Floquet-induced bosonic pair condensate with unconventional symmetry

This paper proposes a dynamical pairing mechanism in a periodically driven two-dimensional hard-core boson model that generates an effective three-site interaction, leading to a bosonic pair condensate with unconventional $s+id$ wave symmetry while completely depleting the single-particle condensate.

The Gist

Imagine you are at a crowded dance party. Usually, people dance alone, or maybe they find a partner and dance together. In the world of quantum physics (the rules that govern tiny particles like atoms), scientists have spent decades studying how particles pair up to create special states of matter, like superconductors (materials that conduct electricity with zero resistance).

Traditionally, these pairs form because the particles naturally attract each other, like magnets snapping together. But this new paper proposes a wildly different way to get particles to dance together: by shaking the dance floor.

Here is the story of their discovery, broken down into simple concepts:

1. The Setup: The Shaking Dance Floor

The researchers imagined a grid of hard-core bosons. Think of these as bouncers on a dance floor. They are "hard-core," meaning they are grumpy and cannot share the same spot (two bouncers can't stand on the same tile).

Usually, these bouncers just hop from one tile to the next. But in this experiment, the scientists didn't let them hop naturally. Instead, they periodically shook the entire dance floor (a process called "periodic driving"). They wiggled the floor horizontally and vertically in a specific rhythm, like a DJ changing the beat.

2. The Magic Trick: The "Ghost" Interaction

When you shake the floor fast enough, something strange happens. The usual rules of hopping disappear. The bouncers can no longer move alone.

Instead, the shaking creates a new, invisible rule: A bouncer can only move if it is holding hands with a neighbor.

• If a bouncer tries to move alone, the floor shakes them back.
• If two bouncers are stuck together as a pair, they can glide across the floor effortlessly.

This is the paper's big breakthrough. Usually, particles pair up because they like each other (attraction). Here, they pair up because the shaking forces them to stick together to survive. It's like a game of musical chairs where the only way to stay in the game is to sit on someone else's lap.

3. The Result: A "Ghost" Condensate

In normal physics, when particles condense (like water turning to ice), they all line up and move in unison as individuals. This is called a Bose-Einstein Condensate (BEC).

In this new "shaken" world, the individual bouncers are frozen. They are completely depleted; they can't move on their own. However, the pairs of bouncers are dancing wildly. They form a "pair condensate."

Think of it like a crowd of people where everyone is standing still, but everyone is holding hands with a partner, and those pairs are zooming around the room at high speed. The individuals are frozen, but the couples are free.

4. The Weird Dance Style: "S + Id"

The paper also discovered that these pairs don't just dance in a simple circle. They have a very specific, complex rhythm.

• In a normal dance, everyone moves in the same direction (like a wave).
• In this new state, the pairs move in a pattern that looks like a mix of a circle and a four-leaf clover.

The scientists call this "s + id wave symmetry."

• Imagine a dancer spinning (the "s" part).
• Now imagine that dancer is also doing a weird, twisting move that changes direction every time they step (the "id" part).
• This creates a unique pattern that has never been seen in equilibrium systems (systems that aren't being shaken).

5. Why Does This Matter?

This is a big deal for a few reasons:

• New Physics: It proves you can create exotic states of matter just by shaking things, without needing the particles to naturally attract each other.
• Quantum Computers: The authors suggest this could be built using superconducting quantum circuits (the kind of chips used in quantum computers). By programming the chips to "shake" the qubits (the quantum bits) in this specific rhythm, we could create these special paired states.
• Future Tech: Understanding these "forced" pairs might help us design better superconductors or new types of quantum sensors.

The Bottom Line

The paper shows that if you shake a quantum system fast enough, you can force particles to ignore their individuality and only exist as pairs. It's a "dance of necessity" rather than a "dance of love," creating a brand new type of quantum matter that is completely different from anything we see in nature under normal conditions.

In short: They turned a quantum dance floor into a place where you can't move unless you have a partner, resulting in a super-fast, super-organized dance of pairs that no one knew was possible.

Technical Summary

1. Problem Statement

The search for macroscopic quantum coherent states with unconventional pairing symmetries (e.g., $d$-wave in cuprates, $p$-wave in $^3$He) has traditionally focused on equilibrium systems where pairing arises from conservative pairwise interactions. However, the existence and mechanisms of such unconventional pairing in non-equilibrium quantum systems remain elusive.

While periodic driving (Floquet engineering) is a powerful tool for creating new quantum phases, most existing proposals rely on modifying effective pairwise interactions. This paper addresses the open question: Can a simple periodic modulation of single-particle hopping induce an unconventional pairing condensate without intrinsic pairwise interactions, and what is the underlying mechanism?

2. Methodology

The authors propose a theoretical framework combining Floquet theory and numerical simulation:

• Model System: A two-dimensional (2D) hard-core boson model on a square lattice.

  • Hamiltonian: The system features a time-dependent hopping amplitude $J(t)$ modulated periodically. Crucially, the horizontal and vertical hopping terms are modulated with the same frequency $\omega$ but with a $\pi/2$ phase lag:
    $$H(t) = J \sum_{i_x, i_y} \left\{ \cos(\omega t) [a^\dagger_{i} a_{i+\hat{x}} + h.c.] + \sin(\omega t) [a^\dagger_{i} a_{i+\hat{y}} + h.c.] \right\}$$
  • Constraint: Only hard-core constraints exist; no intrinsic two-body or three-body interactions are present in the microscopic Hamiltonian.

• Floquet Analysis:

  • Assuming fast driving ($\omega \gg J$), the stroboscopic dynamics are governed by an effective time-independent Floquet Hamiltonian ($H_F$) derived via the Magnus expansion.
  • The zero-order term $H_0$ (time-averaged Hamiltonian) vanishes.
  • The leading order term $H_1$ (order $1/\omega$) is derived, which contains three-site interactions (density-assisted hopping) rather than standard pairwise terms.

• Numerical Simulation:

  • The authors employ the Density Matrix Renormalization Group (DMRG) method on a 2D cylindrical lattice (quasi-1D geometry) to study the ground state properties of the effective Hamiltonian $H_1$.
  • They analyze single-particle and pair correlation functions to determine the nature of the condensate.

3. Key Contributions

A. Emergent Three-Site Interaction Mechanism

The paper demonstrates that periodic driving can completely suppress the bare single-particle hopping (which vanishes in the time-average) and generate an effective Hamiltonian dominated by three-site correlated hopping terms.

• The effective term $H_1$ couples density operators on one site to current operators on diagonal bonds.
• This mechanism creates a kinetic constraint: bosons can only hop if they move as a pair. Single bosons are "stuck" in "dark states" (eigenstates of $H_1$ with zero energy) unless they are adjacent to another boson.

B. Discovery of $s + id_{x^2-y^2}$ Pair Condensate

Through DMRG simulations, the authors reveal a novel ground state where:

• Single-particle Bose-Einstein Condensate (BEC) is completely depleted: Single-particle correlations decay exponentially.
• Pair Condensation occurs: Bosons form pairs that exhibit quasi-long-range order.
• Unconventional Symmetry: The pairing order parameter exhibits $s + id_{x^2-y^2}$ symmetry.
  • This is distinct from the more commonly discussed $p_x + ip_y$ (chiral) or $d_{x^2-y^2}$ states.
  • The symmetry arises from a $\pi/2$ phase difference between pairing amplitudes on horizontal and vertical bonds.
  • Unlike $p$-wave pairing (odd parity), this state possesses even parity, consistent with bosonic statistics.

C. Dark States and Hilbert Space Fragmentation

The study identifies a vast number of "dark Fock states" (states annihilated by $H_1$) that do not evolve under the effective Hamiltonian. This leads to Hilbert space fragmentation, imposing strong kinetic constraints that prevent thermalization and stabilize the pair condensate.

4. Key Results

1. Dynamical Pairing Verification:

   • Numerical evolution of two bosons shows that under fast driving, the average distance between them remains constant ($r(t) \approx 1$), confirming they move only as a bound pair.
   • Under slow driving, higher-order terms in the Magnus expansion break this constraint, allowing the bosons to separate.

2. Correlation Functions:

   • Single-particle correlation $C(r)$: Decays exponentially, confirming the absence of single-particle BEC.
   • Pair correlations $D(r)$: Decay algebraically ($D(r) \sim r^{-\eta}$), indicating quasi-long-range order.
   • Symmetry Analysis:
     • $D_{yy}(r)$ (vertical bonds) is real.
     • $D_{yx}(r)$ (diagonal bonds) is purely imaginary.
     • $D_{xx}(r)$ (horizontal bonds) is real and positive.
     • This pattern confirms the $s + id_{x^2-y^2}$ structure, which breaks time-reversal symmetry but is non-chiral (no circulating edge currents).

3. Response to Impurities:

   • When a non-magnetic impurity is introduced, localized currents emerge around it.
   • Due to the non-chiral nature of the $s+id$ state, these currents form a "two-in, two-out" pattern respecting reflection symmetries, rather than a circulating loop.

4. Density Dependence:

   • At low density (1/8 filling), the system is a pure pair condensate.
   • At high density (half-filling), the entanglement between pairs allows single particles to hop, potentially recovering single-particle BEC alongside pair condensation.

5. Significance and Outlook

• New Pairing Mechanism: The work establishes a dynamical pairing mechanism driven purely by periodic modulation of hopping, distinct from equilibrium interactions. It proves that effective multi-body interactions can dominate over two-body terms in non-equilibrium settings.
• Experimental Feasibility: The authors propose realizing this model using superconducting quantum circuits (transmon qubits). By periodically modulating the frequency of couplers between qubits, the effective hopping can be tuned to realize the required sine/cosine modulation.
• Non-Equilibrium Physics: The system exhibits pre-thermalization, where the pair condensate survives for a long time ($\sim e^{\omega/J}$) before heating to infinite temperature.
• Future Directions: The presence of extensive dark states and kinetic constraints suggests the potential for disorder-free localization and quantum scar states, offering a new platform to study non-ergodic phenomena in quantum many-body systems.

In summary, this paper provides a theoretical blueprint for creating a Floquet-induced bosonic pair condensate with $s + id_{x^2-y^2}$ symmetry in a system devoid of intrinsic interactions, opening new avenues for exploring unconventional quantum phases in driven systems.