Abstract Periodically driven systems provide a novel route to control the topology of quantum materials. In particular, Floquet theory allows an effective band description of periodically-driven systems through the Floquet Hamiltonian. Here, we study the time evolution of d-wave superconductors irradiated with intense circularly-polarized laser light. We consider the Floquet t–J model with time-periodic interactions, and investigate its mean-field dynamics by formulating the time-dependent Gutzwiller approximation. We observe the development of the idxy-wave pairing amplitude along with the original $${d}_{{x}^{2}-{y}^{2}}$$ d x 2 − y 2 -wave order upon gradual increasing of the field amplitude. We further numerically construct the Floquet Hamiltonian for the steady state, with which we identify the system as the fully-gapped d + id superconducting phase with a nonzero Chern number. We explore the low-frequency regime where the perturbative approaches in the previous studies break down, and find that the topological gap of an experimentally-accessible size can be achieved at much lower laser intensities.
Abstract Periodically driven systems provide a novel route to control the topology of quantum materials. In particular, Floquet theory allows an effective band description of periodically-driven systems through the Floquet Hamiltonian. Here, we study the time evolution of d-wave superconductors irradiated with intense circularly-polarized laser light. We consider the Floquet t–J model with time-periodic interactions, and investigate its mean-field dynamics by formulating the time-dependent Gutzwiller approximation. We observe the development of the idxy-wave pairing amplitude along with the original dx2−y2 d x 2 − y 2 -wave order upon gradual increasing of the field amplitude. We further numerically construct the Floquet Hamiltonian for the steady state, with which we identify the system as the fully-gapped d + id superconducting phase with a nonzero Chern number. We explore the low-frequency regime where the perturbative approaches in the previous studies break down, and find that the topological gap of an experimentally-accessible size can be achieved at much lower laser intensities.
Abstract Non-equilibrium engineering is becoming a seminal way for realising novel quantum phases that are unimaginable in equilibrium. In particular, Floquet theory applied to quantum mechanics revealed that we can even control the band topology in semimetallic/insulating systems, while the straightforward application to topological superconductivity fails for typical superconductors because the supercondicting gap function does not couple to the electromagnetic field in a direct manner. Here we show that we can overcome this difficulty by taking account of correlation effects. Namely, we study how a d-wave superconductivity is changed when illuminated by circularly-polarised light (CPL) in the repulsive Hubbard model in the strong-coupling regime. We adopt the Floquet formalism for the Gutzwiller-projected effective Hamiltonian with the time-periodic Schrieffer-Wolff transformation. We find that CPL induces a topological superconductivity with a d + id pairing, which arises from the chiral spin coupling and the three-site term generated by the CPL. The latter term remains significant even for low frequencies and low intensities of the CPL. This is clearly reflected in the obtained phase diagram against the laser intensity and temperature for various frequencies red-detuned from the Hubbard U, with the transient dynamics also examined. The phenomenon revealed here can open a novel, dynamical way to induce a topological superconductivity.
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.
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
Intense time-periodic laser fields can transform the electronic structure of a solid into strongly modified Floquet-Bloch bands. While this suggests multiple pathways to induce electronic orders such as superconductivity or charge density waves, the possibility of preparing low-energy phases of Floquet Hamiltonians remains unclear because of the energy absorption at typical experimentally accessible driving frequencies. Here we investigate a realistic pathway towards laser control of electronic orders, which is the transient enhancement of fluctuating orders. Using a conserving Keldysh Green's function formalism, we simulate the buildup of short range Cooper-pair correlations out of a normal metal in the driven attractive Hubbard model. Even for frequencies only slightly above or within the bandwidth, a substantial enhancement of correlations can be achieved before the system reaches a high electronic temperature. This behavior relies on the nonthermal nature of the driven state. The effective temperature of the electrons at the Fermi surface, which more closely determines the superconducting correlations, remains lower than an estimate from the global energy density. Even though short ranged, the fluctuations can have marked signatures in the electronic spectra.
In this thesis, we theoretically investigate exotic quantum phenomena underlying periodically-driven systems.We discuss three distinct systems, noninteracting Dirac systems, lattice systems with a strong repulsive interaction, and lattice systems with a strong attractive interaction.We reveal various novel phenomena underlying these systems, and provide clear understanding of them.Periodically-driven systems, i.e., systems with a discrete time-translational symmetry, are described by the Floquet formalism, which maps the time-dependent Hamiltonian to a static "photon-dressed Hamiltonian".While a perturbative expansion of this Hamiltonian from the limit of high driving frequency has clarified various novel phenomena peculiar to periodically-driven systems, such an approach suffers from serious heating or damage when it is applied to realistic condensed matters.In this thesis, we focus on a lower-frequency regime, especially when the heating should be suppressed due to the small driving amplitude or an energy gap larger than the driving frequency.There, the high-frequency expansion cannot capture the relevant physics, so that we first formulate various perturbative expansions for three distinct regimes, i.e., high-frequency, small-amplitude, and strong-coupling regimes.We apply the small-amplitude expansion to two-and three-dimensional Dirac electrons driven by a circularly-polarized light to explore their behavior in the lowfrequency regime.We find an emergence of infinite numbers of topological gaps for the former and Weyl nodes for the latter.We also discuss how a regularization of the model affects this topological structure.We then introduce a strong repulsive interaction, and discuss behaviors of spins using the strong-coupling expansion.We predict an emergence of scalar spin chirality when a circularly-polarized light is illuminated, and discuss possible applications to photo-induced chiral spin liquids, and optical probing of spin chirality.Finally we turn to a strong attractive interaction to discuss the fate of the ordinary s-wave superconductivity under the periodic driving.We demonstrate an emergence of long sought-after η-pairing superconductivity, first as a ground state of the photon-dressed Hamiltonian, and then as a consequence of a dynamical phase transition in an isolated system such as an optical lattice system.
In the different chapters of this dissertation we investigate multiple non-perturbative approaches that allow the study of solids subjected to time dependent fields. In the second chapter, we develop a very general non-perturbative and flexible approach that allows to study periodically driven quantum systems via an effective time-independent theory. It makes use of renormalization-group-like flow equations to find highly accurate effective Hamiltonians[Phys. Rev. X 9, 021037 (2019)]. The range of validity of the approach is checked numerically for various 1D spin systems and confirmed to be beyond the reaches of ordinary perturbative methods. It also compares favourably to another non-perturbative approach. In the third chapter we examine the wealth of different approximations to the time evolution operator that are known from the literature. Inspired by Hamilton-Jacobi theory we find a useful reformulation of the equation for the time evolution operator [Phys. Rev. A 100, 012132 (2019)]. This allows us to find the various known approximations of the time evolution operator in a unified fashion and new ones. Interestingly the RG-like flow equations from the second chapter are found to be one specific limit of the more general approach developed here. A periodically driven Ising model is used to verify the range of validity of the different approximations. This allows us to put them into a hierarchy. Up until here the methods that were explored are most valid in the high to mid frequency regime. In the fourth chapter we show how for the case of a weakly driven system a simple approach can be used to treat the low frequency regime [Phys. Rev. B 101, 024303 (2020)]. To achieve this we derive a quasi-energy dependent Hamiltonians and very crucially require self-consistency for the quasi-energies. For the simple two band example of single layer graphene it is found to be highly accurate approach that also allows for accurate descriptions of even sensitive quantities like Chern numbers For the last fifth chapter of the thesis we switch gears and consider a specific system of much recent interest - twisted bilayer graphene. We investigate how light that is altered by the boundary conditions of a metallic waveguide can change the strength of interlayer couplings [arXiv:2001.04416 (2020)]. We investigate how this change affects the twist angles at which flat bands appear and find very simple analytic results that are in good agreement with a numerical treatment. In the sixth chapter of the thesis, we investigate the effects circularly polarized light has on twisted bilayer graphene[ arXiv:2002.05124 (2020)]. We derive two effective Hamiltonians valid in different driving regimes. One is valid for weak drives but low frequencies and the other for relatively strong drives. The Hamiltonian for the strongly driven regime is compared to exact numeric results and found to perform better than other common methods. We analyse the newly generated terms in both Hamiltonians for their symmetry.