In this work, we investigate the one-dimensional $XX$ lattice model and its cousins through the lens of momentum and winding $U(1)$ symmetries. We distinguish two closely related $\mathbb{Z}_2$ symmetries based on their relation to the $U(1)$ symmetries, and establish a web of $\mathbb{Z}_2$ -gauging relations among these models, rooted in two fundamental seeds: the $ XY \pm YX$ models. These two seeds, each self-dual under gauging of the respective $\mathbb{Z}_2$ -symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the $XX$ model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the $c =1$ compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.In this work, we investigate the one-dimensional $XX$ lattice model and its cousins through the lens of momentum and winding $U(1)$ symmetries. We distinguish two closely related $\mathbb{Z}_2$ symmetries based on their relation to the $U(1)$ symmetries, and establish a web of $\mathbb{Z}_2$-gauging relations among these models, rooted in two fundamental seeds: the $ XY \pm YX$ models. These two seeds, each self-dual under gauging of the respective $\mathbb{Z}_2$-symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the $XX$ model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the $c =1$ compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.

arXiv:2503.01831Physical review. B./Physical review. BJun 23, 20254 citations

Self-dual criticality in three-dimensional $\mathbb{Z}_2$ gauge theory with matterSelf-dual criticality in three-dimensional $\mathbb{Z}_2$ gauge theory with matter

A. M. Somoza, Pablo Serna, Adam Nahum

The simplest topologically ordered phase in 2+1D is the deconfined phase of $Z_2$ gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (`` $e$ '' and `` $m$ '' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the $Z_2$ gauge-Higgs model on cubic lattices of linear size $L\leq 96$ . Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators $A$ and $S$ . All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions $x_A$ and $x_S$ and dynamical exponent $z$ . We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.The simplest topologically ordered phase in 2+1D is the deconfined phase of $Z_2$ gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (``$e$'' and ``$m$'' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the $Z_2$ gauge-Higgs model on cubic lattices of linear size $L\leq 96$. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators $A$ and $S$. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions $x_A$ and $x_S$ and dynamical exponent $z$. We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.

arXiv:2012.15845Dec 31, 20202 citations

Self-dual Quantum Electrodynamics on the boundary of 4d Bosonic Symmetry Protected Topological States

Zhen Bi, Kevin Slagle, Cenke Xu

We study $3d$ (or $(3+1)d$ ) Quantum Electrodynamics (QED) realized on the boundary of $4d$ (or $(4+1)d$ ) bosonic symmetry protected topological (BSPT) states, using a systematic nonlinear sigma model (NLSM) field theory description of BSPT states. We demonstrate that many of these QED states have an exact electric-magnetic duality due to the symmetry of the BSPT states in the $4d$ bulk. The gauge charge and Dirac monopole both carry projective representations of the bulk symmetry, and the emergent gapless photons of the QED phase also transform nontrivially under the bulk symmetry. Some of these QED boundary states can be further driven into a $3d$ $\mathbb{Z}_2$ topological order, and the statistics and symmetry transformation of its point particle and vison loop excitations guarantee that this topological order cannot be driven into a trivial confined or Higgs phase. With a finite fourth dimension, the entire system becomes a $3d$ lattice, the self-dual QED and the $\mathbb{Z}_2$ topological order can coexist on two opposite boundaries respectively, which together constitute an exotic $3d$ self-dual "topological photon phase".We study $3d$ (or $(3+1)d$) Quantum Electrodynamics (QED) realized on the boundary of $4d$ (or $(4+1)d$) bosonic symmetry protected topological (BSPT) states, using a systematic nonlinear sigma model (NLSM) field theory description of BSPT states. We demonstrate that many of these QED states have an exact electric-magnetic duality due to the symmetry of the BSPT states in the $4d$ bulk. The gauge charge and Dirac monopole both carry projective representations of the bulk symmetry, and the emergent gapless photons of the QED phase also transform nontrivially under the bulk symmetry. Some of these QED boundary states can be further driven into a $3d$ $\mathbb{Z}_2$ topological order, and the statistics and symmetry transformation of its point particle and vison loop excitations guarantee that this topological order cannot be driven into a trivial confined or Higgs phase. With a finite fourth dimension, the entire system becomes a $3d$ lattice, the self-dual QED and the $\mathbb{Z}_2$ topological order can coexist on two opposite boundaries respectively, which together constitute an exotic $3d$ self-dual "topological photon phase".

arXiv:1504.04373Apr 16, 20156 citations

In this work, we investigate the one-dimensional

XX

lattice model and its cousins through the lens of momentum and winding

U(1)

symmetries. We distinguish two closely related

\mathbb{Z}_2

symmetries based on their relation to the

U(1)

symmetries, and establish a web of

\mathbb{Z}_2

-gauging relations among these models, rooted in two fundamental seeds: the $ XY \pm YX$ models. These two seeds, each self-dual under gauging of the respective

\mathbb{Z}_2

-symmetries, possess manifestly symmetric conserved charges, making transparent the connection between the noninvertible symmetries and the Kramers-Wannier duality. By leveraging the self-dualities of these two seed models, we derive the self-dualities of their cousins, including the

XX

model and the Levin-Gu model, through appropriate gauging procedures. Moreover, under these gauging schemes, the lattice T-duality matrices take the form of the identity matrix. These lattice models flow to the

c =1

compact boson conformal field theory, with a twist that depends on the lattice size modulo four. Finally, we unify the mapping structures of local conserved charges across these models, providing a comprehensive framework for understanding their symmetries and dualities.

Self-dual criticality in three-dimensional

\mathbb{Z}_2

gauge theory with matter

The simplest topologically ordered phase in 2+1D is the deconfined phase of

Z_2

gauge theory, realized for example in the toric code. This phase permits a duality that exchanges electric and magnetic excitations (``

e

'' and ``

m

'' particles). Condensing either particle while the other remains gapped yields a phase transition with 3D Ising exponents. More mysterious, however, is the transition out of the deconfined phase when self-duality symmetry is preserved. If this transition is continuous, which has so far been unclear, then it may be the simplest critical point for which we still lack any useful continuum Lagrangian description. This transition also has a soft matter interpretation, as a multicritical point for classical membranes in 3D. We study the self-dual transition with Monte Carlo simulations of the

Z_2

gauge-Higgs model on cubic lattices of linear size

L\leq 96

. Our results indicate a continuous transition: for example, cumulants show a striking parameter-free scaling collapse. We estimate scaling dimensions by using duality symmetry to distinguish the leading duality-odd/duality-even scaling operators

A

and

S

. All local operators have large scaling dimensions, making standard techniques for locating the critical point ineffective. We develop an alternative using ``renormalization group trajectories'' of cumulants. We check that two- and three-point functions, and temporal correlators in the Monte-Carlo dynamics, are scale-invariant, with scaling dimensions

x_A

and

x_S

and dynamical exponent

z

. We also give a picture for emergence of 1-form symmetries, in some parts of the phase diagram, in terms of ``patching'' of membranes/worldsurfaces. We relate this to the percolation of anyon worldlines in spacetime. Analyzing percolation yields a fourth exponent for the self-dual transition. We propose variations of the model for further investigation.

We study

3d

(or

(3+1)d

) Quantum Electrodynamics (QED) realized on the boundary of

4d

(or

(4+1)d

) bosonic symmetry protected topological (BSPT) states, using a systematic nonlinear sigma model (NLSM) field theory description of BSPT states. We demonstrate that many of these QED states have an exact electric-magnetic duality due to the symmetry of the BSPT states in the

4d

bulk. The gauge charge and Dirac monopole both carry projective representations of the bulk symmetry, and the emergent gapless photons of the QED phase also transform nontrivially under the bulk symmetry. Some of these QED boundary states can be further driven into a

3d

\mathbb{Z}_2

topological order, and the statistics and symmetry transformation of its point particle and vison loop excitations guarantee that this topological order cannot be driven into a trivial confined or Higgs phase. With a finite fourth dimension, the entire system becomes a

3d

lattice, the self-dual QED and the

\mathbb{Z}_2

topological order can coexist on two opposite boundaries respectively, which together constitute an exotic

3d

self-dual "topological photon phase".