We study the competition between two different topological orders in three dimensions by considering the X-cube model and the three-dimensional toric code. The corresponding Hamiltonian can be decomposed into two commuting parts, one of which displaying a self-dual spectrum. To determine the phase diagram, we compute the high-order series expansions of the ground-state energy in all limiting cases. Apart from the topological order related to the toric code and the fractonic order related to the X-cube model, we found two new phases which are adiabatically connected to classical limits with nontrivial sub-extensive degeneracies. All phase transitions are found to be first order.We study the competition between two different topological orders in three dimensions by considering the X-cube model and the three-dimensional toric code. The corresponding Hamiltonian can be decomposed into two commuting parts, one of which displaying a self-dual spectrum. To determine the phase diagram, we compute the high-order series expansions of the ground-state energy in all limiting cases. Apart from the topological order related to the toric code and the fractonic order related to the X-cube model, we found two new phases which are adiabatically connected to classical limits with nontrivial sub-extensive degeneracies. All phase transitions are found to be first order.

Jun 10, 20210 citations

Fracton topological order via coupled layers

Han Ma, Ethan Lake, Xie Chen +1 more

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple $d=2$ topological phases and from stacks of $d=3$ fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

arXiv:1701.00747Physical review. B./Physical review. BJun 21, 2017189 citations

Quantum phase transitions in the K -layer Ising toric code

Lukas Schamriß, Lea Lenke, Matthias Mühlhauser +1 more

We investigate the quantum phase diagram of the $K$ -layer Ising toric code corresponding to $K$ layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays ${\mathbb{Z}}_{2}^{K}$ topological order originating from the toric codes in each layer, the system shows ${\mathbb{Z}}_{2}$ topological order in the high-Ising limit. The latter is demonstrated for general $K$ by deriving an effective low-energy model in $K\mathrm{th}$ -order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases $K=3$ and $K=4$ we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both $K$ separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.We investigate the quantum phase diagram of the $K$-layer Ising toric code corresponding to $K$ layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays ${\mathbb{Z}}_{2}^{K}$ topological order originating from the toric codes in each layer, the system shows ${\mathbb{Z}}_{2}$ topological order in the high-Ising limit. The latter is demonstrated for general $K$ by deriving an effective low-energy model in $K\mathrm{th}$-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases $K=3$ and $K=4$ we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both $K$ separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.

arXiv:2201.10384Physical review. B./Physical review. BMay 26, 20224 citations

Fracton and topological order in the XY checkerboard toric code

Maximilian Vieweg, Kai Phillip Schmidt

We introduce the XY checkerboard toric code. It represents a generalization of the Z 2 toric code with two types of star operators with x and y flavor and two anisotropic star sublattices forming a checkerboard lattice. The quantum phase diagram is deduced exactly by a duality transformation to two copies of self-dual Xu-Moore models, which builds on the existence of a subextensive number of Z 2 conserved parities. For any spatial anisotropy of the star sublattices, the XY checkerboard toric code realizes two quantum phases with Z 2 topological order and an intermediate phase with type-I fracton order. The properties of the fracton phase such as subextensive ground-state degeneracy can be analytically deduced from the degenerate limit of isolated stars. For the spatially isotropic case the extension of the fracton phase vanishes. The topological phase displays anyonic excitations with restricted mobility and dimensional reduction. All phase transitions are first order.We introduce the XY checkerboard toric code. It represents a generalization of the Z 2 toric code with two types of star operators with x and y flavor and two anisotropic star sublattices forming a checkerboard lattice. The quantum phase diagram is deduced exactly by a duality transformation to two copies of self-dual Xu-Moore models, which builds on the existence of a subextensive number of Z 2 conserved parities. For any spatial anisotropy of the star sublattices, the XY checkerboard toric code realizes two quantum phases with Z 2 topological order and an intermediate phase with type-I fracton order. The properties of the fracton phase such as subextensive ground-state degeneracy can be analytically deduced from the degenerate limit of isolated stars. For the spatially isotropic case the extension of the fracton phase vanishes. The topological phase displays anyonic excitations with restricted mobility and dimensional reduction. All phase transitions are first order.

arXiv:2411.19051Physical Review ResearchJun 23, 20254 citations

Topological multicritical point in the phase diagram of the toric code model and three-dimensional lattice gauge Higgs model

I. S. Tupitsyn, Alexei Kitaev, Nikolay V. Prokof'ev +1 more

We construct a mapping between the two-dimensional toric code model in external magnetic fields, ${h}_{z}$ and ${h}_{x}$ , and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional ${\mathbb{Z}}_{2}$ gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to ${60}^{3}$ ) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the ${\mathbb{Z}}_{2}$ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.We construct a mapping between the two-dimensional toric code model in external magnetic fields, ${h}_{z}$ and ${h}_{x}$, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional ${\mathbb{Z}}_{2}$ gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to ${60}^{3}$) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the ${\mathbb{Z}}_{2}$ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.

arXiv:0804.3175Physical Review BAug 17, 2010195 citations

Quantum critical phase transition between two topologically ordered phases in the Ising toric code bilayer

Raymond Wiedmann, Lea Lenke, Matthias R. Walther +2 more

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled ${\mathbb{Z}}_{2}$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional ${\mathrm{Ising}}^{*}$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local ${\mathbb{Z}}_{2}$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly coupled ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ and the strongly coupled ${\mathbb{Z}}_{2}$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional ${\mathrm{Ising}}^{*}$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local ${\mathbb{Z}}_{2}$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.

arXiv:2010.05982Physical review. B./Physical review. BDec 17, 202019 citations

In this work, we develop a coupled layer construction of fracton topological orders in

d=3

spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple

d=2

topological phases and from stacks of

d=3

fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

We investigate the quantum phase diagram of the

K

-layer Ising toric code corresponding to

K

layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays

{\mathbb{Z}}_{2}^{K}

topological order originating from the toric codes in each layer, the system shows

{\mathbb{Z}}_{2}

topological order in the high-Ising limit. The latter is demonstrated for general

K

by deriving an effective low-energy model in

K\mathrm{th}

-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases

K=3

and

K=4

we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both

K

separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.

We construct a mapping between the two-dimensional toric code model in external magnetic fields,

{h}_{z}

and

{h}_{x}

, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional

{\mathbb{Z}}_{2}

gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to

{60}^{3}

) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the

{\mathbb{Z}}_{2}

gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.

{\mathbb{Z}}_{2}

topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the three-dimensional

{\mathrm{Ising}}^{*}

universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local

{\mathbb{Z}}_{2}

conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a nonlocal order parameter.