an SFB/TR 185 magazine
Second Funding Period, Issue 2
Author: Dr. Julian Schmitt
Finite-Temperature Topological Invariant for Interacting Systems
R. Unanyan, M. Kiefer-Emmanouilidis, and M. Fleischhauer
Phys. Rev. Lett. 125, 215701 (2020)
Since the discovery of the quantum Hall effect, topology has emerged as an important paradigm for the classification of phases of matter. Topological systems are characterized by numbers, so-called topological invariants, which describe global properties of the system and are responsible for, e.g., robust quantized bulk transport or edge currents. Invariants such as the Chern number of single-particle Bloch functions are restricted to pure, i.e., zero-temperature states. In recent years, research has focused on extending the concept of topology to the for experiments relevant realm of finite temperatures or nonequilibrium states. Its application to both interacting ensembles and higher dimensions, however, remains a major challenge. For mixed states of noninteracting fermions in 1D, recent work has identified a finite-temperature topological invariant, termed ensemble geometric phase (EGP). A winding of the EGP can have direct physical consequences. For example, it may lead to quantized transport in a coupled auxiliary system. The group of M. Fleischhauer has now extended the concept of EGP to the case of interacting particles. This allowed them to show that for 1D systems with a gapped ground state of fractional filling a topological invariant for finite-temperature states of inter-acting particles can be defined.
In their study, Unanyan et al. focus on the extended superlattice Bose-Hubbard model (Ext-SLBHM) at quarter filling illustrated in Fig. 1, where the ground state is a doubly degenerate Mott insulator. The model poses a specific example that features interaction-induced fractional topological charges and associated fractional winding number. By starting in one of the two many-body ground states at zero temperature and changing the lattice parameters, relative tunneling t1-t2 and staggered potential Δ, along a closed loop (Fig. 1a, bottom), the ground state returns to itself after performing 2 loops with an additional phase. Similarly, this implies an integer-quantized particle current (Fig.1b, left). At finite temperatures, however, the number of transported particles deviates substantially from unity as the temperature approaches the many-body gap. In sharp contrast, the authors show (Fig.1b, right) that the winding of the EGP remains strictly unity and identical to that of the ground state for finite temperatures below a critical temperature (above the gap energy) – even for interacting systems.
Crucially, the EGP introduces a new approach to detect topological properties from finite-temperature measurements, accessible, e.g., in ultracold atom experiments with quantum gas microscopes. The findings can be extended to interacting two-dimensional systems.
Fig.1: (a) In the extended superlattice Bose-Hubbard model interacting bosons move along a 1D Δ-staggered lattice with alternating hopping t1,2, and (next-)nearest neighbor interaction V1 (V2). Cyclic adiabatic variations of t1-t2 and Δ realize a fractionally quantized charge pump. (b) Left: Integer-quantized particle current as a function of time at different temperatures obtained from exact diagonalization. Right: the EGP shows strictly quantized winding at all temperatures.
Thermodynamics of Trapped Photon Gases at Dimensional Crossover from 2D to 1D
E. Stein, and A. Pelster
The emergence of phase transitions between different states of matter delicately depends on dimensionality. E.g., even without interactions, a uniform gas of bosonic particles in three dimensions undergoes Bose-Einstein condensation, a phase transition that relies solely on the quantum statistics of the indistinguishable particles. In lower-dimensional uniform systems, however, no long-range order can emerge; the Mermin-Wagner theorem precludes Bose-Einstein condensation in such systems. The situation changes when an external trapping potential is applied. While for attractive potentials, condensation can occur in two dimensions (2D), in a one dimensional (1D) setting a potential more confining than a quadratic one is necessary. The group of A. Pelster has now addressed the dimensional crossover between 2D and 1D by studying the thermodynamics of an ideal Bose gas confined in highly anisotropic harmonic traps. Their results provide guidance for experiments of ideal gases of photons in patterned optical resonators at the dimensional crossover from 2D to 1D.
In quantum gas experiments, the effective dimensionality is often reduced to a direct comparison of different length scales, e.g., the in-plane radius and axial width of the gas, the healing length ξ and the thermal wavelength λth. A system is effectively 2D, if ξ and λth exceed the axial width, and effectively 1D, if they become larger than the in-plane radius while remaining smaller than the axial width. As the healing length depends on density and interaction strength, and the thermal wavelength on temperature, one can control the dimension by i) compressing the trap, ii) exploiting Feshbach resonances for atomic gases, or by iii) cooling the gas. Stein et al. focus on i) and analytically study a photonic quantum gas in a highly anisotropic, a priori 2D, harmonic potential with tunable aspect ratio. By observing changes in thermodynamic quantities, an effective system dimension can be defined for both the thermodynamic limit and finite-sized systems. Fig. 2a shows the crossover of the temperature-dependent specific heat as the trap is deformed from isotropic (2D) towards anisotropic (1D). Below the critical temperature Tc, the reduced dimensionality is quantitatively revealed in the exponent of CN/(NkB), while above Tc its value of 1 resembles a 1D Dulong-Petit law, indicating that the system has lost 2 degrees of freedom due to the trap compression. Exploiting both signatures, an effective dimension of the system in both the Bose-condensed and the thermal regime can now be defined; see Fig.2b for the corresponding phase diagram.
The reported findings pave the way towards hitherto elusive experiments on photon gases confined in (strongly) anisotropic harmonic trapping potentials, as may be realized by employing lithography techniques.
Fig. 2: (a) Specific heat versus temperature for various trap anisotropies reveals 2D-to-1D-crossover. In the isotropic 2D case (blue), BEC is indicated by a cusp that disappears with increasing anisotropy (orange, purple), eventually vanishing in the 1D limit (green). (b) Effective dimension of the ideal Bose gas at the dimensional crossover versus temperature and trap-aspect ratio λ, derived from the specific heat.
Dissipation engineered directional filter for quantum ratchets
Z. Fedorova, C. Dauer, A. Sidorenko, S. Eggert, J. Kroha, and S. Linden
Systems described by time-periodic Hamiltonians can exhibit intriguing transport phenomena inaccessible under equilibrium conditions. A fascinating example, where a periodic drive is directed into motion without a bias force, are ratchets. Their working principle relies on the breaking of space- and time-reversal symmetry, and such microscopic motors can operate both in classical as well as in quantum systems. In contrast to classical ratchets, which are based on thermal motion and thus independent of the initial conditions, directed transport in quantum ratchets arises from a quantum coherence effect. Therefore, precise initial-state preparation and slow (adiabatic) driving are required, making it difficult to achieve optimal transport efficiency. In joint theory-experimental work, Z. Federova et al. have now demonstrated a scheme for quantized, directional transport in fast (nonadiabatic) Hamiltonian ratchets using dielectric-loaded surface plasmon polariton waveguides (DLSPPW) equipped with a local impurity with dynamic dissipation as a direction-dependent filter.
The theoretically considered Hamiltonian ratchet scheme is based on a periodically driven Su-Schrieffer-Heeger (SSH) model (Fig. 3a, top), known to support quantized transport for certain, nonadiabatic driving frequencies if the space inversion symmetry is broken by initial conditions. By inserting a time-periodic non-Hermititian impurity, the authors predict a novel type of non-reciprocal transport through the impurity – even for mixed initial states. Experimentally, the predictions are tested in DLSPPWs (Fig. 3a, bottom), in which light propagates according to the paraxial Helmholtz equation. Owing to its analogy with the Schrödinger equation, time here directly maps into propagation distance, making waveguides a well-suited platform to mimic time-periodic systems. Each DLSPPW realizes a lattice site, and precise control over the hopping amplitudes is achieved by evanescent coupling of the wriggling adjacent waveguide modes; periodic losses are introduced by deposition of absorbing chromium stripes.
To probe the direction-dependent filter, the team of S. Linden shone a laser at one end of the sample to inject a wave packet propagating in the transverse direction, i.e., along the 1D chain. Its group velocity was then probed for different driving frequencies. Without engineered losses, a resonance occurs at which transport is maximized in one direction. Away from it, the wave packet splits due to hybrid counterpropagating states. When fabricating DLSPPW arrays with modulated local losses, however, the transport properties change: as the wave packet impinges upon the lossy region from top (Fig. 3b), it is strongly damped and disappears. In contrast, when launched from the opposite side, the wave packet is almost fully transmitted. This nonreciprocity shows strikingly the feasibility of novel direction-dependent filters in Hamiltonian quantum ratchets that can be used to rectify mixed initial quantum states.
Fig. 3. (a) Top: SSH model with time-periodic hopping, split by a local dissipative impurity, Bottom: Experimental realization in a plasmonic waveguide array with low and high loss (green, red) directions. (b) Unidirectional filter from local modulated dissipation (dashed) transmits only upward currents.