Deconfined quantum criticality with internal supersymmetry

This paper extends the deconfined quantum criticality paradigm to systems with internal supersymmetry by proposing a supersymmetric deconfined quantum critical point (sDQCP) between an $OSp(1|2)$-breaking phase and a lattice rotation-breaking phase, which is described via a non-linear sigma model on a supersphere and a gauge theory, and shown to continuously connect to the conventional DQCP when supersymmetry is explicitly broken.

The Gist

Imagine a world made of tiny magnets (spins) sitting on a grid, like a chessboard. Usually, when these magnets change how they are arranged, they do so in a predictable way, following a rulebook called the "Landau paradigm." But physicists have discovered a weird, special kind of transition called a Deconfined Quantum Critical Point (DQCP). It's like a magic door where the magnets can switch from one organized pattern to a completely different, unrelated pattern without getting stuck in a messy middle ground.

This paper takes that magic door and adds a new ingredient: Supersymmetry (SUSY).

The New Ingredient: Supersymmetry

In physics, "supersymmetry" is a concept that pairs up two different types of particles: bosons (which like to crowd together, like a choir) and fermions (which hate to be in the same spot, like introverts). Usually, this is a theory about the universe's fundamental forces. But here, the authors are looking at a "toy model" on a computer or a lattice where these pairings happen inside the atoms themselves.

They use a specific mathematical rulebook called OSp(1|2). Think of this as a special set of instructions that forces every single spot on the grid to hold a "super-particle" that is half-boson and half-fermion.

The Two States: The Dance of Order

The paper describes a battle between two different ways these super-particles can organize:

1. The Super-Néel Phase (The Spin Dancers):
   Imagine the magnets lining up in a perfect, alternating pattern (up, down, up, down). In this state, the "super" nature of the particles is broken, but the grid itself looks the same from every angle. It's a rigid, ordered dance.

2. The Super-VBS Phase (The Bonded Pairs):
   Now, imagine the magnets stop dancing individually and instead pair up with their neighbors to form tight couples (like holding hands). This breaks the grid's rotational symmetry (the grid looks different if you turn it 90 degrees), but the "super" nature of the particles remains intact.

The Magic Transition: The "Intertwined" Defect

The core discovery is what happens when the system tries to switch from the "Spin Dancers" to the "Bonded Pairs."

In normal physics, defects (mistakes in the pattern) are boring. But in this Supersymmetric Deconfined Quantum Critical Point (sDQCP), the defects are magical.

• If you make a "vortex" (a swirl) in the "Bonded Pairs" phase, that swirl accidentally carries the charge of the "Spin Dancers."
• If you make a "skyrmion" (a twist) in the "Spin Dancers" phase, that twist accidentally carries the charge of the "Bonded Pairs."

It's as if the two phases are holding hands through their mistakes. When the "Bonded Pairs" start to fall apart (proliferate), they don't just break their own order; they accidentally force the "Spin Dancers" to wake up and organize. This "intertwining" is what makes the transition smooth and continuous, rather than a messy crash.

The Mathematical Magic Trick

To explain this, the authors used two different "languages" (mathematical models):

1. The Geometry Language (Non-linear Sigma Model):
   They imagined the state of the system as a point moving on a strange, multi-dimensional shape called a "supersphere." This shape has regular dimensions (like up/down/left/right) and "ghost" dimensions (fermionic coordinates). They showed that the rules of this shape force the two phases to be connected.

2. The Gauge Theory Language (The "Deconfinement" Story):
   They described the particles as being connected by invisible strings (gauge fields).

   • In the "Bonded Pairs" phase, the strings are tight, and the particles are stuck together.
   • In the "Spin Dancers" phase, the strings are broken, and the particles are free.
   • At the critical point, the strings are loose enough that the particles are "deconfined" (free) but still interacting.

The Big Surprise: 3D XY Criticality

Usually, when you mix bosons and fermions, the math gets incredibly complicated. However, the authors found a beautiful cancellation effect.

• The "boson" part of the system wants to act one way.
• The "fermion" part wants to act the opposite way.
• Because of the supersymmetry, these two effects cancel each other out perfectly, leaving behind a much simpler behavior.

They concluded that despite the complex "super" ingredients, the transition behaves exactly like a well-known, simpler type of phase transition called the 3D XY model. It's like adding a complex spice to a soup, only to find that the spice perfectly neutralizes the salt, leaving you with the exact taste of plain water.

The Connection to the Real World

Finally, the paper shows that if you take away the "super" rules (by breaking the supersymmetry), this fancy new transition smoothly turns back into the standard, non-supersymmetric DQCP that physicists have been studying for years. This proves that their new discovery isn't a totally alien concept; it's just the "supercharged" version of something we already knew.

In summary: The paper proposes a new type of quantum phase transition where bosons and fermions are paired up. These pairs create a "magic door" between two different ordered states. The transition is smooth because the mistakes in one state automatically trigger the order in the other. Surprisingly, this complex super-transition simplifies down to a known, standard behavior, bridging the gap between complex supersymmetry and familiar quantum physics.

Technical Summary

Technical Summary: Deconfined Quantum Criticality with Internal Supersymmetry

Problem Statement
The paper addresses the extension of the Deconfined Quantum Critical Point (DQCP) paradigm to systems possessing internal supersymmetry (SUSY). Conventional DQCP describes direct, non-fine-tuned quantum phase transitions between two ordered phases breaking distinct symmetries (e.g., Néel to Valence Bond Solid), characterized by the interplay where defects of one symmetry carry the charge of the other. While the standard DQCP is well-studied in the context of spin rotation $SU(2)$ and lattice rotation symmetries, the authors investigate whether this paradigm holds when the internal symmetry is generalized to a Lie supergroup, specifically $OSp(1|2)$, which represents an $N=1$ internal SUSY generalization of $SU(2)$. The central challenge is to formulate a consistent theoretical framework for such a transition, accounting for the pseudo-Hermitian nature of lattice Hamiltonians with internal SUSY and the resulting intertwinement of bosonic and fermionic degrees of freedom.

Methodology
The authors employ a combination of lattice model construction, parton theory, non-linear sigma models (NL$\sigma$M), and gauge theory formalisms:

1. Lattice Model Construction: They define a two-dimensional square lattice model where each site hosts a three-dimensional Hilbert space transforming as a spin-1/2 representation of the $OSp(1|2)$ Lie supergroup. The Hamiltonian is constructed from two-site Casimir operators, which are non-Hermitian but pseudo-Hermitian (satisfying $H^\dagger = PHP$), ensuring a real spectrum and unitary time evolution.
2. Phase Identification: They identify two competing ground states:
   • Super-Valence Bond Solid (SVBS): Breaks lattice rotation symmetry ($Z_4$) while preserving internal $OSp(1|2)$.
   • Super-Néel (SN): Breaks internal $OSp(1|2)$ down to $U(1)$ while preserving lattice rotation symmetry.
3. Kinetic Description (NL$\sigma$M): To capture the symmetry intertwinement, they formulate a non-linear sigma model on a supersphere target space $S^{4|2}$. This model unifies the bosonic Goldstone modes (from symmetry breaking) and fermionic Goldstino modes (SUSY partners) into a single field configuration, including a level-1 Wess-Zumino-Witten (WZW) term.
4. Dynamic Description (Gauge Theory): They develop a gauge theory formulation by parameterizing the supersphere $S^{2|2}$ using complex bosonic coordinates ($z_1, z_2$) and a complex fermionic coordinate ($\xi$) coupled to a dynamical $U(1)$ gauge field. This leads to a $CP^{1|1}$ model (a supersymmetric generalization of the $CP^1$ model).
5. Symmetry Breaking Analysis: They analyze the continuous connection between the supersymmetric scenario and the conventional DQCP by explicitly breaking $OSp(1|2)$ down to $SU(2)$.

Key Contributions and Results

• Proposal of sDQCP: The authors propose a supersymmetric deconfined quantum critical point (sDQCP) as the direct transition between the SVBS and SN phases.
• Symmetry Intertwinement via WZW Term: Using the NL$\sigma$M with a WZW term, they demonstrate that an SVBS vortex carries an $OSp(1|2)$ spin-1/2 charge. This confirms the mixed anomaly characteristic of DQCP, where the proliferation of lattice defects (vortices) restores lattice symmetry but spontaneously breaks the internal supersymmetry.
• Gauge Theory and Criticality: The gauge theory analysis reveals that the internal $OSp(1|2)$ symmetry enforces the bosonic spinons ($z$) and fermionic spinons ($\xi$) to be simultaneously gapless at the critical point.
  • The authors argue that the presence of "symplectic fermions" (which contribute negatively to the degrees of freedom due to their non-unitary nature) effectively cancels one complex bosonic degree of freedom.
  • This cancellation reduces the effective critical theory to a single complex boson coupled to a non-compact $U(1)$ gauge field.
  • Consequently, the authors argue that the sDQCP belongs to the 3D XY universality class (specifically a 3D XY* transition), sharing the same critical exponents for correlation length and heat capacity as the standard 3D XY model, despite having different operator content.
• Connection to Conventional DQCP: By explicitly breaking the internal SUSY (making the Hamiltonian Hermitian), the fermionic modes become gapped and decouple from the low-energy spectrum. The theory smoothly reduces to the standard $CP^1$ model and the $O(5)$ NL$\sigma$M, recovering the conventional DQCP between Néel and VBS phases.

Significance
The paper claims to provide a unified framework for deconfined criticality that encompasses both systems with and without internal supersymmetry. By extending the DQCP paradigm to Lie supergroups, the work:

1. Establishes a theoretical route for continuous quantum phase transitions in pseudo-Hermitian systems with internal SUSY.
2. Identifies a specific universality class (3D XY*) for the sDQCP, driven by the unique cancellation of degrees of freedom between bosons and symplectic fermions.
3. Offers a complementary set of continuum descriptions (NL$\sigma$M and gauge theory) that capture both the topological intertwinement of symmetries and the dynamical critical behavior.
4. Suggests that the sDQCP can be viewed as a deformation of the conventional DQCP, bridging the gap between standard quantum criticality and supersymmetric critical phenomena.

The authors conclude by noting that while the theoretical arguments point to 3D XY criticality, the specific operator content and anomalous dimensions of the sDQCP differ from the standard XY model, and these distinctions would require numerical verification (e.g., Monte Carlo simulations) to fully substantiate.