phys.Rev.Lett. 119 (2017) 25, 257202
•
e-Print:
- 1707.02308
•
DOI:
- 10.1103/PhysRevLett.119.257202

[18]

Simple anisotropic three-dimensional quantum spin liquid with fractonlike topological order

- Phys.Rev.B 96 (2017) 22, 224429
•
e-Print:
- 1709.10094
•
DOI:
- 10.1103/PhysRevB.96.224429

[19]

Quantum memories at finite temperature

Benjamin J. Brown
(
- Imperial Coll., London and
- Bohr Inst.
)
,
Daniel Loss
(
- Basel U.
)
,
Jiannis K. Pachos
(
- Leeds U.
)
,
Chris N. Self
(
- Imperial Coll., London and
- Bohr Inst.
)
,
James R. Wootton
(
- Basel U.
)

- Rev.Mod.Phys. 88 (2016) 4, 045005
•
e-Print:
- 1411.6643
•
DOI:
- 10.1103/RevModPhys.88.045005

[20]

However, since fracton orders are not topologically invariant, including the word “topological” in “fracton topological order” is misleading We propose that fracton geometric order would be a more accurate name

[21]

Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids

Michael Pretko
(
- MIT, Cambridge, Dept. Phys.
)

- Phys.Rev.B 95 (2017) 11, 115139
•
e-Print:
- 1604.05329
•
DOI:
- 10.1103/PhysRevB.95.115139

[22]

Other examples of topological excitations include ordinary electrons and the charge and flux excitations of toric code

[23]

Correlation function diagnostics for type-I fracton phases

Trithep Devakul
(
- Princeton U.
)
,
S.A. Parameswaran
(
- Oxford U., Theor. Phys.
)
,
S.L. Sondhi
(
- Princeton U.
)

- Phys.Rev.B 97 (2018) 4, 041110
•
e-Print:
- 1709.10071
•
DOI:
- 10.1103/PhysRevB.97.041110

[24]

Fracton Models on General Three-Dimensional Manifolds

- Phys.Rev.X 8 (2018) 3, 031051
•
e-Print:
- 1712.05892
•
DOI:
- 10.1103/PhysRevX.8.031051

[25]

Deciphering the nonlocal entanglement entropy of fracton topological orders

Bowen Shi
(
- Ohio State U.
)
,
Yuan-Ming Lu
(
- Ohio State U.
)

- Phys.Rev.B 97 (2018) 14, 144106
•
e-Print:
- 1705.09300
•
DOI:
- 10.1103/PhysRevB.97.144106

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