Higher-form symmetry breaking at Ising transitions

Global symmetries are instrumental in organizing our understanding of phases of matter. The celebrated Landau paradiagm classifies phases according to broken symmetries, which also determines the universality classes of transitions between phases. Symmetry principles become even more powerful from the point of view of long wavelength, low-energy physics, as the renormalization group fixed points (i.e. IR) often embody more symmetries than the microscopic lattice model (i.e. UV), which is the phenomenon of emergent symmetry Moessner and Sondhi (2001); Isakov and Moessner (2003); Senthil et al. (2004); He et al. (2016); Ma et al. (2019). A common example is the emergence of continuous space-time symmetries in the field-theoretical description of a continuous phase transition Altland and Simons (2010). It is even plausible that a critical point is determined up to finite choices by its full emergent symmetry, which is the basic philosophy (or educated guess) behind the conformal bootstrap program Poland et al. (2019).

Modern developments in quantum many-body physics have significantly broadened the scope of quantum phases beyond the Landau classification Wen (2019a). For these exotic phases, more general notions of global symmetry are called for to completely characterize the phases and the associated phase transitions. Intuitively, these “beyond Landau” phases do not have local order parameters. Instead, non-local observables are often needed to characterize them. For a well-known example, confined and deconfined phases of a gauge theory are distinguished by the behavior of the expectation value of Wilson loop operators Fradkin (2013); Gregor et al. (2011). To incorporate such extended observables into the symmetry framework, higher-form symmetries Nussinov and Ortiz (2009a, b); Gaiotto et al. (2015), and more generally algebraic symmetries Ji and Wen (2019a); Kong et al. (2020) have been introduced. These are symmetries whose charged objects are spatially extended, e.g. strings and membranes. In other words, their symmetry transformations only act nontrivially on extended objects. Most notably, spontaneous breaking of such higher symmetries can lead to highly entangled phases, such as topological order Gaiotto et al. (2015). Therefore, even though topologically ordered phases are often said to be beyond the Landau paradiagm, they can actually be understood within a similar conceptual framework once higher symmetries are included. In addition, just as the usual global symmetries, higher-form symmetries can have quantum anomalies Gaiotto et al. (2015), which lead to strong non-perturbative constraints on low-energy dynamics Gaiotto et al. (2017).

In this work, we make use of the prototypical continuous quantum phase transition, the Ising transition, to elucidate the functionality of the higher-form symmetry. The motivation to re-examine the well-understood Ising transition is the following: in addition to the defining 0-form $\mathbb{Z}_{2}$ symmetry, the topological requirement that $\mathbb{Z}_{2}$ domain walls must be closed (in the absence of spatial boundary) can be equivalently formulated as having an unbreakable $\mathbb{Z}_{2}$ $(D-1)$-form symmetry, where $D$ is the spatial dimension. Gapped phase on either side of the transition spontaneously breaks one and only one of the two symmetries. Therefore to correctly determine the full emergent internal symmetry in the Ising CFT, the $\mathbb{Z}_{2}$ higher-form symmetry should be taken into account. For $D=2$, the $1$-form symmetry manifests more clearly in the dual formulation Wegner (1971), namely as the confinement-deconfinement transition of a $\mathbb{Z}_{2}$ gauge theory, which will shed light on higher-form symmetry breaking transitions in a concrete setting.

A basic question about a global symmetry is whether it is broken spontaneously or not in the ground state. For clarity, let us focus on the $D=2$ case. It is well-known that the Ising symmetric, or “quantum disordered” phase, spontaneously breaks the higher-form symmetry, and the opposite in the Ising symmetry-breaking phase. The fate at the critical point remains unclear to date. To diagonose higher-form symmetry breaking, we compute the ground state expectation value of the “order parameter” for the higher-form symmetry – commonly known as the disorder operator in literature Kadanoff and Ceva (1971); Fradkin (2017); Wu et al. (2020); Wang et al. (2021); Wu et al. (2021), which creates a domain wall in the Ising system. Spontaneous breaking of the $\mathbb{Z}_{2}$ $1$-form symmetry is signified by the perimeter law for the disorder operator. In the dual formulation, the corresponding object is the Wilson loop operator. Through large-scale QMC simulations, we find numerically that at the transition, the disorder operator defined on a rectangular region scales as $l^{s}e^{-a_{1}l}$, where $l$ is the perimeter of the region, and $s>0$ is a universal constant. We thus conclude that the 1-form symmetry is spontaneously broken at the (2+1)d Ising transition, and it remains so in the disordered phase of the model. This is in stark contrast with the $D=1$ case, where the disorder operator has a power-law decay.

To corroborate the numerical results, we consider generally disorder operator corresponding to a 0-form $\mathbb{Z}_{2}$ symmetry in a free scalar theory in $D$ dimensions, which is a stable fixed point for $D\geq 3$. We show that for the kind of $\mathbb{Z}_{2}$ symmetry in this case, the disorder operator can be related to the 2nd Renyi entropy. Therefore, the disorder operator also obeys a “perimeter” (i.e. volume of the boundary) scaling, with possibly multiplicative power-law correction. Whether the higher-form symmetry is broken or not is determined by the subleading power-law corrections. We also discuss other free theories, such as a Fermi liquid, where the decay of the disorder operator is in between the “perimeter” and the “area” laws, and therefore no higher-form symmetry breaking.

The rest of the paper is organized as follows. In Sec. II we review higher-form symmetry and its spontaneous breaking, and its relevancy in conventional phases. We also consider higher-form symmetry breaking in free and interacting conformal field theories. In Sec. III we specialize to the setting of quantum Ising model in (2+1)d and define the disorder operator. Sec. IV presents the main numerical results from quantum Monte Carlo simulations, which reveal the key evidence of the 1-form symmetry breaking at the $(2+1)$d Ising transition. Sec. V outlines a few immediate directions about the higher-form symmetry breaking and their measurements in unbiased numerical treatments in other quantum many-body systems.

Consider a quantum many-body system in $D$ spatial dimensions. Global symmetries are unitary transformations which commute with the Hamiltonian. Typically the symmetry transformation is defined over the entire system, and charges of the global symmetry are carried by particle-like objects.

An important generalization of global symmetry is the higher-form symmetry Gaiotto et al. (2015). For an integer $p\geq 0$, $p$-form symmetry transformations act nontrivially on $p$-dimensional objects. In other words, “charges” of $p$-form symmetry are carried by extended objects. In this language, the usual global symmetry is 0-form as the particle-like object is of 0-dimension. $p$-form symmetry transformations themselves are unitary operators supported on each codimension-$p$ (i.e. spatial dimension $(D-p)$) closed submanifold $M_{D-p}$. In particular, it means that there are infinitely many symmetry transformations in the thermodynamic limit. In this work we will only consider discrete, Abelian higher-form symmetry, so for each submanifold $M_{D-p}$ the associated unitary operators form a finite Abelian group $G$. Physically, higher-form symmetry means that the certain $p$-dimensional objects are charged under the group $G$, and the quantum numbers they carry constrain the processes of creation, annihilation and splitting etc. In particular, these extended objects are “unbreakable”, i.e. they are always closed and can not end on $(p-1)$-dimensional objects.

For a concrete example, let us consider (2+1)$d$ $\mathbb{Z}_{2}$ gauge theory definend on a square lattice. Each edge of the lattice is associated with a $\mathbb{Z}_{2}$ gauge field (i.e. a qubit), subject to the Gauss’s law at each site $v$:

Here $e$ runs over edges ending on $v$.

The divergence-free condition implies that there are no electric charges in the gauge theory. In other words, all $\mathbb{Z}_{2}$ electric field lines must form loops. An electric loop can be created by applying the following operator along any closed path $\gamma$ on the lattice:

The corresponding $\mathbb{Z}_{2}$ 1-form symmetry operator is defined as

for any closed path $\gamma^{\star}$ on the dual lattice. Here the subscript $m$ in $W_{m}$ indicates that this is actually the string operator for $\mathbb{Z}_{2}$ flux excitations. In field theory parlance, $W_{e}$ is the Wilson operator of the $\mathbb{Z}_{2}$ gauge theory, and $W_{m}$ is the corresponding Gukov-Witten operator Gukov and Witten (2008).

We notice that the $W_{m}(\gamma^{\star})$ operator is in fact the product of Gauss’s law term $\prod_{v\in e}\tau_{e}^{x}$ for all $v$ in the region enclosed by $\gamma^{\star}$. In other words, the smallest possible $\gamma^{\star}$ is a loop around one vertex $v$, and the fact tht $W_{m}(\gamma^{\star})$ is conserved by the dynamics means that the gauge charge at site $v$ must be conserved (mod 2) as well. Therefore, the $\mathbb{Z}_{2}$ gauge theory with electric 1-form symmetry is one with completely static charges, including the case with no charges at all. For applications in relativistic quantum field theories, it is usually further required that the 1-form symmetry transformation is “topological”, i.e. not affected by local deformation of the loop $\gamma^{\star}$, which is equivalent to the absence of gauge charge as given in Eq. (1).

It is instructive to consider how the 1-form charge of an electric loop can be measured. This is most clearly done in space-time: to measure a $p$-dimensional charge, one “wraps” around the charge by a $(D-p)$-dimensional symmetry operator. Appying the symmetry transformation is equivalent to shrinking the symmetry operator, and in $(D+1)$ spacetime, because of the linking the two must collide, and the non-commutativity (e.g. between $W_{e}$ and $W_{m}$) measures the charge value. We illustrate the process for $p=0$ (Fig. 1(a)) and $p=1$ (Fig. 1(b)), in three-dimensional space-time.

Now consider the following Hamiltonian of Ising gauge theory:

where $J,K>0$. When $J\ll K$, the ground state is in the deconfined phase, which can be viewed as an equal-weight superposition of all closed $\mathbb{Z}_{2}$ electric loops. In this phase, the $\mathbb{Z}_{2}$ 1-form symmetry is spontaneously broken. When $J\gg K$, the ground state is a product state with $\tau_{e}^{x}=1$ everywhere, and the 1-form symmetry is preserved. This is the confined phase. Similar to the usual boson condensation, the expectation value of the electric loop creation operator $W_{e}(\gamma)$ can be used to characterize the 1-form symmetry breaking phase, which obeys perimeter law in the deconfined phase.

This example shows that higher-forms symmetry naturally arises in gauge theories. In condensed matter applications, gauge theories are usually emergent Senthil et al. (2004); Ma et al. (2018), which means that dynamical gauge charges are inevitably present and the electric 1-form symmetry is explicitly broken. Even under such circumstances, at energy scales well below the electric charge gap, the theory still has an emergent 1-form symmetry Wen (2019b).

Let us now discuss more generally the spontaneous breaking of higher-form symmetry Gaiotto et al. (2015); Lake (2018); Hofman and Iqbal (2019). We will assume that the symmetry group is discrete. For a $p$-form symmetry, a charged object is created by an extended operator $W(C)$ defined on a $p$-dimensional manifold $C$. When the symmetry is unbroken, we have

where $\mathrm{Area}(C)$ is the volume of a minimal $(p+1)$-dimensional manifold whose boundary is $C$. $t_{p+1}$ can be understood as the “tension” of the $(p+1)$-dimensional manifold. This generalizes the exponential decay of charged local operator for the 0-form case. On the other hand, when the symmetry is spontaneously broken,

where $\mathrm{Perimeter}(C)$ denotes the “volume” of $C$ itself. Importantly the expectation value only depends locally on $C$, which is the analog of the factorization of the correlation function of local order parameter $\langle O(x)O^{\dagger}(y)\rangle\approx\langle O(x)\rangle\langle O^{\dagger}(y)\rangle$ for $0$-form symmetry. One can then redefine the operator $W(C)$ to remove the perimeter scaling and in that case $\langle W(C)\rangle$ would approach a constant in the limit of large $C$ Hastings and Wen (2005). At critical point, however, subleading corrections become important, which will be examined below.

The $\mathbb{Z}_{2}$ gauge theory is famously dual to a quantum Ising model Kogut (1979). In fact, more generally, there is a duality transformation which relates a system with global $\mathbb{Z}_{2}$ 0-form symmetry (in the $\mathbb{Z}_{2}$ even sector) to one with global $\mathbb{Z}_{2}$ $(D-1)$-form symmetry, a generalization of the Kramers-Wannier duality in (1+1)d.

Let us now review the duality in (2+1)d. The dual Ising spins are defined on plaquettes, whose centers form the dual lattice. For a given edge $e$ of the original lattice, denote the two adjacent plaquettes by $p$ and $q$, as shown in the figure below:

The duality map is defined as follows:

Note that the expression automatically ensures $\prod_{p}\sigma^{x}_{p}=1$ in a closed system, so the dual spin system has a $\mathbb{Z}_{2}$ 0-form symmetry generated by $S=\prod_{p}\sigma_{p}^{x}$, and the map can only be done in the $\mathbb{Z}_{2}$ even sector with $S=1$ ¹¹1In a sense the $\mathbb{Z}_{2}$ symmetry is gauged. In fact one way to derive the duality is to first gauge the $\mathbb{Z}_{2}$ symmetry and then perform gauge transformations to eliminate the Ising matter.. Conversely, the mapping also implies $\prod_{v\in e}\tau_{x}^{e}=1$, and in fact $W_{m}(\gamma^{\star})=1$ for any $\gamma^{*}$, i.e. the $\mathbb{Z}_{2}$ 1-form symmetry is strictly enforced.

In the dual model, the electric field line of the $\mathbb{Z}_{2}$ gauge theory becomes the domain walls separating regions with opposite Ising magnetizations. Therefore, a Wilson loop $W_{e}(\gamma)$ maps to

where $\partial M=\gamma$, i.e. $M$ is the region enclosed by $\gamma$. Physically $X_{M}$ flips all the Ising spins in the region $M$, thus creating a domain wall along the boundary $\gamma$. It is called the disorder operator for the Ising system, which will be the focus of our study below.

Under the duality map, the Hamiltonian becomes

The phases of the gauge theory can be readily understood in the dual representation. For $K\gg J$, the $\mathbb{Z}_{2}$ gauge theory is in the deconfined phase, which means that the ground state contains arbitrarily large electric loops. For the dual Ising model, the ground state is disordered, with all $\sigma_{p}^{x}=1$. If we work in the $\sigma^{z}$ eigenbasis (which is natural to discuss symmetry breaking), the ground state wavefunction is given by

Namely we pick any basis state and apply the ground state projector. Expanding out the projector, one can see that the wavefunction is an equal superposition of all domain wall configurations, i.e. a condensation of domain walls. Since the domain walls carry $\mathbb{Z}_{2}$ 1-form charges, the condensation breaks the 1-form symmetry spontaneously, much like the Bose condensation spontaneously breaks the conservation of particle numbers

In the other limit $K\ll J$, the gauge theory is confined. Correspondingly, the dual Ising model is in the ferromagnetically ordered phase: there are two degenerate ground states $\ket{\uparrow\cdots\uparrow}$ and $\ket{\downarrow\cdots\downarrow}$. There are no domain walls at all in the limit $K\rightarrow 0$. When a small but finite $K/J$ is turned on, quantum fluctuations create domain walls on top of the fully polarized ground states, but these domain walls are small and sparse.

In the following we study $1$-form symmetry breaking in the transverse field Ising (TFI) model which gives rise to the $(2+1)$d Ising transition. We have reviewed the connection with the $\mathbb{Z}_{2}$ gauge thory in Sec. II, as well as the 1-form symmetry in the Ising spin system. We will now focus more on the quantitative aspects of the TFI model. Even though the TFI model and the $\mathbb{Z}_{2}$ lattice gauge theory are equivalent by the duality map, we choose to work with the TFI model here because the numerical simulation is more straightforward.

We will now consider a square lattice with one Ising spin per site, and the global Ising symmetry is generated by $S=\prod_{\mathbf{r}}\sigma_{\mathbf{r}}^{x}$. There are, generally speaking, two phases: a “disordered” phase, where the Ising symmetry is preserved by the ground state ²²2We note that there are in fact two distinct types of Ising-disordered phases in 2D, one trivial paramagnet and the other one a nontrivial Ising symmetry-protected topological phase., and an ordered phase where the ground states spontaneously break the symmetry. They are separated by a quantum phase transition, described by a conformal field theory with $\mathbb{Z}_{2}$ symmetry. It is well-understood how to characterize the Ising symmetry breaking (and its absence) in the three cases: consider the two-point correlation function of the order parameter $\sigma^{z}_{\mathbf{r}}$. The asympotic forms of the correlation function $\langle\sigma_{\mathbf{r}}^{z}\sigma_{\mathbf{r^{\prime}}}^{z}\rangle$ for large $|\mathbf{r}-\mathbf{r}^{\prime}|$ distinguish the three cases:

In both the disordered phase and the quantum critical point, the Ising symmetry is preserved because of the absence of long-range order. The prototypical lattice model that displays all these features is the TFI model defined on a square lattice:

Note that this is the same as Eq. (9), but we have set $J=1$ and renamed $K$ by $h$, to align with the standard convention in literature. The model is in the ordered (disordered) phase for $h\ll 1$ ($h\gg 1$). The precise location of the critical point varies with dimension, $h_{c}=1$ in $D=1$ and $h_{c}=3.044$ in $D=2$ Blöte and Deng (2002); Liu et al. (2019).

We will be interested in the disorder operator:

where $M$ is a rectangle region in the lattice, illustrated in Fig. 2. In Ref. Ji and Wen, 2019a this operator is called the patch symmetry operator.

When $X_{M}$ is applied to e.g. $\ket{\uparrow\cdots\uparrow}$, a domain wall is created along the boundary of the region $M$. These operators are charged under the dual $\mathbb{Z}_{2}$ 1-form symmetry. One can easily see that $\bra{\psi_{h=\infty}}X_{M}\ket{\psi_{h=\infty}}=1$, and $\bra{\psi_{h=0}}X_{M}\ket{\psi_{h=0}}=0$. More generally,

when $M$ is sufficiently large compared to the correlation length. Here $l$ is the perimeter of the boundary of $M$, and $A$ is the area of $M$. The coefficients $a$ and $b$ can be computed perturbatively in the limit of large and small $h$. In 2D, take $M$ to be a square of perimeter $l$, so Perimeter$(M)=l$ and the Area$(M)=l^{2}/16$. We can find that for large $l$:

In this section we study the disorder operator in the (2+1)d TFI model. We employ the Stochastic Series Expansion (SSE) quantum Monte Carlo method Sandvik (2003); Syljuåsen and Sandvik (2002); Syljuåsen (2003); Da Liao et al. (2021) to simulate the Hamiltonian in Eq. (27). In particular, to be able to directly access the disorder operator in Eq. (28), instead of implementing the algorithm in the conventional $\sigma^{z}$ basis we choose to work in the $\sigma^{x}$ basis and construct the highly efficient directed loop algorithm therein Syljuåsen and Sandvik (2002). The implementation details of the SSE-QMC algorithm are given in Appendix A.

In our numerical simulations, we choose $M$ to be a rectangular region of size $R_{1}\times R_{2}$ (i.e. the region contains $R_{1}R_{2}$ sites), and denote the perimeter $l=2(R_{1}+R_{2})$. As shown in Fig. 2 (a) and (b), for finite-size studies, we fix the aspect ratio $R_{2}/R_{1}=1$ of square shape and $2$ of rectangle shape. The linear system size of the lattice is $L$ and at the critical point we scale the inverse temperature $\beta=1/T\sim L$ to access the thermodynamic limit.

As discussed in the beginning of the paper, in recent years, new types of quantum phases and phase transitions that are “beyond Laudau” are flourishing, exhibiting topological order, emergent gauge field and fractionalization. Higher-form symmetries and their spontaneous breaking are new conceptual tools introduced to provide an unified framework for both conventional and exotic phases. A quantum phase, gapped or gapless, is fundamentally characterized by its emergent symmetry and the associated anomaly. While the philosophy went back to the Landau classification of phase transitions, the power of this perspective has only begun to unfold recently with the introduction of generalized global symmetries.

Here we re-examine the familiar Ising symmetry-breaking transition, arguably the simplest conformal field theory, from the emergent symmetry perspective. A $D$-dimensional Ising system has a “hidden” $\mathbb{Z}_{2}$ $(D-1)$-form symmetry, whose charges are Ising domain walls. Gapped phases in this system are associated to the spontaneous breaking of the enlarged symmetry (0-form and $(D-1)$-form symmetries). It is then of great interest to determine the symmetry breaking pattern at the critical point, to complete our understanding of the global phase diagram from the emergent symmetry point of view.

In this work we determine the scaling form of the disorder operator in Ising CFTs when $D>1$. The most challenging case is $D=2$ where the transition is described by the interacting Wilson-Fisher fixed point, and we exploit large-scale quantum Monte Carlo simulations. We use the disorder operator of the Ising system to probe the breaking of the dual higher-form symmetry. We find numerically that at the critical point of the 2D quantum Ising model, the one-form disorder operator exhibits sponatenous symmetry breaking as in the disordered phase, whereas in the ordered phase, the one-form symmetry is intact.

The disorder operator is intimately related to a line defect (also called a twist operator) in a Ising CFT, around which the spin operator sees an anti-periodic boundary condition. In fact, a line defect is nothing but the boundary of a disorder operator. It is believed that in general such a line defect can flow to a conformal one at low energy, which is indeed consistent with a perimeter law scaling for the expectation value of the disorder operator ³³3We are grateful for Shu-Heng Shao for discussions on this point.. Local properties of disorder line defects have been previously investigated in Ref. [Billó et al., 2013] and [Gaiotto et al., 2014]. It will be interesting to understand the relation between the local properties with the universal corner contributions to the disorder operator Bueno et al. (2015).

Our findings, besides elucidating the physics of quantum Ising systems from a new angle, provides a working example of higher-form symmetry at practical use. Similar physical systems can be studied, for example, the disordered operator constructed in this work is readily generalized to the $(2+1)$d XY transition and can be measured with unbiased QMC simulations. Another important direction is to study other higher-form symmetry breaking transitions, such as 1-form symmetry breaking transition in 3D systems. It would also be interesting to investigate the ultility of the disorder operator in the topological Ising paramagnetic phase. More applications in quantum lattice models are awaiting to be explored, and will certainly lead to new insight for a new framework that unifies our understanding of the exotic quantum phases and transitions going beyond the Landau paradigm and those within.

Note added.- We would like to draw the reader’s attention to few closely related recent works by X.-C. Wu, C.-M. Jian and C. Xu Wu et al. (2020, 2021) and by some of the present author on scaling of disorder oeprator at $(2+1)$d U(1) quantum criticality Wang et al. (2021).

JRZ, ZY and ZYM thank the enjoyable discussions with Yan-cheng Wang and Yang Qi and acknowledge the support from the RGC of Hong Kong SAR of China (Grant Nos. 17303019 and 17301420), MOST through the National Key Research and Development Program (Grant No. 2016YFA0300502) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB33000000). We are grateful for Xiao-Gang Wen, Shu-Heng Shao and William William-Krempa for helpful comments. MC would like to thank Zhen Bi, Wenjie Ji and Chao-Ming Jian for enlightening discussions and acknowledges support from NSF (DMR-1846109). We thank the Computational Initiative at the Faculty of Science and the Information Technology Services at the University of Hong Kong, and the Tianhe-1A, Tianhe-2 and Tianhe-3 prototype platforms at the National Supercomputer Centers in Tianjin and Guangzhou for their technical support and generous allocation of CPU time.