Non-relativistic Floquet Conformal Field Theory

Arnab Kundu; Diptarka Das; Krishnendu Sengupta; Sumit R. Das

Non-relativistic Floquet Conformal Field Theory

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Pith's one-line read This paper claims that any non-relativistic conformal many-body system driven periodically by the Hamiltonian, dilation, and special conformal generators exhibits three stroboscopic phases — exponential, oscillatory, and power-law — classif

desk verdict Exact SO(2,1) Floquet phase classification for non-relativistic CFTs, with a falsifiable experimental diagnostic; solid and worth refereeing despite being a modest extension of prior work. read the letter →

arxiv 2607.27668 v1 pith:7VMA4OWT submitted 2026-07-30 hep-th cond-mat.quant-gascond-mat.str-el

Diptarka Das , Sumit R. Das , Arnab Kundu , Krishnendu Sengupta This is my paper · ORCID

classification hep-thcond-mat.quant-gascond-mat.str-el

keywords non-relativisticconformalfieldtheoryFloquetdynamicsSO(21)algebradynamicalphasesunitaryFermigastrappedultracoldfermionsfidelityholography

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that periodically driven non-relativistic conformal field theories are exactly solvable and display three distinct dynamical phases: exponential growth, oscillation, and power-law growth. Everything is controlled by a single parameter, the sign of ρ², which is built from the drive amplitudes and durations through the SO(2,1) algebra. The result is claimed to be universal for any system with non-relativistic conformal invariance, including trapped ultracold fermions near unitarity and resonant anyons. The paper also maps the hyperbolic phase to an ergosurface and the parabolic phase to an extremal Killing horizon in a holographic bulk. The relevance is that these phases are measurable, e.g. through the breathing-mode amplitude of a trapped Fermi gas.

What carries the argument

The load-bearing structure is the SO(2,1) subalgebra generated by {H, D, C} inside the non-relativistic conformal algebra, with commutators [H,C]=−iD, [D,C]=−2iC, [D,H]=2iH. Each step of the drive is H + μᵢ²C, and the single-cycle evolution is computed in a 2×2 Pauli-matrix representation. The Floquet Hamiltonian is H_F = α₊H + γD + α₋C, and the phase is decided by ρ² = T²(α₊α₋ − γ²). The initial state is a highest-weight primary |Δ₀⟩_μ1 satisfying L₋|Δ⟩=0, with ⟨D⟩=0 and ⟨H⟩=μ1²⟨C⟩=Δμ1/2, which fixes the one-point functions exactly.

What would settle it

Measure the breathing-mode amplitude A₀ in a trapped unitary Fermi gas driven by the cosine protocol at fixed μ1, scanning μ2 and ω_D across the predicted tongue in Fig. 4(a). The phase boundary is where A₀ switches from bounded oscillation to exponential growth with drive time, and the parabolic line is where it grows linearly in t_s. If the boundary location disagrees with the parameter-free SO(2,1) prediction beyond experimental uncertainty, the primary-state assumption is violated.

Watch

Extended reading notes

Core claim

The central discovery is that the stroboscopic expectation value of the special conformal generator C(ℓ) in a primary state takes the exact form (Δμ1/2)(Tα₊/ρ)² sin²(ρℓ) + (Δ/2μ1)(cos ρℓ + γT/ρ sin ρℓ)². The large-ℓ behavior is set entirely by whether ρ² is positive (elliptic — oscillatory), negative (hyperbolic — exponentially growing), or zero (parabolic — power law). Equivalently, the Floquet Hamiltonian is SO(2,1)-conjugate to H + κ²C, so the three phases correspond to a harmonic trap, an inverted oscillator, and free evolution. This classification is exact for square-pulse protocols and is reproduced numerically for continuous cosine drives.

Load-bearing premise

The initial many-body state is a highest-weight primary of SO(2,1) — L₋ annihilates it and ⟨D⟩=0 with ⟨H⟩=μ1²⟨C⟩=Δμ1/2 — and the paper asserts, rather than proves, that the interacting unitary Fermi gas ground state has this property; off unitarity or in an imperfect trap the formulas fail after a finite number of drive cycles.

Editorial extensions

If this is right

For any square-pulse or continuous cosine drive built from H+μ²C, the stroboscopic response of C, H, D, the autocorrelators, and the fidelity is determined exactly by the half-trace cosρ, so the phase diagram is parameter-free.
In trapped fermions near unitarity, the cloud radius scales with C(ℓ), so the elliptic phase appears as bounded oscillation, the hyperbolic phase as exponential expansion, and the parabolic surface as linear growth.
The fidelity decays exponentially in the hyperbolic phase, revives periodically in the elliptic phase, and decays only linearly at the transition surface, giving an experimentally accessible scrambling diagnostic.
The holographic dual of the hyperbolic phase contains a timelike stationary-limit (ergo)surface, the elliptic phase has a globally timelike Killing vector, and the parabolic phase corresponds to an extremal Killing horizon.
The time-dependent drive is unitarily equivalent to a static trap H+W²C via the Ermakov-Pinney equation, so stable, unstable, and marginal solutions of that equation are the same three phases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct stress test is to measure ⟨D⟩ and ⟨H⟩/⟨C⟩ in a trapped Fermi gas slightly off unitarity; once the primary-state expectation values acquire corrections, the predicted phase boundary should shift, giving a controlled test of how far the universality extends.
The same SO(2,1) reduction suggests that quasiperiodic or aperiodic drives will still show a three-phase classification ruled by the sign of the stroboscopic Casimir, though the paper does not analyze such protocols.
For resonant anyons, the prediction transfers directly, so a driven anyon trap could serve as a second platform to search for the parabolic transition.
A sharp experimental falsifier is the oscillation amplitude A₀ after the drive is stopped: it should grow with drive time t_s in the hyperbolic phase, stay bounded in the elliptic phase, and grow linearly exactly at the parabolic line.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the thing to know: this paper is a clean, exact algebraic result. The stroboscopic behavior of the conformal generator C(ℓ) in a non-relativistic CFT driven by square pulses is fully determined by ρ (Eq. 7): exponential growth for ρ²<0, oscillations for ρ²>0, power-law at ρ=0. That classification is derived from the SO(2,1) algebra alone, no fitting, no hidden assumptions beyond the initial state being primary. The formulas are explicit and the numerics for the cosine protocol are backed by a shared code with convergence checks. The experimental connection to trapped fermions at unitarity is concrete: the amplitude of the breathing-mode oscillations after the drive is predicted to grow exponentially in the hyperbolic phase and stay bounded in the elliptic phase. That is a sharp, testable statement.

What's genuinely new: the application of the known SO(2,1) Floquet machinery to non-relativistic CFTs, where there is a unique SO(2,1) subalgebra, plus the explicit one-point and fidelity expressions for NR primaries, and the link to existing unitary Fermi gas experiments. The paper also explains why the recent experiment [100] only saw heating: their resonant modulation sits on a cut through the hyperbolic phase. That is a useful observation.

Where are the soft spots? The main one is that the initial state is assumed to be a primary, and the paper asserts rather than proves that the interacting N-particle ground state has this property. The stress-test note is right that for a bounded-below Hamiltonian H+μ²C, the ground state is necessarily annihilated by the lowering operator, so the assumption is standard and not a real gap, but the paper would be healthier with a one-line justification. Second, the holographic section is explicitly kinematic and the authors admit a full dual requires backreaction; that section does not carry the weight of the rest. Third, the novelty relative to the authors' own higher-dimensional relativistic FCFT papers [75,76] is modest — the algebra is the same, the classification is the same, and the new content is the NRCFT realization and the experimental consequences. That is still a worthwhile contribution. The 'universal' claim should be read as universal within the SO(2,1)-invariant sector and for unitarity or near-unitarity for a finite number of cycles; the paper itself acknowledges the latter.

Bottom line: the central argument holds up. The paper is not a breakthrough, but it is a solid, honest, and useful piece of work. I'd send it to a competent referee with confidence that it will survive contact.

Referee Report

0 major / 4 minor

Summary. The paper develops a Floquet formalism for non-relativistic CFTs. For a square-pulse protocol alternating H+μ1²C and H+μ2²C, with an initial primary state, it obtains exact stroboscopic expectation values of C(ℓ), H(ℓ), D(ℓ) and the fidelity. The single-cycle monodromy parameter ρ defined by Eq. (7) classifies the stroboscopic dynamics into elliptic (ρ²>0, oscillatory), hyperbolic (ρ²<0, exponential), and parabolic (ρ=0, polynomial) phases. The paper extends the classification to continuous cosine drives numerically, relates the general time-dependent SO(2,1) problem to the Ermakov–Pinney equation, and gives a holographic interpretation in Schrödinger geometries where the hyperbolic class is associated with an ergosurface and the parabolic with an extremal horizon. It also proposes trapped ultracold fermions near unitarity as an experimental platform and analyzes the resonant protocol of a recent experiment.

Significance. If correct, the central result is an exact, parameter-free classification of Floquet phases in any non-relativistic CFT with an SO(2,1) subalgebra, conditional on a primary initial state. The derivation is transparent: Eq. (17) follows from the algebra and the primary expectation values (3), with no fitting parameters. Strengths include the explicit analytic formulas (6)–(9), the numerical Trotter-convergence analysis for the cosine protocol in the SM, and the publicly available code. The paper also gives a falsifiable experimental diagnostic (growth versus boundedness of the post-drive oscillation amplitude A0 in Eq. (26)). The main recognized limitations—exact conformality and primary ground state, and the kinematic nature of the holographic picture—are stated in the text. Within the ideal SO(2,1) sector, I find the central claim sound.

minor comments (4)

[End Matter, after Eq. (23)] The statement that the trapped unitary-Fermi-gas ground state is a primary state is asserted but not justified. A one-line standard argument would close the gap: L− lowers the L0 eigenvalue, and since H+μ²C is bounded below in the fixed-particle-number sector, the ground state must be annihilated by L−. This is the only physical input beyond the algebra, so it deserves an explicit sentence.
[Holography, Eq. (22)] The extremal-horizon identification is derived for the generic parabolic representative with α−≠0. The parabolic conjugacy class also contains representatives with α−=0, γ=0 (e.g. a simple evolution generated by H only), for which the Killing vector K_X has no vanishing locus. The statement that the parabolic phase corresponds to an extremal Killing horizon should be qualified to generic points of the parabolic class.
[SM bibliography] SM bibliography item [2] cites the main text itself as a separate 2026 work. This self-reference should be replaced by a proper citation or by a clear statement that the main text is the companion letter.
[General presentation] Several typos should be corrected: “Fig. 2(c) indicate shows” (main text), “opscillatory phase” (End Matter), “ellptic” (SM), and the duplicated reference markers “[98] [79]” in the Ermakov–Pinney paragraph. These do not affect the physics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central phase classification is an exact SO(2,1) computation with no fitted parameters and no load-bearing self-citation.

full rationale

The derivation of the central result is self-contained. The paper defines H_i = H + μ_i^2 C within the SO(2,1) subalgebra (Eqs. 1, 4), computes the single-cycle Floquet operator in a Pauli representation, obtaining a_l, b_l, c_l, d_l and ρ (Eqs. 5–9), and then evaluates ⟨C(l)⟩ using only the primary-state identities L_-|Δ⟩=0, ⟨Δ|L_+=0, ⟨D⟩=0, ⟨H⟩=μ_1^2⟨C⟩=Δμ_1/2 (Eqs. 2–3, 16–17). The exponential/oscillatory/power-law classification is read off from the sign of ρ^2 in Eq. (17), and ρ is determined by the drive parameters through Eq. (7); no fitted parameter is introduced. The comparison with the experiment of ref. [100] uses the independently reported μ_0, β, and modulation period to compute the Floquet half-trace (|T2|>1), rather than tuning theory to match observed heating. The only substantive non-algebraic input is the assumption that the trapped Fermi-gas ground state is primary; the paper explicitly acknowledges that the analysis holds only near unitarity and for a finite number of drive cycles. Self-citations ([75], [76], and the SM's references to the main text) are contextual background, not load-bearing for the algebraic derivation. No quoted equation reduces to its own input, so no circular step can be exhibited; the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles, forces, or degrees of freedom. The drive parameters μ1, μ2, T1, T are external controls, not fitted constants; the scaling dimension Δ0 is an input from the specific CFT. The ergosurface and extremal horizon are existing geometric notions applied to a known Schrödinger background.

assumptions (4)

domain assumption The SO(2,1) algebra (1): [H,C]=-iD, [D,C]=-2iC, [D,H]=2iH holds for the driven system's generators.
The whole calculation is built on the premise that the driven Hamiltonians H1, H2 are linear combinations of H, C, D satisfying these SU(1,1)/SO(2,1) relations. This is standard for non-relativistic CFTs at unitarity (Nishida–Son), but it is imported from prior literature, not derived here (Eqs. 1–5).
domain assumption There exists a primary (highest-weight) state |Δ0>_μ1 annihilated by L−, with ⟨D⟩=0 and ⟨H⟩=μ1²⟨C⟩=Δμ1/2 (Eq. 3).
Used to evaluate all one-point functions and fidelity. For the trapped Fermi gas this is asserted ('the fermionic ground state ... is a primary state of the CFT', End Matter), but no derivation for the interacting N-particle state is provided.
domain assumption The periodic drive realizes exactly H1=H+μ1²C and H2=H+μ2²C with Hamiltonians bounded from below.
Assumes the only effect of the periodic modulation is to change the harmonic trap frequency while preserving SO(2,1). Real atoms deviate near unitarity; the authors note the analysis holds only for a finite number of drive cycles.
domain assumption Holographic dictionary: the NRCFT is dual to the Schrödinger geometry (84) with sl(2,R) Killing vectors (85).
The ergosurface/extremal-horizon classification uses this background metric from prior literature (Blau–Hartong–Rollier etc.). The paper states the analysis is kinematic and the backreacted metric is not computed.

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Pith. "Pith review of Non-relativistic Floquet Conformal Field Theory." pith.science (2026). https://pith.science/paper/7VMA4OWT

@misc{pith2026260727668,
  author       = {Pith},
  title        = {Pith review of: Non-relativistic Floquet Conformal Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VMA4OWT}},
  note         = {Machine review of arXiv:2607.27668}
}

abstract

We develop a formalism for studying Floquet dynamics for systems with {\it non-relativistic} conformal invariance in $d$ spatial dimensions. Our analysis indicates the existence of two dynamical phases, hyperbolic and elliptic, separated by a parabolic transition surface. We demonstrate this by studying the expectation value of a conformal generator in the many body ground state during the drive, as well as the fidelity of this state. Stroboscopically, they behave exponentially in the hyperbolic phase, show oscillatory behavior in the elliptic phase, and exhibit power-law on the transition surface. Our analysis is completely universal and can be directly applied to several systems including trapped fermions near unitarity and resonant anyons. The former can provide experimental signatures of these dynamical phases. We also comment on a holographic perspective of such driven non-relativistic CFTs and demonstrate that the hyperbolic phase is associated with a timelike stationary-limit surface, such as an ergosphere, in the bulk, while the parabolic phase corresponds to an extremal Killing horizon.

Figures

Figures reproduced from arXiv: 2607.27668 by the authors.

**Figure 2.** (d) exhibits its linear decay in the hyperbolic phase. The inset of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4. (a) Schematic representation of an experimentally [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 1.** Figure 1: FIG. 1. Plots of absolute value of Γ( [PITH_FULL_IMAGE:figures/full_fig_p015_1.png]

**Figure 2.** Figure 2: FIG. 2. Plots of absolute value of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png]

**Figure 3.** Figure 3: FIG. 3. Plot of expectation value of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png]

**Figure 4.** Figure 4: FIG. 4. Cosine-drive phase diagram for [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: FIG. 5. Protocol-matched NRCFT computation for the trap modulation used in Ref. [1]. The upper panel shows the imposed [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

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