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Register a new userHigher-form symmetry breaking at Ising transitions
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In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking are mountaining, and to catch up with this fast development, new notions of global symmetry are introduced. Among them, the higher-form symmetry, whose symmetry charges are spatially extended, can be used to describe topologically ordered phases as the spontaneous breaking of the symmetry, and consequently unify the unconventional and conventional phases under the same conceptual framework. However, such conceptual tools have not been put into quantitative test except for certain solvable models, therefore limiting its usage in the more generic quantum manybody systems. In this work, we study Z2 higher-form symmetry in a quantum Ising model, which is dual to the global (0-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a non-local order parameter for the higher-form symmetry, analytically in free scalar theories and through unbiased quantum Monte Carlo simulations for the interacting fixed point in (2+1)d. From the scaling form of this extended object, we confirm that the higher-form symmetry is indeed spontaneously broken inside the paramagnetic, or quantum disordered phase (in the Landau sense), but remains symmetric in the ferromagnetic/ordered phase. At the Ising critical point, we find that the higher-form symmetry is also spontaneously broken, even though the 0-form symmetry is preserved. We discuss examples where both the global 0-form symmetry and the dual higher-form symmetry are preserved, in systems with a codimension-1 manifold of gapless points in momentum space. These results provide non-trivial working examples of higher-form symmetry operators, including the first computation of one-form order parameter in an interacting conformal field theory, and open the avenue for their generic implementation in quantum many-body systems.
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Using a specially designed Monte Carlo algorithm with directed loops, we investigate the triangular lattice Ising antiferromagnet with coupling beyond nearest neighbour. We show that the first-order transition from the stripe state to the paramagnet can be split, giving rise to an intermediate nematic phase in which algebraic correlations coexist with a broken symmetry. Furthermore, we demonstrate the emergence of several properties of a more topological nature such as fractional edge excitations in the stripe state, the proliferation of double domain walls in the nematic phase, and the Kasteleyn transition between them. Experimental implications are briefly discussed.
We investigate the behavior of higher-form symmetries at various quantum phase transitions. We consider discrete 1-form symmetries, which can be either part of the generalized concept categorical symmetry (labelled as $tilde{Z}_N^{(1)}$) introduced recently, or an explicit $Z_N^{(1)}$ 1-form symmetry. We demonstrate that for many quantum phase transitions involving a $Z_N^{(1)}$ or $tilde{Z}_N^{(1)}$ symmetry, the following expectation value $ langle left( log O_mathcal{C} right)^2 rangle$ takes the form $langle left( log O_mathcal{C} right)^2 rangle sim - frac{A}{epsilon} P+ b log P $, where $O_mathcal{C}$ is an operator defined associated with loop $mathcal{C}$ (or its interior $mathcal{A}$), which reduces to the Wilson loop operator for cases with an explicit $Z_N^{(1)}$ 1-form symmetry. $P$ is the perimeter of $mathcal{C}$, and the $b log P$ term arises from the sharp corners of the loop $mathcal{C}$, which is consistent with recent numerics on a particular example. $b$ is a universal microscopic-independent number, which in (2+1)d is related to the universal conductivity at the quantum phase transition. $b$ can be computed exactly for certain transitions using the dualities between (2+1)d conformal field theories developed in recent years. We also compute the strange correlator of $O_mathcal{C}$: $S_{mathcal{C}} = langle 0 | O_mathcal{C} | 1 rangle / langle 0 | 1 rangle$ where $|0rangle$ and $|1rangle$ are many-body states with different topological nature.
Close to the quantum critical point of the transverse-field Ising spin-chain model, an exotic dynamic spectrum was predicted to emerge upon a perturbative longitudinal field. The dynamic spectrum consists of eight particles and is governed by the symmetry of the $E_8$ Lie algebra. Here we report on high-resolution terahertz spectroscopy of quantum spin dynamics in the ferromagnetic Ising-chain material CoNb$_2$O$_6$. At 0.25 K in the magnetically ordered phase we identify characteristics of the first six $E_8$ particles, $mathbf{m}_1$ to $mathbf{m}_6$, and the two-particle ($mathbf{m}_1+mathbf{m}_2$) continuum in an applied transverse magnetic field of $B_c^{1D}=4.75$ T, before the three-dimensional magnetic order is suppressed above $B_c^{3D}approx 5.3$ T. The observation of the higher-energy particles ($mathbf{m}_3$ to $mathbf{m}_6$) above the low-energy two-particle continua features quantum many-body effects in the exotic dynamic spectrum.
We study the Ising model two-point diagonal correlation function $ C(N,N)$ by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable $lambda$, the $j$-particle contributions, $ f^{(j)}_{N,N}$. The corresponding $ lambda$ extension of the two-point diagonal correlation function, $ C(N,N; lambda)$, is shown, for arbitrary $lambda$, to be a solution of the sigma form of the Painlev{e} VI equation introduced by Jimbo and Miwa. Linear differential equations for the form factors $ f^{(j)}_{N,N}$ are obtained and shown to have both a ``Russian doll nesting, and a decomposition of the differential operators as a direct sum of operators equivalent to symmetric powers of the differential operator of the elliptic integral $ E$. Each $ f^{(j)}_{N,N}$ is expressed polynomially in terms of the elliptic integrals $ E$ and $ K$. The scaling limit of these differential operators breaks the direct sum structure but not the ``Russian doll structure. The previous $ lambda$-extensions, $ C(N,N; lambda)$ are, for singled-out values $ lambda= cos(pi m/n)$ ($m, n$ integers), also solutions of linear differential equations. These solutions of Painleve VI are actually algebraic functions, being associated with modular curves.
Near the transverse-field induced quantum critical point of the Ising chain, an exotic dynamic spectrum consisting of exactly eight particles was predicted, which is uniquely described by an emergent quantum integrable field theory with the symmetry of the $E_8$ Lie algebra, but rarely explored experimentally. Here we use high-resolution terahertz spectroscopy to resolve quantum spin dynamics of the quasi-one-dimensional Ising antiferromagnet BaCo$_2$V$_2$O$_8$ in an applied transverse field. By comparing to an analytical calculation of the dynamical spin correlations, we identify $E_8$ particles as well as their two-particle excitations.
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