Local Strong-to-Weak Spontaneous Symmetry Breaking

Abstract: We propose a local notion of strong-to-weak spontaneous symmetry breaking (SW-SSB), through a local one-point fidelity correlator. Compared with the previous definition in terms of a two-point fidelity correlator, our local formulation offers two key advantages: (1) it is easier to detect in large systems: for a system of size $N$ and with ${\rm poly}(N)$ amount of resources, one can detect the local fidelity order up to volume scale $O(\log(N))$; and (2) the local SW-SSB order remains well defined in the thermodynamic limit, where the density matrix itself is not well defined. We show that key features of SW-SSB, including stability under finite-depth symmetric channels and long-range conditional mutual information, persist within this local framework. Our definition is conceptually analogous to local thermalization, as exemplified by pure states obeying the eigenstate thermalization hypothesis (ETH). For critical states, the local one-point fidelity correlator defines an interesting class of defect problems. We demonstrate the applicability of the local formulation through several concrete examples, and derive the universal scaling behavior of the local fidelity correlator in a range of critical systems, including ground states of conformal field theories as well as ballistic and diffusive free-fermion metals.

• The paper introduces a novel local order parameter using one-point fidelity correlators to diagnose strong-to-weak symmetry breaking in quantum matter.
• It demonstrates computational efficiency and a well-defined thermodynamic limit, bridging global and local symmetry diagnostics via rigorous inequalities.
• Numerical and analytical results in systems like CFT ground states and Fermi metals reveal distinct scaling behaviors critical for phase classification.

Local Strong-to-Weak Spontaneous Symmetry Breaking: A Technical Overview

Introduction and Motivation

Spontaneous symmetry breaking (SSB) is central to delineating phases of quantum matter, but in open quantum systems or mixed states, robustness and characterization of symmetry breaking phenomena become fundamentally more involved. This is due to a crucial distinction: strong symmetry, defined as $U \rho U^\dagger = \rho$ together with $U\rho \propto \rho$, can be reduced to weak symmetry when acting on ensembles. This dichotomy has motivated the focus on "strong-to-weak spontaneous symmetry breaking" (SW-SSB), a property extensively relevant for classifying mixed-state phases, protected quantum information, emergent hydrodynamics, and robust quantum memory (2605.28967).

Conventional diagnostics—two-point correlators—are insufficient to probe SW-SSB, as their sensitivity is only to linear averages over the density matrix. Detecting SW-SSB demands nonlinear functionals, typically quantum fidelities, requiring access to the global density matrix $\rho$ and entailing exponential overhead with system size. This research introduces a local notion of SW-SSB using one-point fidelity correlators, making the diagnosis both scalable and well-defined intrinsically in the thermodynamic limit.

Local One-Point Fidelity Correlator: Definition and Properties

The local measure of SW-SSB is formulated in terms of the one-point local fidelity correlator (LFC)

$F(\rho_A; O_x) = F(\rho_A, O_x \rho_A O_x^\dagger)$

where $\rho_A = \operatorname{Tr}_{\bar{A}}[\rho]$ is the reduced density matrix on a finite region $A$ containing $x$ and $O_x$ is a local charged operator. The key parameter is the minimal distance $\ell = \operatorname{dist}(x, \partial A)$ between the insertion point and the region boundary.

A state exhibits local SW-SSB if, in the infinite region limit,

$$\lim_{|A|\to\infty} F(\rho_A, O_x \rho_A O_x^\dagger) > 0$$

In contrast to global (two-point) fidelity correlators, this local approach

• Is computationally efficient ($U\rho \propto \rho$0 resources for $U\rho \propto \rho$1-dimensional systems),
• Retains sharply defined meaning in the thermodynamic ($U\rho \propto \rho$2) limit where the global $U\rho \propto \rho$3 ceases to exist,
• Yields a phase diagnostic robust to finite-depth symmetric quantum channels.

(Figure 1)

Figure 1: Schematic of the local fidelity correlator $U\rho \propto \rho$4, probing the response of the local reduced state to symmetry action at a site $U\rho \propto \rho$5.

Universal and Robust Features

Implications of Local SW-SSB

The local SW-SSB order parameter exhibits several universal and robust aspects:

• Implication from Global to Local: Global SW-SSB implies local SW-SSB for any finite region, guaranteed by monotonicity of fidelity (the data processing inequality).
• Conditional Mutual Information (CMI) Obstruction: In strongly symmetric states, nonzero local SW-SSB implies persistent long-range CMI, quantifying a fundamental obstruction to state recovery from local data. This follows from the Fawzi-Renner bound on quantum recoverability.
• Robustness Under Finite-Depth Channels: Local SW-SSB is preserved under application of any finite-depth, strongly symmetric quantum channel, ensuring that the order parameter is intrinsic to the phase and not a sensitive artifact.

(Figure 2)

Figure 2: Tripartition geometry illustrating CMI, with the inability to locally reconstruct global charge information due to SW-SSB.

Distinction Between Local and Global SW-SSB

While global SW-SSB always yields local SW-SSB, the converse need not hold. The paper constructs explicit examples:

• Eigenstate Thermalization Hypothesis (ETH) Eigenstates: Highly excited pure states show locally thermal reduced density matrices (hence robust local SW-SSB), while their global two-point correlators decay exponentially.
• Pseudo-SWSSB States: Ensembles where individual states lack global SW-SSB, but no efficient protocol can distinguish them from states with SW-SSB on polynomial resource budgets, resulting in effective local SW-SSB.

The notion of local indistinguishability emerges as central: physical observables cannot distinguish between different charge sectors within local regions, consistent with the practical and conceptual robustness expected of a phase.

Infinite-Volume Limit and Operator-Algebraic Approach

For infinite systems, the total Hilbert space and global $U\rho \propto \rho$6 are ill-defined. The local SW-SSB order parameter is constructed via a covering sequence of finite regions centered on $U\rho \propto \rho$7, and monotonicity ensures existence and independence from the covering choice:

$U\rho \propto \rho$8

where $U\rho \propto \rho$9 is the state over the quasi-local algebra and $\rho$0 the local reduced state.

This covering limit is respected only by quantities satisfying the data processing inequality, notably fidelity and the R\'enyi-1 correlator, but not generic R\'enyi-$\rho$1 or higher-order correlators.

(Figure 3)

Figure 3: Construction of coverings for defining the local order parameter in the infinite system.

Alternate Diagnostics: R\'enyi-1 and Two-Point Local Correlators

The R\'enyi-1 correlator,

$\rho$2

is shown to be monotonic and equivalent to the one-point fidelity order parameter. Two-point local fidelity and R\'enyi-1 correlators are established to probe the same phases as the one-point version, with rigorous inequalities connecting their scaling behavior.

Averaged inequalities based on Uhlmann's theorem are technically established, ensuring that nonvanishing one-point local fidelity always guarantees a nonvanishing two-point local fidelity for widely separated points.

Numerical and Analytical Results in Physical Systems

Scaling in Critical States and Metals

The local order parameter allows systematic characterization of scaling in gapless and critical states:

• CFT Ground States:

$\rho$3

with $\rho$4 the scaling dimension.

• Fermi Metals:

$\rho$5

regardless of spatial dimension $\rho$6.

• Diffusive Metals (with Disorder):

$\rho$7

also dimension-independent.

These distinct scalings are nontrivially different from ordinary two-point correlators and illustrate the utility of the local SW-SSB measure as a critical diagnostic.

(Figure 4)

Figure 4: Numerical verification of the $\rho$8 and $\rho$9 scalings of the one-point R\'enyi-1 correlator in clean and diffusive Fermi systems, respectively.

Examples: Decohered Ising Models and Random Gaussian States

The formalism is illustrated in explicit models including:

• ZZ-decohered Ising Paramagnets: The transition in local SW-SSB tracks the random-bond Ising model's Nishimori line, and explicit connections between local fidelity and classical defect/partition functions are made.
• Random Gaussian Fermionic States: Realizations with maximal local (but not global) SW-SSB, leveraging ETH-like behavior, are constructed and analyzed.

Order-Disorder Inequalities

A notable analytical result is the order-disorder trade-off:

$F(\rho_A; O_x) = F(\rho_A, O_x \rho_A O_x^\dagger)$0

for $F(\rho_A; O_x) = F(\rho_A, O_x \rho_A O_x^\dagger)$1 symmetry, relating the local SW-SSB order parameter to the expectation value of disorder (string) operators. This provides a quantum generalization of classical order-disorder relations, with practical consequences for phase diagnostics in both pure and mixed states.

Non-Abelian Symmetry Generalization

The local SW-SSB framework is extended to non-Abelian symmetry groups, by considering irreducible representation multiplets and maximizing fidelity over all linear combinations within a multiplet. The phase criterion is then basis-invariant and compatible with Schur's lemma, yielding a robust methodology for diagnosing non-Abelian SW-SSB.

Implications, Open Directions, and Outlook

The local formulation of SW-SSB bridges a gap between robust measurement (theoretical and experimental) and conceptual foundations of symmetry breaking in open quantum systems and mixed-state phases. It provides:

• A scalable, operational diagnostic for (strong-to-weak) phase structure,
• An order parameter amenable to infinite systems and operator-algebraic rigor,
• A unifying perspective capable of distinguishing phases in complex, open, or random environments.

Future directions highlighted in the work include:

• Deeper analogies between SW-SSB and local thermalization,
• Exploring connections to charge scrambling, information spreading, and monitored dynamics,
• Investigating defect interpretations of the local fidelity in critical and non-equilibrium states, and
• Optimizing computational and experimental approaches to detecting SW-SSB in large-scale quantum platforms.

Conclusion

This work establishes a technically precise and scalable approach to the diagnosis of strong-to-weak spontaneous symmetry breaking in both finite and infinite quantum systems. The local fidelity correlator, and its R\'enyi-1 equivalent, provide a robust, practical, and physically meaningful order parameter for phases previously inaccessible due to global resource constraints. As mixed-state and open quantum phases become increasingly central in quantum information science, condensed matter physics, and quantum simulation, these results form an essential component of the modern theory of symmetry in quantum matter.

Reference: F. Divi, L.A. Lessa, C. Wang, "Local Strong-to-Weak Spontaneous Symmetry Breaking" (2605.28967)

Figure 5: One-point R correlator in physical examples, illustrating distinct scaling behaviors in different quantum phases.