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arXiv:2107.13080 · Nuclear Physics B

Periodically driven perturbed CFTs: The sine-Gordon model

Zoltán Bajnok, Robin Oberfrank

Published Feb 28, 2022• 6 citations
arXivHTMLPDF

Abstract

We analyze a version of the sine-Gordon model in which the strength of the cosine potential has a periodic dependence on time. This model can be considered as the continuum limit of the many-body generalization of the Kapitza pendulum. Based on the perturbed CFT point of view, we develop a truncated conformal space approach (TCSA) to investigate the Floquet quasienergy spectrum. We focus on the effective behaviour for large driving frequencies, which we also derive exactly. Depending on the driving protocol, we can recover the original sine-Gordon model or its two-frequency version. The rich structure of the two-frequency model implies that the time-periodic drive can break integrability, can lead to new states in the spectrum, or can result in a phase transition. Our method is applicable for any periodically driven perturbed conformal field theories.

DOI
10.1016/j.nuclphysb.2022.115717

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We analyze a version of the sine-Gordon model in which the strength of the cosine potential has a periodic dependence on time. This model can be considered as the continuum limit of the many-body generalization of the Kapitza pendulum. Based on the perturbed CFT point of view, we develop a truncated conformal space approach (TCSA) to investigate the Floquet quasienergy spectrum. We focus on the effective behaviour for large driving frequencies, which we also derive exactly. Depending on the driving protocol, we can recover the original sine-Gordon model or its two-frequency version. The rich structure of the two-frequency model implies that the time-periodic drive can break integrability, can lead to new states in the spectrum, or can result in a phase transition. Our method is applicable for any periodically driven perturbed conformal field theories.
Abstract We study the properties of a conformal field theory (CFT) driven periodically with a continuous protocol characterized by a frequency ωD. Such a drive, in contrast to its discrete counterparts (such as square pulses or periodic kicks), does not admit exact analytical solution for the evolution operator U. In this work, we develop a Floquet perturbation theory which provides an analytic, albeit perturbative, result for U that matches exact numerics in the large drive amplitude limit. We find that the drive yields the well-known heating (hyperbolic) and non-heating (elliptic) phases separated by transition lines (parabolic phase boundary). Using this and starting from a primary state of the CFT, we compute the return probability (Pn), equal (Cn) and unequal (Gn) time two-point primary correlators, energy density(En), and the mth Renyi entropy ( Snm {S}_n^m Snm​ S n m ) after n drive cycles. Our results show that below a crossover stroboscopic time scale nc, Pn, En and Gn exhibits universal power law behavior as the transition is approached either from the heating or the non-heating phase; this crossover scale diverges at the transition. We also study the emergent spatial structure of Cn, Gn and En for the continuous protocol and find emergence of spatial divergences of Cn and Gn in both the heating and non-heating phases. We express our results for Snm {S}_n^m Snm​ S n m and Cn in terms of conformal blocks and provide analytic expressions for these quantities in several limiting cases. Finally we relate our results to those obtained from exact numerics of a driven lattice model.
We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems.Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory.The final results are equivalent to those obtained within a different approach (Rahav et al 2003 Phys.Rev. A 68 013820), (Goldman and Dalibard 2014 Phys.Rev. X 4 031027) and can also be related to the Floquet-Magnus expansion (Casas et al 2001 J. Phys.A 34 3379).We discuss that the dependence on the driving phase, which plagues the latter, can lead to artifactual symmetry breaking.The high-frequency approach is illustrated using the example of a periodically driven Hubbard model.Moreover, we discuss the nature of the approximation and its limitations for systems of many interacting particles.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.
We construct a systematic high-frequency expansion for periodically driven quantum systems based on the Brillouin-Wigner (BW) perturbation theory, which generates an effective Hamiltonian on the projected zero-photon subspace in the Floquet theory, reproducing the quasienergies and eigenstates of the original Floquet Hamiltonian up to desired order in $1/\ensuremath{\omega}$, with $\ensuremath{\omega}$ being the frequency of the drive. The advantage of the BW method is that it is not only efficient in deriving higher-order terms, but even enables us to write down the whole infinite series expansion, as compared to the van Vleck degenerate perturbation theory. The expansion is also free from a spurious dependence on the driving phase, which has been an obstacle in the Floquet-Magnus expansion. We apply the BW expansion to various models of noninteracting electrons driven by circularly polarized light. As the amplitude of the light is increased, the system undergoes a series of Floquet topological-to-topological phase transitions, whose phase boundary in the high-frequency regime is well explained by the BW expansion. As the frequency is lowered, the high-frequency expansion breaks down at some point due to band touching with nonzero-photon sectors, where we find numerically even more intricate and richer Floquet topological phases spring out. We have then analyzed, with the Floquet dynamical mean-field theory, the effects of electron-electron interaction and energy dissipation. We have specifically revealed that phase transitions from Floquet-topological to Mott insulators emerge, where the phase boundaries can again be captured with the high-frequency expansion.
The phase spaces of the two- and three-frequency sine-Gordon models are examined in the framework of the truncated conformal space approach. The focus is mainly on a tricritical point in the phase space of the three-frequency model. We give substantial evidence that this point exists. We also find the critical line in the phase space and present TCSA data showing the change of the spectrum on the critical line as the tricritical endpoint is approached. We find a few points of the line of first-order transition as well.
Conformal field theories (CFT) play an important role in quantum field theories as they allow for exact studies that are hard to access without the conformal symmetries in place. A remarkable recent development is the application of conformal field theory to periodically driven systems, the so-called Floquet-driven CFTs. A number of recent works have found that there are classes of periodically driven (1+11+11+1)-dimensional CFTs that can be analytically solved, and that show both heating and nonheating phases, as well as unusual properties in terms of information scrambling. Hitherto, most results have focused on the dynamics of such models at stroboscopic times, i.e., samples taken at each period of the time evolution. In this paper, we consider time evolution at arbitrary real times and derive formulas for the exact time propagation of heating and nonheating classes of Floquet-driven CFTs. This requires a subtle treatment of nonstroboscopic time that leads to different interpretations in terms of curved space-time dynamics in these systems compared to previous results in the literature.