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 nonperturbative technique for confining interactions between instantons that introduce irreparable infrared 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 nonperturbative technique for confining interactions between instantons that introduce irreparable infrared divergences in the standard perturbative approaches.

Physical review. B./Physical review. BApr 16, 20244 citations

Pseudogap phases, chiral anomaly, and topological order with quantum loop entanglement

Predrag Nikolić

A many-body quantum system whose topological defects are conserved, abundant, and mobile is a correlated quantum liquid. Since topological defects can be classified by homotopy groups, each homotopy identifies a class of quantum liquids. Here we explore the quantum liquids based on the ${\ensuremath{\pi}}_{3}({S}^{2})$ homotopy group, i.e., Hopf fibration. Their topologically nontrivial dynamics emerges from the interlinking between magnetic flux or skyrmion loops in the charge and spin sectors respectively. We lay down a field theory foundation for analyzing such states by naturally incorporating the well-known framing regularization into the theory, and constructing the appropriate topological Lagrangian terms. We show that at least two strongly correlated phases of interlinked loops can exist in $d=3$ spatial dimensions at zero and low-finite temperatures. These phases are closely related to the chiral quantum anomaly and do not have an obvious topological order, but they are distinguished from the trivial disordered phase by a generalization of the Wilson loop operator. In $d=4$ spatial dimensions, interlinked loops are able to produce topological order at zero temperature, featuring charge, angular momentum, and braiding fractionalization. We discuss some possible experimental signatures of loop entanglement in the quantum noise of charge currents.A many-body quantum system whose topological defects are conserved, abundant, and mobile is a correlated quantum liquid. Since topological defects can be classified by homotopy groups, each homotopy identifies a class of quantum liquids. Here we explore the quantum liquids based on the ${\ensuremath{\pi}}_{3}({S}^{2})$ homotopy group, i.e., Hopf fibration. Their topologically nontrivial dynamics emerges from the interlinking between magnetic flux or skyrmion loops in the charge and spin sectors respectively. We lay down a field theory foundation for analyzing such states by naturally incorporating the well-known framing regularization into the theory, and constructing the appropriate topological Lagrangian terms. We show that at least two strongly correlated phases of interlinked loops can exist in $d=3$ spatial dimensions at zero and low-finite temperatures. These phases are closely related to the chiral quantum anomaly and do not have an obvious topological order, but they are distinguished from the trivial disordered phase by a generalization of the Wilson loop operator. In $d=4$ spatial dimensions, interlinked loops are able to produce topological order at zero temperature, featuring charge, angular momentum, and braiding fractionalization. We discuss some possible experimental signatures of loop entanglement in the quantum noise of charge currents.

arXiv:2310.01493Physical review. B./Physical review. BAug 26, 20241 citation

Diagnosing deconfinement and topological order

Karol Gregor, David A. Huse, Roderich Moessner +1 more

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

Diagnosing deconfinement and topological order

Karol Gregor, Huse, D., Roderich Moessner +1 more

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.)

Feb 11, 201163 citations

Frustrated Kondo impurity triad: A toy model of deconfinement

Elio J. König, Piers Coleman, Yashar Komijani

The concepts of deconfinement and topological order are of great current interest for quantum information science and for our understanding of quantum materials. Here, we introduce a simple model of three antiferromagnetically coupled Kondo impurities, a ``Kondo triangle'', which can be used to further extend the application of these concepts to electronic systems. We show that by tuning the magnetic frustration, the Kondo triangle undergoes a quantum phase transition between two phases of unbroken symmetry, signaling a phase transition beyond the Landau paradigm. We demonstrate that the frustrated ``spin liquid'' phase is described by a three-channel Kondo fixed point and thus displays an irrational ground state degeneracy. Using an Abrikosov pseudofermion representation this quantum state is categorized by an emergent U(1) gauge field and its projective symmetry group. The gauge theory is deconfining in the sense that a miniature Wilson loop orders and that topological defects (instantons in the gauge field) are expelled. This phase persists in presence of moderate Kondo screening until the proliferation of topological defects leads to a quantum phase transition to an unfrustrated Fermi liquid phase. Based on this evidence, we propose that the three-channel Kondo phase displays topological order in a similar sense as gapless spin liquids.The concepts of deconfinement and topological order are of great current interest for quantum information science and for our understanding of quantum materials. Here, we introduce a simple model of three antiferromagnetically coupled Kondo impurities, a ``Kondo triangle'', which can be used to further extend the application of these concepts to electronic systems. We show that by tuning the magnetic frustration, the Kondo triangle undergoes a quantum phase transition between two phases of unbroken symmetry, signaling a phase transition beyond the Landau paradigm. We demonstrate that the frustrated ``spin liquid'' phase is described by a three-channel Kondo fixed point and thus displays an irrational ground state degeneracy. Using an Abrikosov pseudofermion representation this quantum state is categorized by an emergent U(1) gauge field and its projective symmetry group. The gauge theory is deconfining in the sense that a miniature Wilson loop orders and that topological defects (instantons in the gauge field) are expelled. This phase persists in presence of moderate Kondo screening until the proliferation of topological defects leads to a quantum phase transition to an unfrustrated Fermi liquid phase. Based on this evidence, we propose that the three-channel Kondo phase displays topological order in a similar sense as gapless spin liquids.

Feb 27, 20203 citations

Long-Range Toric Code: Finite-Temperature Topological Order in Two Dimensions

Jacopo Gliozzi, Taylor L. Hughes, Kyung-Su Kim

We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.

cond-mat.str-elCross-listedarXiv:2610.37765Oct 29, 20260 citations

d=3

spatial dimensions at zero and low-finite temperatures. These phases are closely related to the chiral quantum anomaly and do not have an obvious topological order, but they are distinguished from the trivial disordered phase by a generalization of the Wilson loop operator. In

d=4

spatial dimensions, interlinked loops are able to produce topological order at zero temperature, featuring charge, angular momentum, and braiding fractionalization. We discuss some possible experimental signatures of loop entanglement in the quantum noise of charge currents.