Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2509.18845 · Physical review. B./Physical review. B

Interacting-cluster spin liquids with robust flat bands evolving into higher-rank half-moon phases and topological Lifshitz transitions

Naïmo Davier, Ludovic D. C. Jaubert

Published Jan 28, 2026• 1 citation
arXivHTMLPDF

Abstract

Classical spin liquids are disordered magnetic phases, governed by local constraints that often give rise to flat-band ground states. When constraints take the form of a zero-divergence field within a cluster of spins, the spin liquid is often described by an emergent Coulomb gauge theory. Here we introduce an interaction $η$ between these clusters of spins which compete with the zero-divergence field. Using a framework embracing both the connectivity matrices of graph theory and the topology of band structures, we develop a generic theory of interacting-cluster Hamiltonians. We show how flat bands remain at zero energy up to finite interaction $η$, until a dispersive band becomes negative, stabilizing a spiral spin liquid with a hypersurface of ground-state manifold in reciprocal space. This hypersurface can be interpreted as an effective Fermi surface in the spectrum of the parent system, acting as a tunable energy selector despite the absence of particle filling. This effective Fermi surface serves as a mold for the apparition of the half-moon patterns in the equal-time structure factor. Our generic approach enables to extend the notion of half moons to the perturbation of higher-rank Coulomb fields and pinch-line spin liquids. In particular, multi-fold half moons appear when unconventional gauge charges, such as potential fractons, are stabilized in the ground state. Finally, half-moon phases can be tuned across the equivalent of a Lifshitz transition, when the hypersurface manifold changes topology.

DOI
10.1103/np33-ngyr

Related papers

Pinch line spin liquids as layered Coulomb phases and applications to cubic models

Naïmo Davier, F. A. Gómez Albarracín, H. D. Rosales +2 more

Spin liquids form fluctuating magnetic textures which have to obey certain rules imposed by frustration. These rules can often be written in the form of a Gauss law, indicating the local conservation of an emergent electric field. In reciprocal space, these emergent Gauss laws appear as singularities known as pinch points, that are accessible to neutron-scattering measurements. But more exotic forms of electromagnetism have been stabilized in spin liquids, and in a few rare instances, these zero-dimensional singularities have been extended into one-dimensional pinch lines. Here we propose a simple framework for the design of pinch-line spin liquids in a layered structure of two-dimensional algebraic spin liquids. A plethora of models can be build within this framework, as exemplified by several concrete examples where our theory is confirmed by simulations, and where the rank of the tensorial gauge field is continuously varied along the pinch line, opening new avenues in fractonic matter. Then we use our framework to understand how the evolution of the singularity pinch point along the pinch line can be understood as the interference pattern of two emergent electric fields. Finally, we apply our intuition on these emergent electric fields in real space to generic pinch line models beyond our layered framework, and revisit the recently proposed pinch line model on the octochlore lattice.

arXiv:2502.16978•Physical review. B./Physical review. B•Aug 19, 2025•2 citations

Topological Route to New and Unusual Coulomb Spin Liquids

Owen Benton, Roderich Moessner

Coulomb spin liquids are topological magnetic states obeying an emergent Gauss law. Little distinction has been made between different kinds of Coulomb liquids. Here we show how a series of distinct Coulomb liquids can be generated straightforwardly by varying the constraints on a classical spin system. This leads to pair creation, and coalescence, of topological defects of an underlying vector field. The latter makes higher-rank spin liquids, of recent interest in the context of fracton theories, with attendant multifold pinch points in the structure factor, appear naturally. New Coulomb liquids with an abundance of pinch points also arise. We thus establish a new and general route to uncovering exotic Coulomb liquids, via the manipulation of topological defects in momentum space.

arXiv:2103.10817•Physical Review Letters•Sep 3, 2021•40 citations

Magnetic properties of the spiral spin liquid and surrounding phases in the square lattice XY model

M. G. Gonzalez, Anna Fancelli, Han Yan +1 more

Spiral spin liquids possess a subextensively degenerate ground-state manifold, represented by a continuum of energy minima in reciprocal space. Since a small change of the spiral state wavevector requires a global change of the spin configuration in real space, it is a priori unclear how such systems can fluctuate within the degenerate ground state manifold. Only recently it was proposed that momentum vortices are responsible for the liquidity of the spiral phase and that these systems are closely related to an emergent rank-2 U(1) gauge theory [H. Yan and J. Reuther, Phys. Rev. Research 4, 023175 (2022)]. As a consequence of this gauge structure, four-fold pinch-point singularities were found in a generalized spin correlator. In this article, we use classical Monte Carlo and molecular dynamics calculations to embed the previously studied spiral spin liquid into a broader phase diagram of the square lattice XY model. We find a multitude of unusual phases and phase transitions surrounding the spiral spin liquid such as an effective four-state Potts transition into a colinear double-striped phase resulting from the spontaneous breaking of two coupled $\mathbb{Z}_2$ symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a re-entrance phenomenon at low temperatures into a nematic spiral phase. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the four-fold pinch point singularities across the phase diagram

arXiv:2404.00100•Mar 29, 2024•0 citations

Low-energy structure of spiral spin liquids

Han Yan, Johannes Reuther

In this paper we identify a previously unexplored type of topological defect in spiral spin liquids---the momentum vortex---and reveal its dominant role in shaping the low-energy physics of such systems. Spiral spin liquids are a class of classical spin liquids featuring subextensively degenerate ground states. They are distinct from spin liquids on geometrically frustrated lattices, in which the ground-state degeneracy is extensive and connected by local spin flips. Despite a handful of experimental realizations and many theoretical studies, a concrete physical picture of their spin liquidity has not been established so far. In this paper, we study a 2D spiral spin liquid model to answer this question. We find that the local momentum vector field can carry topological defects in the form of vortices, which, however, have very different properties from the commonly known spin vortices. The fluctuations of such vortices lead the system into a liquid phase at intermediate temperatures. Furthermore, the effective low-energy theory of such vortices indicates their equivalence to quadrupoles of fractons in a rank-2 U(1) gauge theory or, alternatively, to quadrupoles of disclinations in elasticity theory. At very low temperatures, the system freezes into a glassy state in which these vortices form a rigid network with straight-line domain walls. Our paper sheds light on the nature of spiral spin liquids and also paves the way toward understanding their quantum limit.

arXiv:2112.10676•Physical Review Research•Jun 1, 2022•27 citations

Unified non-Fermi liquid and superconductivity in a model of strongly-correlated systems

Yi Zhang

We study a model of strongly-correlated systems that incorporates phases such as Fermi liquids, non-Fermi liquids, and superconductivity, in addition to potential intertwined orders. The model describes Fermi surfaces of spinful electron gas or electron liquid coupled to bosons via an intermediate gauge field. Effectively, the coupling imposes constraints and interactions between the fermion spin and the local boson density. This grants the bosons the meaning as the Schwinger boson of the magnetic order and allows us to probe a larger phase space, rather than around the quantum critical point. We design the initial model so that after the boson and gauge fields are integrated out exactly, the resulting fermion-only effective theory only consists of several local interactions, allowing controlled weak-coupling interpretation for certain parameter regions. Consequently, we find a non-Fermi liquid whose degrees of freedom in the effective theory take the form of an \emph{emergent} Fermi surface that has a different Luttinger volume from the original fermions, therefore explicitly violates the Luttinger theorem. Further, this allows a $d$-wave superconducting instability if the effective interacting induced by boson coupling is in the relevant pairing channel. When the emergent Fermi surface undergoes a Lifshitz transition, the carrier type changes. In addition, we attribute the Mott insulating behavior in the strong coupling limit to the loss of emergent Fermi surface and argue the possibility of coexisting magnetic order. Our effective theory also suggests the possibility of charge density waves as a consequence of \emph{effective} strong repulsion and is independent of the Fermi-surface-nesting scenario. We discuss the possible optimal conditions for higher superconducting transition temperature and potential relevances and implications for realistic materials.

arXiv:1711.03981•Nov 10, 2017•0 citations

Magnetic properties of the spiral spin liquid and surrounding phases in the square lattice XY model

M. G. Gonzalez, Anna Fancelli, Han Yan +1 more

Spiral spin liquids possess a subextensively degenerate ground-state manifold, represented by a continuum of energy minima in reciprocal space. Since a small change of the spiral state wave vector requires a global change of the spin configuration in real space, it is a priori unclear how such systems can fluctuate within the degenerate ground-state manifold. Only recently it was proposed that momentum vortices are responsible for the liquidity of the spiral phase and that these systems are closely related to an emergent rank-2 U(1) gauge theory H. Yan et al. [Phys. Rev. Res. 4, 023175 (2022)]. As a consequence of this gauge structure, fourfold pinch-point singularities were found in a generalized spin correlator. In this paper, we use classical Monte Carlo and molecular dynamics calculations to embed the previously studied spiral spin liquid into a broader phase diagram of the square lattice XY model. We find a multitude of unusual phases and phase transitions surrounding the spiral spin liquid such as an effective four-state Potts transition into a collinear double-striped phase resulting from the spontaneous breaking of two coupled ${\mathbb{Z}}_{2}$ symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a second phase transition at low temperatures into a nematic spiral phase which only breaks one ${\mathbb{Z}}_{2}$ symmetry. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the fourfold pinch-point singularities across the phase diagram.

Physical review. B./Physical review. B•Aug 5, 2024•5 citations
Classical spin liquids are disordered magnetic phases, governed by local constraints that often give rise to flat-band ground states. When constraints take the form of a zero-divergence field within a cluster of spins, the spin liquid is often described by an emergent Coulomb gauge theory. Here we introduce an interaction ηηη between these clusters of spins which compete with the zero-divergence field. Using a framework embracing both the connectivity matrices of graph theory and the topology of band structures, we develop a generic theory of interacting-cluster Hamiltonians. We show how flat bands remain at zero energy up to finite interaction ηηη, until a dispersive band becomes negative, stabilizing a spiral spin liquid with a hypersurface of ground-state manifold in reciprocal space. This hypersurface can be interpreted as an effective Fermi surface in the spectrum of the parent system, acting as a tunable energy selector despite the absence of particle filling. This effective Fermi surface serves as a mold for the apparition of the half-moon patterns in the equal-time structure factor. Our generic approach enables to extend the notion of half moons to the perturbation of higher-rank Coulomb fields and pinch-line spin liquids. In particular, multi-fold half moons appear when unconventional gauge charges, such as potential fractons, are stabilized in the ground state. Finally, half-moon phases can be tuned across the equivalent of a Lifshitz transition, when the hypersurface manifold changes topology.
Spin liquids form fluctuating magnetic textures which have to obey certain rules imposed by frustration. These rules can often be written in the form of a Gauss law, indicating the local conservation of an emergent electric field. In reciprocal space, these emergent Gauss laws appear as singularities known as pinch points, that are accessible to neutron-scattering measurements. But more exotic forms of electromagnetism have been stabilized in spin liquids, and in a few rare instances, these zero-dimensional singularities have been extended into one-dimensional pinch lines. Here we propose a simple framework for the design of pinch-line spin liquids in a layered structure of two-dimensional algebraic spin liquids. A plethora of models can be build within this framework, as exemplified by several concrete examples where our theory is confirmed by simulations, and where the rank of the tensorial gauge field is continuously varied along the pinch line, opening new avenues in fractonic matter. Then we use our framework to understand how the evolution of the singularity pinch point along the pinch line can be understood as the interference pattern of two emergent electric fields. Finally, we apply our intuition on these emergent electric fields in real space to generic pinch line models beyond our layered framework, and revisit the recently proposed pinch line model on the octochlore lattice.
Coulomb spin liquids are topological magnetic states obeying an emergent Gauss law. Little distinction has been made between different kinds of Coulomb liquids. Here we show how a series of distinct Coulomb liquids can be generated straightforwardly by varying the constraints on a classical spin system. This leads to pair creation, and coalescence, of topological defects of an underlying vector field. The latter makes higher-rank spin liquids, of recent interest in the context of fracton theories, with attendant multifold pinch points in the structure factor, appear naturally. New Coulomb liquids with an abundance of pinch points also arise. We thus establish a new and general route to uncovering exotic Coulomb liquids, via the manipulation of topological defects in momentum space.
Spiral spin liquids possess a subextensively degenerate ground-state manifold, represented by a continuum of energy minima in reciprocal space. Since a small change of the spiral state wavevector requires a global change of the spin configuration in real space, it is a priori unclear how such systems can fluctuate within the degenerate ground state manifold. Only recently it was proposed that momentum vortices are responsible for the liquidity of the spiral phase and that these systems are closely related to an emergent rank-2 U(1) gauge theory [H. Yan and J. Reuther, Phys. Rev. Research 4, 023175 (2022)]. As a consequence of this gauge structure, four-fold pinch-point singularities were found in a generalized spin correlator. In this article, we use classical Monte Carlo and molecular dynamics calculations to embed the previously studied spiral spin liquid into a broader phase diagram of the square lattice XY model. We find a multitude of unusual phases and phase transitions surrounding the spiral spin liquid such as an effective four-state Potts transition into a colinear double-striped phase resulting from the spontaneous breaking of two coupled Z2\mathbb{Z}_2Z2​ symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a re-entrance phenomenon at low temperatures into a nematic spiral phase. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the four-fold pinch point singularities across the phase diagram
In this paper we identify a previously unexplored type of topological defect in spiral spin liquids---the momentum vortex---and reveal its dominant role in shaping the low-energy physics of such systems. Spiral spin liquids are a class of classical spin liquids featuring subextensively degenerate ground states. They are distinct from spin liquids on geometrically frustrated lattices, in which the ground-state degeneracy is extensive and connected by local spin flips. Despite a handful of experimental realizations and many theoretical studies, a concrete physical picture of their spin liquidity has not been established so far. In this paper, we study a 2D spiral spin liquid model to answer this question. We find that the local momentum vector field can carry topological defects in the form of vortices, which, however, have very different properties from the commonly known spin vortices. The fluctuations of such vortices lead the system into a liquid phase at intermediate temperatures. Furthermore, the effective low-energy theory of such vortices indicates their equivalence to quadrupoles of fractons in a rank-2 U(1) gauge theory or, alternatively, to quadrupoles of disclinations in elasticity theory. At very low temperatures, the system freezes into a glassy state in which these vortices form a rigid network with straight-line domain walls. Our paper sheds light on the nature of spiral spin liquids and also paves the way toward understanding their quantum limit.
We study a model of strongly-correlated systems that incorporates phases such as Fermi liquids, non-Fermi liquids, and superconductivity, in addition to potential intertwined orders. The model describes Fermi surfaces of spinful electron gas or electron liquid coupled to bosons via an intermediate gauge field. Effectively, the coupling imposes constraints and interactions between the fermion spin and the local boson density. This grants the bosons the meaning as the Schwinger boson of the magnetic order and allows us to probe a larger phase space, rather than around the quantum critical point. We design the initial model so that after the boson and gauge fields are integrated out exactly, the resulting fermion-only effective theory only consists of several local interactions, allowing controlled weak-coupling interpretation for certain parameter regions. Consequently, we find a non-Fermi liquid whose degrees of freedom in the effective theory take the form of an \emph{emergent} Fermi surface that has a different Luttinger volume from the original fermions, therefore explicitly violates the Luttinger theorem. Further, this allows a ddd-wave superconducting instability if the effective interacting induced by boson coupling is in the relevant pairing channel. When the emergent Fermi surface undergoes a Lifshitz transition, the carrier type changes. In addition, we attribute the Mott insulating behavior in the strong coupling limit to the loss of emergent Fermi surface and argue the possibility of coexisting magnetic order. Our effective theory also suggests the possibility of charge density waves as a consequence of \emph{effective} strong repulsion and is independent of the Fermi-surface-nesting scenario. We discuss the possible optimal conditions for higher superconducting transition temperature and potential relevances and implications for realistic materials.
Spiral spin liquids possess a subextensively degenerate ground-state manifold, represented by a continuum of energy minima in reciprocal space. Since a small change of the spiral state wave vector requires a global change of the spin configuration in real space, it is a priori unclear how such systems can fluctuate within the degenerate ground-state manifold. Only recently it was proposed that momentum vortices are responsible for the liquidity of the spiral phase and that these systems are closely related to an emergent rank-2 U(1) gauge theory H. Yan et al. [Phys. Rev. Res. 4, 023175 (2022)]. As a consequence of this gauge structure, fourfold pinch-point singularities were found in a generalized spin correlator. In this paper, we use classical Monte Carlo and molecular dynamics calculations to embed the previously studied spiral spin liquid into a broader phase diagram of the square lattice XY model. We find a multitude of unusual phases and phase transitions surrounding the spiral spin liquid such as an effective four-state Potts transition into a collinear double-striped phase resulting from the spontaneous breaking of two coupled Z2{\mathbb{Z}}_{2}Z2​ symmetries. Since this phase is stabilized by entropic effects selecting the momenta away from the spiral manifold, it undergoes a second phase transition at low temperatures into a nematic spiral phase which only breaks one Z2{\mathbb{Z}}_{2}Z2​ symmetry. We also observe a region of parameters where the phase transition into the spiral spin liquid does not break any symmetries and where the critical exponents do not match those of standard universality classes. We study the importance of momentum vortices in driving this phase transition and discuss the possibility of a Kosterlitz-Thouless transition of momentum vortices. Finally, we explore the regime where the rank-2 U(1) gauge theory is valid by investigating the fourfold pinch-point singularities across the phase diagram.