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| Fig. 1: 4 step lattice-hopping Floquet system. (Source: J. Montana-Lopez, after Rudner et al. [2]) |
This is a review of Harper et al. [1] Most systems that occur naturally have time-independent Hamiltonians. The common procedure is to plug in the Hamiltonian in Schrodinger's equation, obtain a basis of time-independent eigenstates and express the time-evolution of the system in terms of this basis of eigenstates. In the case of Floquet systems, the Hamiltonian is time-dependent and periodic, with period T:
This makes our procedure more complex, since the eigenstates will not be time-independent anymore, and we will need to consider the time-dependent unitary operator U(t). Moreover, the energy E of a state will only be defined up to the periodic condition E = E + n(2pi)/(T), for integer n, and it will be called "quasienergy" because of this. Due to Floquet's theorem, we can decompose the unitary time-evolution operator as
where HF is the Floquet Hamiltonian (different from H!), an effective Hamiltonian that will be most useful for us, since it is time independent and applying it evolves the system forward by T. This added complexity gives us new phenomena which may not have a non-periodic counterpart: single-particle Floquet systems can display novel topology in their band structures, while interacting many-body Floquet phases can arive from broken symmetries or nontrivial topology (in the presence of many-body localization to prevent heating)
As an example, take a system consisting of a two-dimensional lattice with a time-dependent Hamiltonian that makes a particle hop from its current point in the lattice to a nearest neighbor, from Rudner et al. [2] This can be defined for any lattice size, but just for the purpose of this example we will use a 6 × 4-dimensional lattice.
If this Hamiltonian has period T, every T/4 interval the Hamiltonian will change which nearest neighbor the particle should hop to: 1. to the left, 2. upwards, 3. to the right, 4. downwards. After the 4th step, we would go back to 1, since the Hamiltonian will have completed the full period T. This is illustrated in Fig. 1. If we start at a point in the center of the lattice, after a full period we will go back to the starting point (blue arrow). If we start at the third point in the upper edge of the lattice, diagram 1 tells us to move left, 2 tells us to move up (but we cannot, since it is the upper edge), so we do not move, 3 tells us to move to the left (we now move from the second on the top to the first), 4 tells us not to do anything (the first vertex doesn't have a downward yellow line in step 4). So on the edge, we have a leftwards movement (green arrow), and a similar reasoning gives us a rightwards movement on the bottom (red arrow). This gives us a chirality (a tendency to rotate) on the edge modes, which is one of the properties used to characterize systems.
This chirality cannot be characterized by the Floquet Hamiltonian: if we take any point not on the edge of the lattice (i.e. in the bulk, the middle), after one full period we are back to the starting point (blue line). Since the Floquet Hamiltonian only measures by timesteps of length T, if the system is the same after one period T, nothing has changed. So the unitary time-evolution operator is the identity, for periodic boundary conditions:
Since the Floquet Hamiltonian is time-independent, this would mean that it's zero for all times, for periodic boundary conditions. But the zero Hamiltonian does not make things change, and certainly the edge points still have chirality in a system with open boundaries. In static system, there is a bulk-edge correspondence that allows us to find the edge properties from the invariants of the bulk (with periodic boundary conditions), but this does not hold in time-dependent Hamiltonians. Therefore, the Floquet Hamiltonian cannot capture topological behavior like chirality, so we will need new beautiful mathematical tools like K-theory and cohomology. [3]
Fig. 1c describes the Floquet energy spectrum in terms of the crystal momentum, of this lattice model. The red and green lines correspond to the chiral modes, which are linear in momentum (dE/dk = ± 1/T), while the blue line corresponds to the bulk modes, with Floquet operator the identity and quasienergy 0. [2] We see that the difference in quasienergy in the edge lines in one period (this is called the "quasienergy gap") is high, so small perturbations will still have chirality. If we consider the lattice as a whole, the edge modes are localized at different ends of the lattice, so local perturbations can not mix the edge modes to open a gap. This means that the system has a stable phase with topologically protected edge modes. This just means that the edge modes are preserved as long as the topology of the system is preserved. In practice, this means that the perturbations are not big enough to close the quasienergy gap, i.e. make the difference of quasienergies 0.
This will be a recurrent topic in Floquet systems: they will have some topological invariants (analogs to the chirality we just saw) that will not change upon small perturbations (hence invariants). Therefore, they are something really qualitative and intrinsic to the system, and we will be interested in classifying Floquet systems according to their topology. [4] Many-Body Floquet localization and localized interacting phases will also be related to the topology of the system and the symmetries within it. [5,6]
For example, in a closed Many-Body Floquet system, one could expect that at long times we would achieve a state of maximal entropy, i.e. that in which the value of all local expectation values is time-independent and independent of the starting state. Since all states are available, we consider this the infinite temperature state. However, there are mechanisms to prevent this thermalization. One of them is Floquet Many-Body Localization (MBL). The idea is that there will be subsystems that interact with each other by receiving and transferring energy in a stable way, so that the system does not reach thermalization, but rather keeps this local exchange of energy going. The operators in these subsystems are called "l-bits" and they allow for Floquet MBL systems that are stable under small perturbations.
In the extended report we will expand on the topics above and introduce ways to study and classify Floquet Topological Insulators according to their topology, and introduce different phases of matter that arise in a binary drive Floquet system.
© Jordi Montana-Lopez. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] F. Harper et al., "Topology and Broken Symmetry in Floquet Systems," Annu. Rev. Condens. Matter Phys. 11, 354 (2020).
[2] M. S. Rudner et al., "Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems," Phys. Rev. X 3, 031004 (2013).
[3] R. Roy and F. Harper, "Periodic Table for Floquet Topological Insulators," Phys. Rev. B 96, 155118 (2017).
[4] F. Nathan and M. S. Rudner, "Topological Singularities and the General Classification of Floquet-Bloch Systems," New J. Phys. 17, 125014 (2015).
[5] P. Ponte et al., "Many-Body Localization in Periodically Driven Systems," Phys. Rev. Lett. 114, 140401 (2015).
[6] V. Khemani et al., "Phase Structure of Driven Quantum Systems," Phys. Rev. Lett. 116, 250401 (2016).