Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness

TL;DR

This work develops a holographic Symm/TO framework that unifies gapped and gapless phases under generalized symmetries by using symmetry TOs and condensable algebras. It shows that phase structure and continuous transitions are dictated by condensation patterns and the reduced symmetry TO M_{/A}, with gaplessness protected whenever M_{/A} is nontrivial. The authors introduce holographic modular bootstrap (holoMB) to extract boundary data from the bulk symmetry TO and apply the method to 1+1D systems, notably S3 and Z2×Z2′ with and without anomalies, uncovering dualities, phase diagrams, and CFT descriptions of critical points. They provide concrete examples—1+1D anomaly-free and anomalous S3, as well as anomalous Z2×Z2′—to illustrate how condensable algebras constrain possible phases, transitions, and their low-energy theories, offering a general route to classify beyond-Landau continuous transitions. The framework promises a powerful, algebraic route to understand gapless states and critical points in higher dimensions as well, through generalized condensations and symmetry TO structures.

Abstract

Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that: (1) For systems with a symmetry described by symmetry TO $M$, their gapped and gapless states are classified by condensable algebras $A$, formed by elementary excitations in $M$ with trivial self/mutual statistics. Such classified states (called $A$-states) can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc., in a unified way. (2) The local low-energy properties of an $A$-state can be calculated from its reduced symmetry TO $M_{/A}$, using holographic modular bootstrap (holoMB) which takes $M_{/A}$ as an input. Here $M_{/A}$ is obtained from $M$ by condensing excitations in $A$. Notably, an $A$-state must be gapless if $M_{/A}$ is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or non-invertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. (4) 1+1D bosonic systems with $S_3$ symmetry have four gapped phases with unbroken symmetries $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, and $\mathbb{Z}_1$. We find a duality between two transitions $S_3 \leftrightarrow \mathbb{Z}_1$ and $\mathbb{Z}_3 \leftrightarrow \mathbb{Z}_2$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT).

Figures (27)

• Figure 1: A 1+1D lattice model with emergent Fibonacci symmetry at low energies. The 1+1D lattice model is constructed from a slab of 2+1D lattice. In the bulk, we have a commuting-projector Hamiltonian that realizes a double-Fibonacci topological order LW0510 with large energy gap. The top boundary $\t\cR$ is a gapped boundary of the double-Fibonacci topological order with large energy gap. The lower boundary is described by an anomalous low energy theory LET$_{ano}$. The low energy theory LET$_{af}$ of the slab has an emergent Fibonacci symmetry below the energy gaps of the bulk and top boundary.
• Figure 2: A 1+1D $\Z_2$-symmetric system (which also has a dual $\t \Z_2$ symmetry JW191213492) is a $\eG\mathrm{au}_{\Z_2}$-system, the system has a categorical symmetry$^{\bigcirc h }$$\Z_2\vee \t \Z_2$, which is described by symmetry TO $\eG\mathrm{au}_{\Z_2}$ -- the quantum double of $\Z_2$ group. Physically, the above statement means that the $\Z_2$ symmetric system (when restricted to its symmetric sub-Hilbert space) can be exactly low-energy simulated by a boundary of bulk $\Z_2$ topological order (TO), described by $\Z_2$ gauge theory. The symmetry TO $\eG\mathrm{au}_{\Z_2}$ has four anyons $\one,e,m,f=e\otimes m$. The possible condensation-induced states in $\eG\mathrm{au}_{\Z_2}$-system are given by the condensable algebras of the symmetry TO, $\cA = \one, \one \oplus e, \one \oplus m$. (a) The $\one \oplus m$-state, corresponding to the $\one \oplus m$-condensed boundary, is the $\Z_2$-symmetric state. (c) The $\one \oplus e$-state is the state with spontaneous $\Z_2$ symmetry breaking. (b) The $\one$-state is the gapless critical point at the continuous transition between $\one \oplus m$-state and $\one \oplus e$-state.
• Figure 3: A $\cA$-state in a $\eM$-system corresponds to a $\cA$-condensed boundary of symmetry TO $\eM$. Such a boundary can be obtained by attaching $\cA$-condensation induced topological order $\eM_{/\cA}$ with $\one$-condensed boundary. The $\cA$-condensation changes $\eM$ to $\eM_{/\cA}$, causing a symmetry TO reduction (an analogue of spontaneous symmetry breaking). $\eM_{/\cA}$ is the reduced symmetry TO (an analogue of unbroken symmetry) of the $\cA$-state.
• Figure 4: A $\Z_2\times \Z_2'$ symmetric system has a symmetry described by 2+1D $\Z_2\times \Z_2'$ topological order $\eG\mathrm{au}_{\Z_2\times \Z_2'}$. Here, the 6 Lagrangian condensable algebras of $\eG\mathrm{au}_{\Z_2\times \Z_2'}$ are represented by the vertices of the hexagon. The gapped phases they correspond to are described in terms of their symmetry-breaking/SPT order. A connecting edge between any pair represents a non-Lagrangian condensable algebra which is strictly included in both of them, and therefore corresponds to a phase transition between them. There is also a trivial condensable algebra $\one$, which is not shown in this picture; it corresponds to a multicritical point between any two gapped phases.The inclusion relations between condensable algebras have implication on the structure of phase diagram. For example, the edge labeled by $\one\oplus m_2$ connecting vertices labeled by $\one\oplus m_1\oplus m_2\oplus m_1m_2$ and $\one\oplus e_1\oplus m_2\oplus e_1m_2$ suggests that the former (gapless) $\one\oplus m_2$-state describes a critical point for the stable continuous transition between the latter two gapped states. For this transition to be non-fine-tuned, the gapless $\one\oplus m_2$-state must have only one symmetric relevant operator, which is indeed the case here (see main text for details).
• Figure 5: The arrows denote possible continuous phase transitions. They are labeled by the associated non-Lagrangian condensable algebras that describe the phase transition. The three lower arrows depict possible symmetry breaking cascades from the two $\Z_2 \times \Z_2'$-symmetric states, the trivial paramagnet and the cluster state SPT. By replacing $\Z_2'$-symmetric state in the middle of the cascade by $\Z_2$- and $\Z_2^d$-symmetric state, we can obtain two other such symmetry breaking cascades. The top arrow labeled by the minimal condensable algebra $\one$ represents a family of possible continuous quantum phase transition between the trivial symmetric phase and the nontrivial SPT phase.TTVV211007599 Note that this condensable algebra can also describe fine-tuned continuous phase transitions between any two of the six gapped phases of this system.