Diagnosing deconfinement and topological order

Topological or deconfined phases are characterized by emergent, weakly fluctuating, gauge fields. In condensed matter settings, they inevitably come coupled to excitations that carry the corresponding gauge charges which invalidate the standard diagnostic of deconfinement—the Wilson loop. Inspired by a mapping between symmetric sponges and the deconfined phase of the Z 2 gauge theory, we construct a diagnostic for deconfinement that has the interpretation of a line tension. One operator version of this diagnostic turns out to be the Fredenhagen–Marcu order parameter known to lattice gauge theorists and we show that a different version is best suited to condensed matter systems. We discuss generalizations of the diagnostic, use it to establish the existence of finite temperature topological phases in d ⩾3 dimensions and show that multiplets of the diagnostic are useful in settings with multiple phases, such as U (1) gauge theories with charge q matter. (Additionally, we present an exact reduction of the partition function of the toric code in general dimensions to a well-studied problem.)Topological or deconfined phases are characterized by emergent, weakly fluctuating, gauge fields. In condensed matter settings, they inevitably come coupled to excitations that carry the corresponding gauge charges which invalidate the standard diagnostic of deconfinement—the Wilson loop. Inspired by a mapping between symmetric sponges and the deconfined phase of the Z 2 gauge theory, we construct a diagnostic for deconfinement that has the interpretation of a line tension. One operator version of this diagnostic turns out to be the Fredenhagen–Marcu order parameter known to lattice gauge theorists and we show that a different version is best suited to condensed matter systems. We discuss generalizations of the diagnostic, use it to establish the existence of finite temperature topological phases in d ⩾3 dimensions and show that multiplets of the diagnostic are useful in settings with multiple phases, such as U (1) gauge theories with charge q matter. (Additionally, we present an exact reduction of the partition function of the toric code in general dimensions to a well-studied problem.)

arXiv:1011.4187New Journal of PhysicsFeb 11, 201173 citations

Topological ordering in the Majorana toric code

Alexander Ziesen, Fabian Hassler, Ananda Roy

At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase transitions separating them. While the field theory provides details on the properties of the system close to the phase transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of this work. We describe a lattice gauge theory of the Majorana toric code in terms of $\mathrm{U}(1)$ matter fields coupled to an emergent ${\mathbb{Z}}_{2}$ gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge- $2e$ superconductor phases, while showing that the toric code phase disappears in the charge- $e$ superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase transitions separating them. While the field theory provides details on the properties of the system close to the phase transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of this work. We describe a lattice gauge theory of the Majorana toric code in terms of $\mathrm{U}(1)$ matter fields coupled to an emergent ${\mathbb{Z}}_{2}$ gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-$2e$ superconductor phases, while showing that the toric code phase disappears in the charge-$e$ superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

arXiv:1906.05158Physical review. B./Physical review. BSep 6, 20195 citations

Deconfined Thermal Phase Transitions with $Z_2$ Gauge StructuresDeconfined Thermal Phase Transitions with $Z_2$ Gauge Structures

Eun-Gook Moon

Fathoming deconfined phases is one of the key issues in modern condensed matter. Striking many-body effects including massive quantum entanglement and coherence may be realized as manifested in quantum spin liquids and topological orders. Here, we demonstrate that deconfined phases even host exotic thermal phase transitions, dubbed deconfined thermal transitions. Constructing a $Z_2$ lattice gauge model with strong interactions between $Z_2$ gauge fluxes, we prove the existence of a thermal phase transition between deconfined and confined phases in two spatial dimensions in sharp contrast to its absence in the Wegner model. Incorporating deconfined fermions, it is shown that gapless excitations from Fermi surfaces endow line-tension to $Z_2$ gauge fluxes at zero temperature, and we argue that a deconfined thermal transition with deconfined fermions may be interpreted as a hidden order transition with thermal gap-opening in Fermi surfaces. Moreover, it is shown that symmetry breaking transitions in deconfined phases may be unconventional. Global $Z_2$ and $U(1)$ symmetry breaking transitions in deconfined phases may be in the same universality class, which is impossible under the conventional Landau-Ginzburg-Wilson paradigm. Characteristic signatures of the transitions in experiments and candidate strongly correlated systems such as Kitaev materials are also discussed.Fathoming deconfined phases is one of the key issues in modern condensed matter. Striking many-body effects including massive quantum entanglement and coherence may be realized as manifested in quantum spin liquids and topological orders. Here, we demonstrate that deconfined phases even host exotic thermal phase transitions, dubbed deconfined thermal transitions. Constructing a $Z_2$ lattice gauge model with strong interactions between $Z_2$ gauge fluxes, we prove the existence of a thermal phase transition between deconfined and confined phases in two spatial dimensions in sharp contrast to its absence in the Wegner model. Incorporating deconfined fermions, it is shown that gapless excitations from Fermi surfaces endow line-tension to $Z_2$ gauge fluxes at zero temperature, and we argue that a deconfined thermal transition with deconfined fermions may be interpreted as a hidden order transition with thermal gap-opening in Fermi surfaces. Moreover, it is shown that symmetry breaking transitions in deconfined phases may be unconventional. Global $Z_2$ and $U(1)$ symmetry breaking transitions in deconfined phases may be in the same universality class, which is impossible under the conventional Landau-Ginzburg-Wilson paradigm. Characteristic signatures of the transitions in experiments and candidate strongly correlated systems such as Kitaev materials are also discussed.

arXiv:1812.05621Dec 13, 20182 citations

Critical behavior of Fredenhagen-Marcu string order parameters at topological phase transitions with emergent higher-form symmetries

Wen-Tao Xu, Frank Pollmann, Michael Knap

Abstract A nonlocal string order parameter detecting topological order and deconfinement has been proposed by Fredenhagen and Marcu (FM). However, due to the lack of exact internal symmetries for lattice models and the nonlinear dependence of the FM string order parameter on ground states, it is a priori not guaranteed that it is a genuine order parameter for topological phase transitions. In this work, we find that the FM string order parameter exhibits universal scaling behavior near critical points of charge condensation transitions, by directly evaluating the FM string order parameter in the infinite string-length limit using infinite Projected Entangled Pair States (iPEPS) for the toric code in a magnetic field. Our results thus demonstrate that the FM string order parameter represents a quantitatively well-behaved order parameter. We find that only in the presence of an emergent 1-form symmetry the corresponding FM string order parameter can faithfully detect topological transitions.Abstract A nonlocal string order parameter detecting topological order and deconfinement has been proposed by Fredenhagen and Marcu (FM). However, due to the lack of exact internal symmetries for lattice models and the nonlinear dependence of the FM string order parameter on ground states, it is a priori not guaranteed that it is a genuine order parameter for topological phase transitions. In this work, we find that the FM string order parameter exhibits universal scaling behavior near critical points of charge condensation transitions, by directly evaluating the FM string order parameter in the infinite string-length limit using infinite Projected Entangled Pair States (iPEPS) for the toric code in a magnetic field. Our results thus demonstrate that the FM string order parameter represents a quantitatively well-behaved order parameter. We find that only in the presence of an emergent 1-form symmetry the corresponding FM string order parameter can faithfully detect topological transitions.

arXiv:2402.00127npj Quantum InformationMay 9, 202519 citations

Instanton confinement-deconfinement transitions: The stability of pseudogap phases and topological order

Predrag Nikolić

We explore the stability of certain many-body quantum states which may exist at zero or finite temperatures, may lack long-range order and even topological order, and still are thermodynamically distinct from uncorrelated disordered phases. We sharply characterize such states by the conservation of topological charge, or equivalently confinement of instantons, using a generalization of the Wilson loop and the correlation length of an emergent gauge field. Our main conclusions are: (i) topological orders can exist at finite temperatures, (ii) relativistic liquids of topological defects can also exist as stable phases at finite temperatures, and (iii) there are two universality classes of instanton suppression. We also relate the instanton dynamics to the problem of the pseudogap state in underdoped cuprates. A universal experimental signature of the instanton deconfinement transition is a change of the quantum noise spectrum, which can perhaps be measured in some situations, for example via a quantum anomaly, or indirectly detected with a specific heat jump. The method of analysis is a functional renormalization group that generalizes the Coulomb gas treatment of Kosterlitz and Thouless to arbitrary interactions and dimensions. In particular, we construct an exact non-perturbative technique for confining interactions between instantons, which introduce irreparable infra-red divergences in the standard perturbative approaches.We explore the stability of certain many-body quantum states which may exist at zero or finite temperatures, may lack long-range order and even topological order, and still are thermodynamically distinct from uncorrelated disordered phases. We sharply characterize such states by the conservation of topological charge, or equivalently confinement of instantons, using a generalization of the Wilson loop and the correlation length of an emergent gauge field. Our main conclusions are: (i) topological orders can exist at finite temperatures, (ii) relativistic liquids of topological defects can also exist as stable phases at finite temperatures, and (iii) there are two universality classes of instanton suppression. We also relate the instanton dynamics to the problem of the pseudogap state in underdoped cuprates. A universal experimental signature of the instanton deconfinement transition is a change of the quantum noise spectrum, which can perhaps be measured in some situations, for example via a quantum anomaly, or indirectly detected with a specific heat jump. The method of analysis is a functional renormalization group that generalizes the Coulomb gas treatment of Kosterlitz and Thouless to arbitrary interactions and dimensions. In particular, we construct an exact non-perturbative technique for confining interactions between instantons, which introduce irreparable infra-red divergences in the standard perturbative approaches.

arXiv:2309.14424Sep 25, 20230 citations

Pauli stabilizer formalism for topological quantum field theories and generalized statistics

Yitao Feng, Hanyu Xue, Ryohei Kobayashi +2 more

Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the $e^2m^2$ -loop in the (4+1)D $\mathbb Z_4$ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of $-1$ . Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by $H^5(B^2G,U(1))$ . Beyond (4+1)D, the fermionic-loop toric codes form a family of $\mathbb Z_2$ topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using $\mathbb Z_4$ qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of $\mathbb Z_2$ topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that $p$ -dimensional excitations in $2p+2$ spatial dimensions can support anyonic $p$ -brane statistics for only even $p$ .Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the $e^2m^2$-loop in the (4+1)D $\mathbb Z_4$ toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of $-1$. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by $H^5(B^2G,U(1))$. Beyond (4+1)D, the fermionic-loop toric codes form a family of $\mathbb Z_2$ topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using $\mathbb Z_4$ qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of $\mathbb Z_2$ topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that $p$-dimensional excitations in $2p+2$ spatial dimensions can support anyonic $p$-brane statistics for only even $p$.

quant-pharXiv:2601.00064Journal of High Energy PhysicsJan 2, 20261 citation

\mathrm{U}(1)

matter fields coupled to an emergent

{\mathbb{Z}}_{2}

gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-

2e

superconductor phases, while showing that the toric code phase disappears in the charge-

e

superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.

Deconfined Thermal Phase Transitions with

Z_2

Gauge Structures

Z_2

lattice gauge model with strong interactions between

Z_2

gauge fluxes, we prove the existence of a thermal phase transition between deconfined and confined phases in two spatial dimensions in sharp contrast to its absence in the Wegner model. Incorporating deconfined fermions, it is shown that gapless excitations from Fermi surfaces endow line-tension to

Z_2

gauge fluxes at zero temperature, and we argue that a deconfined thermal transition with deconfined fermions may be interpreted as a hidden order transition with thermal gap-opening in Fermi surfaces. Moreover, it is shown that symmetry breaking transitions in deconfined phases may be unconventional. Global

Z_2

and

U(1)

symmetry breaking transitions in deconfined phases may be in the same universality class, which is impossible under the conventional Landau-Ginzburg-Wilson paradigm. Characteristic signatures of the transitions in experiments and candidate strongly correlated systems such as Kitaev materials are also discussed.

e^2m^2

-loop in the (4+1)D

\mathbb Z_4

toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of

-1

. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by

H^5(B^2G,U(1))

. Beyond (4+1)D, the fermionic-loop toric codes form a family of

\mathbb Z_2

topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using

\mathbb Z_4

qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of

\mathbb Z_2

topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that

p

-dimensional excitations in

2p+2

spatial dimensions can support anyonic

p

-brane statistics for only even

p