[challenge]: How Far Can Strong-to-Weak Spontaneous Symmetry Breaking Go in Its Applications?

[JunkaiWang-TheoPhy]

Released by

Junkai Wang

Contact email

WangTheoPhys@outlook.com

Method

Exact Diagonalization; Others

Challenge issue

How many roads must a man walk down
Before you call him a man?
How many times must a man look up
Before he can see the sky?
— Bob Dylan, Blowin' in the Wind (1963)

This Challenge is related to Issue #150 .

In short: We have witnessed the theoretical success of spontaneous symmetry breaking in bilayers, along with its rich, fascinating theoretical predictions and novel physical phenomena. However, its experimental signatures and verification remain unclear.

In reality, this concept is highly versatile and spans many different areas, including:

1. Strongly correlated states of matter
2. Quantum information
3. Quantum gravity and wormholes
4. Quantum Hall effects

Given this broad relevance, the key question we must ask is: how can we better integrate these concepts and phenomena with current, concrete physical systems? Specifically, how can we apply and investigate them within actual physical setups, both theoretically and observationally, to fully demonstrate the power of this framework?

---

How many physical systems must strong-to-weak spontaneous symmetry
breaking pass through before we can call it a general organizing
principle of mixed-state physics?

Background

For a mixed quantum state $\rho$, a unitary symmetry $U_g$ can be
realized in two inequivalent ways. A weak symmetry satisfies

$$ U_g\rho U_g^\dagger=\rho, $$

whereas a strong symmetry satisfies

$$ U_g\rho=e^{i\alpha_g}\rho. $$

Strong symmetry requires the state to lie within a definite symmetry
charge sector, while weak symmetry allows an incoherent mixture of
different charge sectors.

Strong-to-weak spontaneous symmetry breaking (SWSSB) occurs when the
strong symmetry develops spontaneous order while the weak symmetry
remains unbroken. For a charged local operator $O_x$, this is commonly
expressed through the coexistence of

$$ \lim_{|x-y|\rightarrow\infty} \operatorname{Tr}!\left(\rho O_xO_y^\dagger\right)=0 $$

and

$$ \lim_{|x-y|\rightarrow\infty} F!\left( \rho,, O_xO_y^\dagger\rho O_yO_x^\dagger \right)>0, $$

where $F$ denotes the fidelity between density matrices.

Equivalent or approximate formulations have been proposed using
Rényi-1, Rényi-2, Wightman, Choi-state, purification and doubled-space
correlators. However, the logical relations among these diagnostics
may depend on positivity, locality, thermodynamic limits, operator
normalization and other assumptions.

Central challenge

Determine how far SWSSB can be extended as a physically predictive and
operationally meaningful principle.

The goal is not merely to identify additional systems with a
non-vanishing nonlinear correlator. The challenge is to determine
whether SWSSB produces common, model-independent consequences across
apparently different settings, including:

1. decohered quantum Hall and topological phases;
2. fermionic matter, projected wavefunctions and emergent order;
3. quantum memories and information recoverability;
4. thermalization and emergent hydrodynamics;
5. holography, quantum gravity and bra–ket or replica wormholes.

A successful solution should clarify when these phenomena represent
the same physical mechanism and when the resemblance is only formal.

I. Quantum Hall states and topological quantum matter

Fractional quantum Hall states provide a particularly nontrivial
setting because they simultaneously contain global charge symmetry,
topological ground-state degeneracy, anyonic excitations and
nonlocal quantum information.

Recent work on Laughlin and Moore–Read states under local density
decoherence finds regimes in which topological information remains
recoverable at arbitrarily strong decoherence, as well as regimes with
decoherence-induced BKT transitions and critical mixed states.

The following questions should be addressed:

• Does the onset of charge-$U(1)$ SWSSB coincide with the loss or
  recoverability of topological quantum information?

• Is SWSSB necessary, sufficient, both or neither for the breakdown of
  topological memory?

• How does SWSSB depend on filling fraction, quasiparticle charge,
  Abelian versus non-Abelian statistics and the structure of the
  topological ground-state manifold?

• Can SWSSB distinguish a decohered topological phase from a trivial
  mixed state that has the same ordinary local correlators?

• What is the relation between ordinary $0$-form SWSSB, $1$-form
  SWSSB, Wilson-loop order and intrinsically mixed topological order?

A minimal numerical benchmark could begin with:

1. a Chern-insulator or Haldane-model ground state under local density
   dephasing;
2. the $\nu=1/3$ Laughlin state on a finite torus;
3. the $\nu=1/2$ Moore–Read state, if computationally feasible;
4. the toric code or doubled-semion model as a higher-form control
   example.

For each system, one should compare:

• ordinary charged correlators;
• Rényi-2 correlators;
• fidelity or Rényi-1 correlators;
• topological-sector distinguishability;
• quantum-information recoverability;
• finite-size and long-distance scaling.

II. Stress-testing the fermionic inequality approach

Recent work by Sarma and Xu derives an inequality in dephased
fermionic systems in which a class of fermion-bilinear correlators is
upper-bounded by a Rényi-2 correlator associated with doubled-space
interlayer order.

This result should be treated as an important test case rather than as
a completed proof of generic SWSSB.

The challenge is to determine precisely what follows from the
inequality:

• Does the inequality establish actual SWSSB, or only identify the
  channel in which the slowest-decaying correlation may occur?

• Can one obtain a lower bound, a saturation condition or an
  independent proof that the proposed SWSSB correlator is nonzero at
  long distances?

• Does quasi-long-range order in the doubled state imply true SWSSB,
  or only a critical analogue of SWSSB?

• Do the conclusions survive when the determinant measure is not
  manifestly nonnegative?

• How essential are the assumptions of even flavor number, traceless
  flavor matrices, connected diagrams and spatial coarse-graining?

• Can the Rényi-2 result be confirmed using the fidelity or Rényi-1
  correlator, which provides a more direct SWSSB diagnostic?

Exact diagonalization, determinant methods or controlled free-fermion
calculations should be used to search for both confirming examples and
counterexamples.

A valuable negative result would be a model in which the proposed
Rényi-2 correlator is long-ranged while the fidelity diagnostic does
not exhibit SWSSB, or vice versa.

III. Holography, quantum gravity and wormholes

A recent holographic construction studies a fixed-charge thermal
state in an $\mathrm{AdS}_3/\mathrm{CFT}_2$ model. The ordinary charged
two-point correlator is short-ranged, while Rényi-2 and Wightman
correlators remain nonzero at large separation. In the bulk, the
cross-copy correlation is associated with a connected geometry linking
the two thermofield-double copies.

This suggests a possible geometric interpretation:

SWSSB on the boundary may diagnose connectivity between bra and ket,
replica or thermofield-double sectors in the bulk.

However, the existence of a connected wormhole geometry should not be
identified with SWSSB automatically. The following questions remain
open:

• Is a bra–ket or replica wormhole necessary for holographic SWSSB?

• Is such a wormhole sufficient, or must one additionally identify a
  charged symmetry sector and demonstrate the absence of conventional
  charged long-range order?

• Can a connected bulk saddle produce cross-copy correlations without
  breaking a strong symmetry to a weak symmetry?

• Can SWSSB occur in a holographic theory whose dominant bulk saddle
  is disconnected?

• Is the apparent weak symmetry a property of an individual boundary
  theory or only of an ensemble-averaged theory?

• What changes at finite $N$, finite volume or beyond the dominant
  semiclassical saddle?

• How do charge projection, gauge constraints, factorization and
  ensemble averaging enter the proposed dictionary?

• Is the proposed relation compatible with the absence of exact global
  symmetries in quantum gravity?

Useful extensions include JT gravity, SYK-like ensembles,
higher-dimensional AdS/CFT, non-Abelian symmetries, higher-form
symmetries and replica-wormhole calculations.

A successful holographic analysis should provide an explicit
boundary-to-bulk dictionary containing:

Boundary concept	Bulk candidate
Strong symmetry	Independent charge constraints on the two copies
Weak symmetry	Diagonal or ensemble-averaged symmetry
Fidelity/Rényi order	Cross-copy charged correlator
Charge delocalization	Inability to assign charge locally to one boundary region
SWSSB	Connected charged channel without ordinary charged order
Failure of SWSSB	Vanishing charged cross-copy channel or explicit inter-copy breaking

IV. Operational and experimental applications

SWSSB is invisible to observables linear in $\rho$. Its use as a
physical principle therefore depends on whether its nonlinear
diagnostics can be measured or reconstructed efficiently.

Important questions include:

• Can local fidelity or Rényi diagnostics detect SWSSB with resources
  polynomial in system size?

• Which SWSSB observables are accessible in quantum gas microscopes,
  randomized measurements, quantum simulators or ancilla-assisted
  protocols?

• Can SWSSB be used to identify error-correction thresholds or
  distinguish recoverable from unrecoverable mixed-state information?

• Does SWSSB define the timescale at which a discrete quantum system
  admits a continuum hydrodynamic description?

• Which predictions of SWSSB are experimentally distinct from ordinary
  decoherence, thermalization or loss of purity?

The recent observation of SWSSB in a dephased Fermi gas provides an
experimental starting point. The challenge is to move from observation
to application: control, decoding, phase identification or dynamical
prediction.

Required output

A complete solution should produce an SWSSB application map. Each
candidate application should specify:

Item	Required information
Physical system	Hamiltonian, channel, ensemble or gravitational path integral
Strong symmetry	Exact charge-sector condition
Weak symmetry	Surviving diagonal or ensemble symmetry
Charged operator	Operator used to probe the breaking
Exact diagnostic	Fidelity or Rényi-1 criterion
Proxy diagnostic	Rényi-2, Wightman or doubled-space correlator
Physical consequence	Recoverability, topology, hydrodynamics, geometry, etc.
Verification method	ED, QMC, tensor networks, field theory or holography
Failure mode	Explicit coupling, finite-size artifact, proxy mismatch, etc.

Success criteria

A successful contribution should accomplish at least one of the
following:

1. Prove a theorem or construct a counterexample relating SWSSB to
   topological information recoverability.

2. Establish SWSSB directly in a nontrivial quantum Hall or
   topologically ordered state using fidelity or Rényi-1 diagnostics,
   supported by controlled finite-size scaling.

3. Demonstrate that a commonly used Rényi-2 or doubled-space diagnostic
   is not sufficient for genuine SWSSB.

4. Prove or falsify a necessary or sufficient relation between
   holographic SWSSB and bra–ket or replica wormholes.

5. Develop an experimentally scalable protocol that distinguishes
   SWSSB from conventional symmetry breaking and ordinary decoherence.

6. Identify a universal consequence shared by at least two genuinely
   different settings, such as a topological phase and a holographic
   model.

A well-established negative result is also a valid solution. For
example, it would be important to show that SWSSB does not universally
track topological-memory loss, that a wormhole does not necessarily
imply SWSSB, or that different nonlinear diagnostics define inequivalent
phase structures.

Why this may lead to research output

At present, SWSSB is rapidly appearing in open systems, topological
matter, hydrodynamics, experiments and holography. The field now needs
a separation between:

• exact definitions and convenient proxies;
• universal consequences and model-specific observations;
• formal doubled-space order and operationally measurable physics;
• genuine applications and reinterpretations of already known effects.

A systematic benchmark containing proofs, counterexamples and
reproducible numerical models could establish the actual domain of
validity of SWSSB and determine whether it is a general organizing
principle or a family of distinct phenomena sharing similar notation.

References

1. Lessa et al., “Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States,” arXiv:2405.03639

2. C. Wang, “Strong-to-Weak Spontaneous Symmetry Breaking,” arXiv:2606.02555

3. Z. Wang et al., “Fractional Quantum Hall States under Density Decoherence,” arXiv:2510.08490

4. Sarma and Xu, “Inequality for Strong-Weak Spontaneous Symmetry Breaking in Fermionic Open Quantum Systems,” arXiv:2603.24671

5. Zhang et al., “Strong-to-Weak Spontaneous Breaking of 1-Form Symmetry and Intrinsically Mixed Topological Order,” arXiv:2409.17530

6. S. Wang et al., “Observation of Strong-to-Weak Spontaneous Symmetry Breaking in a Dephased Fermi Gas,” arXiv:2604.16137

7. Divi, Lessa and Wang, “Local Strong-to-Weak Spontaneous Symmetry Breaking,” arXiv:2605.28967

8. Hauser et al., “Strong-to-Weak Symmetry Breaking in Open Quantum Systems: From Discrete Particles to Continuum Hydrodynamics,” arXiv:2602.16045

9. Kawamoto, Tasuki and Yamazaki, “Strong-to-Weak Spontaneous Symmetry Breaking from Wormholes in Holography,” arXiv:2607.12022

10. Feng, Cheng and Ippoliti, “Hardness of Observing Strong-to-Weak Symmetry Breaking,” arXiv:2504.12233