Periodically Modulated LMG Model

• The periodically modulated LMG model is a driven collective-spin system where periodic modulation of interactions or fields yields rich nonequilibrium dynamics and time-crystalline order.
• Various modulation techniques, including monochromatic driving, piecewise kicking, and smooth Floquet interpolation, are employed to engineer effective Hamiltonians for studying dynamical phase transitions and quantum sensing.
• Analyses reveal that localization, Floquet heating, and counterdiabatic control significantly influence order parameters, entanglement dynamics, and the emergence of metastable phases in these systems.

Searching arXiv for recent and foundational papers on periodically modulated/driven Lipkin-Meshkov-Glick models. The periodically modulated Lipkin-Meshkov-Glick model denotes a family of driven collective-spin systems derived from the Lipkin-Meshkov-Glick (LMG) Hamiltonian by imposing periodic modulation on interaction terms, transverse fields, or global spin rotations. In the cited literature, this modulation is realized through monochromatic driving, smooth Floquet interpolation, piecewise kicking, and rotating-frame engineering, all within all-to-all interacting or closely related collective-spin settings. These constructions have been used to study nonequilibrium quantum phase transitions, multistability, discrete time-crystalline order, Floquet heating and its suppression, quantum-enhanced sensing, and generalized LMG implementations in cold-atom and cavity-style platforms (Ghosh et al., 3 Oct 2025, Gangopadhay et al., 2024, Engelhardt et al., 2012, Opatrný et al., 2014).

1. Core formulations of periodic modulation

A standard periodically modulated LMG setup is an all-to-all interacting spin-$1/2$ ensemble of size $N$ with total spin operators $S_\alpha=\sum_i \sigma_i^\alpha/2$, $\alpha\in\{x,y,z\}$. In the discrete-time-crystal sensing construction, the Hamiltonian within one drive period $\tau$ is piecewise modulated as

$$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$

with $J=1$, $\delta\ll 1$, and $\phi=(1-\epsilon)\pi$. The corresponding one-period Floquet operator is

$U_F=e^{-i\phi S_x}e^{-iH_1\tau},$

and the stroboscopic state after $N$0 periods is $N$1, where $N$2 is the symmetry-broken ground state of $N$3 (Ghosh et al., 3 Oct 2025).

A distinct Floquet LMG protocol uses a smooth time dependence

$N$4

with

$N$5

period $N$6, and $N$7 in the numerics. In that formulation, the fully symmetric sector $N$8 with $N$9 is invariant, and the drive interpolates between a transverse-field-dominated regime at $S_\alpha=\sum_i \sigma_i^\alpha/2$0 and an interaction-dominated regime at $S_\alpha=\sum_i \sigma_i^\alpha/2$1 (Gangopadhay et al., 2024).

A monochromatically driven variant modulates the interaction directly:

$S_\alpha=\sum_i \sigma_i^\alpha/2$2

or equivalently

$S_\alpha=\sum_i \sigma_i^\alpha/2$3

Here the periodic field induces nonequilibrium phase structure that is naturally analyzed through rotating-frame and Floquet methods (Engelhardt et al., 2012).

A broader usage appears in a toroidal Bose-Einstein condensate with a circular lattice and spatially modulated nonlinearity. After a two-mode truncation and a rotating-wave approximation, the effective spin Hamiltonian is

$S_\alpha=\sum_i \sigma_i^\alpha/2$4

which realizes a generalized LMG Hamiltonian with tunable quadratic coefficients (Opatrný et al., 2014).

2. Floquet structure, order parameters, and nonequilibrium criticality

In the kicked driven LMG model, perfect $S_\alpha=\sum_i \sigma_i^\alpha/2$5-kicks, $S_\alpha=\sum_i \sigma_i^\alpha/2$6 and $S_\alpha=\sum_i \sigma_i^\alpha/2$7, swap the two $S_\alpha=\sum_i \sigma_i^\alpha/2$8 ground states and induce robust period-doubling oscillations, while imperfect kicks $S_\alpha=\sum_i \sigma_i^\alpha/2$9 reduce rigidity. The stroboscopic magnetization

$\alpha\in\{x,y,z\}$0

is monitored through the even-odd average

$\alpha\in\{x,y,z\}$1

This serves as an order parameter: for $\alpha\in\{x,y,z\}$2, $\alpha\in\{x,y,z\}$3 characterizes the DTC phase, whereas for $\alpha\in\{x,y,z\}$4, $\alpha\in\{x,y,z\}$5 identifies a trivial phase. The susceptibility $\alpha\in\{x,y,z\}$6 displays a pronounced peak at $\alpha\in\{x,y,z\}$7 for $\alpha\in\{x,y,z\}$8 and $\alpha\in\{x,y,z\}$9, which is reported as confirmation of a second-order transition. Close to criticality, the finite-size scaling ansatz

$\tau$0

yields $\tau$1, $\tau$2, and $\tau$3 from a data collapse over $\tau$4 (Ghosh et al., 3 Oct 2025).

In the monochromatically driven formulation, the symmetric phase admits a Holstein-Primakoff bosonization and reduces to a parametrically driven harmonic oscillator with undriven gap

$\tau$5

Parametric resonances or dynamical stabilization occur when

$\tau$6

For high driving frequency, a rotating-wave approximation is implemented through

$\tau$7

leading to the effective Floquet Hamiltonian

$\tau$8

Its mean-field expansion defines a quasienergy landscape $\tau$9 whose local minima define metastable phases. At the central point $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$0, the Hessian eigenvalues are

$$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$1

The condition $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$2 reproduces the undriven second-order QPT line $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$3, while $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$4 gives the driving-induced boundary

$$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$5

Numerical scans reveal regions with $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$6 local minima of $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$7, corresponding to $$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$8-$$H(t)= \begin{cases} H_1=-2(J/N)S_z^2-2hS_x+\delta S_z, & 0<t<\tau\ H_2=\phi S_x, & t=\tau \text{ (instantaneous kick)} \end{cases}$$9 dynamically stabilized phases (Engelhardt et al., 2012).

A common simplification is to identify periodic modulation only with a harmonic drive. The cited work shows a broader Floquet taxonomy: piecewise kick protocols support DTC melting transitions, while monochromatic interaction modulation produces quasienergy landscapes and multistability.

3. Localization, Floquet heating, and counterdiabatic control

Localization diagnostics play a central role in periodically modulated LMG dynamics. In the DTC sensing model, the instantaneous inverse participation ratio in the computational basis $J=1$0 is

$J=1$1

and the time-averaged quantity is

$J=1$2

For $J=1$3, the state remains localized in two symmetry-broken components and $J=1$4 is large; for $J=1$5, the state delocalizes over the Hilbert space and $J=1$6 drops to $J=1$7. The finite-size behavior of the IPR minimum is fitted by

$J=1$8

and the sharp decay of $J=1$9 is reported to approach $\delta\ll 1$0 (Ghosh et al., 3 Oct 2025).

In the smoothly driven Floquet LMG model, the central control strategy is counterdiabatic driving:

$\delta\ll 1$1

where the exact adiabatic gauge potential is

$\delta\ll 1$2

The local variational approximation

$\delta\ll 1$3

is implemented up to $\delta\ll 1$4 (“CD1”) or $\delta\ll 1$5 (“CD2”), with coefficients fixed by minimizing

$\delta\ll 1$6

Without CD driving, the system is reported to become chaotic for $\delta\ll 1$7, with level statistics approaching Wigner-Dyson and rapid Floquet heating to infinite temperature. With CD1 or CD2 added, the effective Floquet operator becomes nearly diagonal in the instantaneous eigenbasis for $\delta\ll 1$8, and one finds

$\delta\ll 1$9

so that the return amplitude remains $\phi=(1-\epsilon)\pi$0 and approximate stroboscopic freezing occurs (Gangopadhay et al., 2024).

The same work analyzes Floquet eigenstate localization through the Husimi-$\phi=(1-\epsilon)\pi$1 representation. The eigenstate IPR is defined by

$\phi=(1-\epsilon)\pi$2

with $\phi=(1-\epsilon)\pi$3 for highly localized states and $\phi=(1-\epsilon)\pi$4 for fully delocalized states. A Wehrl-entropy-based localization ratio

$\phi=(1-\epsilon)\pi$5

is also used, with $\phi=(1-\epsilon)\pi$6 for ergodic states. In the ergodic $\phi=(1-\epsilon)\pi$7 regime, the bare Floquet eigenstates have $\phi=(1-\epsilon)\pi$8 and $\phi=(1-\epsilon)\pi$9, while CD2 and CD1 reduce these values; the level-spacing ratio $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$0 shifts from $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$1 toward $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$2, signaling partial integrability or local chaos (Gangopadhay et al., 2024).

4. Quantum sensing, entanglement, and dynamical response

The DTC transition in the periodically modulated LMG model has been used as a sensing resource by treating the kick imperfection $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$3 as the estimated parameter. For a pure probe state $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$4, the quantum Fisher information is

$U_F=e^{-i\phi S_x}e^{-iH_1\tau},$5

Numerically, $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$6 peaks sharply at $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$7, and the peak value scales as

$U_F=e^{-i\phi S_x}e^{-iH_1\tau},$8

which is reported to beat the standard quantum limit while remaining below the Heisenberg limit $U_F=e^{-i\phi S_x}e^{-iH_1\tau},$9. A combined finite-size and dynamic scaling ansatz,

$N$00

gives $N$01, $N$02, and $N$03, and the numerical summary includes $N$04 and $N$05 (Ghosh et al., 3 Oct 2025).

The entanglement structure of periodically driven LMG dynamics is treated explicitly in the counterdiabatic Floquet model. For the one-spin reduced density matrix,

$N$06

the one-spin von Neumann entropy is

$N$07

Without CD driving, $N$08 approaches $N$09 at stroboscopic times, indicating maximal mixing. With CD1 or CD2, an initial fully $N$10-polarized state $N$11 satisfies $N$12 at each integer $N$13, while $N$14 becomes large during the micromotion and the state overlaps with the Dicke state $N$15 with amplitude $N$16. The oscillation period is $N$17 and the amplitude is reported to be weakly dependent on $N$18 for systems studied up to $N$19 (Gangopadhay et al., 2024).

A closely related but static effective realization appears in the toroidal BEC mapping. There, the twisting tensor has eigenvalues $N$20, and the two-axis counter-twisting limit occurs when

$N$21

The phase $N$22 rotates the principal squeezing axes in the $N$23-$N$24 plane through

$N$25

This suggests a direct link between periodically engineered LMG couplings and metrological state generation, although the cited work formulates that connection primarily in terms of spin squeezing rather than Floquet-critical sensing (Opatrný et al., 2014).

5. Implementations, rotating frames, and effective Hamiltonian engineering

The periodically modulated LMG model is not tied to a single laboratory realization. In the toroidal condensate scheme, a weak circular lattice

$N$26

and a nonlinearity modulated at twice the lattice spatial frequency,

$N$27

generate an effective two-mode description in the $N$28 subspace. The rotating-frame definitions

$N$29

remove fast phases, and the resulting Hamiltonian becomes strictly time-independent after neglecting fast oscillating terms. The usual LMG form is recovered, for example, by setting $N$30, $N$31, and $N$32, then identifying

$N$33

which yields

$N$34

The same work states that the scheme naturally implements the full three-parameter LMG Hamiltonian and allows continuous tuning between one-axis and two-axis twisting (Opatrný et al., 2014).

The driven nonequilibrium transition studied under monochromatic modulation was explicitly presented as a route to engineer multiple metastable Floquet phases, with potential application to cavity-QED and circuit-QED realizations (Engelhardt et al., 2012). In the DTC sensing construction, the robustness of DTC order to disorder and weak symmetry breaking $N$35 is stated to suggest experimental feasibility in cavity-QED or trapped-ion platforms (Ghosh et al., 3 Oct 2025).

A common point across these realizations is the use of rotating-frame or Floquet-effective descriptions to isolate the slow collective dynamics. In the toroidal BEC case, off-resonant processes are suppressed when

$N$36

which are described as rotating-wave or secular approximations (Opatrný et al., 2014). In the monochromatically driven spin model, the analogous simplification is the high-frequency rotating-wave approximation yielding $N$37 (Engelhardt et al., 2012). The repeated appearance of such effective descriptions suggests that “periodic modulation” in LMG physics is often operationally defined by the emergent Floquet generator rather than by the bare laboratory-frame Hamiltonian alone.

6. Extensions to coupled LMG systems and topological structure

An important extension is a one-dimensional lattice of coupled LMG models with alternating couplings between nearest-neighbor sites, described by

$N$38

Although this is not a single collectively driven spin, the cited work explicitly refers to it as a “periodically modulated LMG” Hamiltonian and analyzes its hybrid semiclassical-topological structure (Sorokin et al., 2016).

In the semiclassical limit $N$39, a Holstein-Primakoff expansion followed by a displacement

$N$40

produces

$N$41

The stationary points of the classical energy landscape $N$42 fall into five families, with phase I disordered and phases II-V ordered. The bifurcation condition is

$N$43

so that only the trivial minimum exists below this line, whereas two symmetry-broken minima appear above it; the cited work identifies this as a second-order quantum phase transition (Sorokin et al., 2016).

Quantum fluctuations around the mean field are governed by a bosonic BdG problem. Because the BdG Hamiltonian satisfies the inversion symmetry $N$44, one can define the symplectic polarization

$N$45

which is quantized to integer or integer plus $N$46 under inversion symmetry. The topological transition occurs at

$N$47

where the gap between the two bands closes; for $N$48, $N$49 jumps from $N$50 to $N$51 when $N$52 crosses $N$53. On open chains, the nontrivial regime $N$54 supports pairs of exponentially localized mid-gap modes at the edges, while the symmetry-broken phases can also exhibit non-topological impurity-type states below the lower band (Sorokin et al., 2016).

This extension indicates that periodically modulated LMG physics is not limited to uniform collective-spin dynamics. A plausible implication is that the combination of bifurcation theory, Floquet engineering, and bosonic BdG topology provides a broader framework in which LMG-type nonlinearities can support both nonequilibrium criticality and boundary-localized excitations.