Probing the Topology of Density Matrices

The mixedness of a quantum state is usually seen as an adversary to topological quantization of observables. For example, exact quantization of the charge transported in a so-called Thouless adiabatic pump is lifted at any finite temperature in symmetry-protected topological insulators. Here, we show that certain directly observable many-body correlators preserve the integrity of topological invariants for mixed Gaussian quantum states in one dimension. Our approach relies on the expectation value of the many-body momentum-translation operator and leads to a physical observable—the “ensemble geometric phase” (EGP)—which represents a bona fide geometric phase for mixed quantum states, in the thermodynamic limit. In cyclic protocols, the EGP provides a topologically quantized observable that detects encircled spectral singularities (“purity-gap” closing points) of density matrices. While we identify the many-body nature of the EGP as a key ingredient, we propose a conceptually simple, interferometric setup to directly measure the latter in experiments with mesoscopic ensembles of ultracold atoms.

Popular Summary

Since the discovery of topological effects in the 1980s, studies at the intersection of topology and quantum physics have led to the identification of multiple robust quantum phenomena. Topology makes it possible for physical observables such as currents or pumped charges to be quantized to a remarkable degree. It provides a way for quantum effects to be resilient against disorder and the various perturbations that shake the quantum world. In recent years, a paradigm has emerged whereby topological properties of quantum systems are identified via their ground-state wave function. Although this captures a wide variety of low-temperature equilibrium phenomena, ground states of quantum systems are often superimposed with excited states, making it much more difficult to identify topological properties. We identify a new type of physical observable—the “ensemble geometric phase” (EGP)—which indicates topological order even in such mixed states.

Conceptually, the EGP is a unimodular phase variable extracted from the many-body wave function of a quantum system. Its many-body nature is essential and is the key to its success as an indicator of topological order in generalized quantum states. We demonstrate the geometric origin of the EGP, identify a corresponding topological invariant, and provide examples of one-dimensional systems of fermions both in and out of equilibrium where this invariant is nonzero. We also give the EGP a concrete physical meaning by proposing an explicit measurement scheme. In particular, we show that many-body observables are accessible in experiments based on ultracold atoms.

The EGP provides a new tool for probing the topology of quantum systems in mixed-state systems beyond traditional condensed-matter quantities such as currents and pumped charges.

Figure 1

Key results. (a) Topologically quantized pumping for Gaussian mixed states with translation invariance. The EGP ${\phi }_E$ of a mixed state [Eq. (7)] reduces to the Zak phase of a pure state $|u_{k,0}⟩$ as the system size $N$ is increased. The state $|u_{k,0}⟩$ corresponds to the lowest band in the so-called “purity spectrum” [see (b)] of the state density matrix $\rho$. For thermal mixed states $\rho \sim e^{-\beta H}$, it coincides with the zero-temperature ground state of the Hamiltonian $H$. When varying a parameter $\varphi$ along a loop from 0 to $2\pi$, the phase ${\phi }_E(\varphi )$ changes by a topologically quantized value ($2\pi$ in the above illustration). This holds irrespective of $N$ and of the state mixedness. The only requirement is a gap in the state purity spectrum. (b) Schematic purity spectrum showing the relevant gap $\mathrm{\Delta }$. The density matrix of the state can be expressed as $\rho =e^{-G}$, where $G=-\ln \rho$ is Hermitian [see, e.g., Eq. (8)]. This allows us to define the state “purity” eigenvalues and eigenvectors in analogy with spectral features of a Hamiltonian (where “purity” refers to the fact that eigenvalues indicate the degree of purity of $\rho$ in each eigenspace [19]). (c),(d) Crucial difference between conventional few-body observables and the many-body EGP considered in this work. While the former take the symbolic form approximately $\mathrm{Tr}[\rho (\dots )]=\mathrm{Tr}[e^{-G}(\dots )]$, the phase ${\phi }_{\mathrm{E}}$ has a more rigid structure approximately $\mathrm{Tr}[e^{-G_k}(\dots )e^{-G_{k+1}}(\dots )\dots ]$, where $G_k$ denotes the matrix $G$ in the momentum sector $k$. This structure corresponds to a path-ordered product over points $k=1,\dots ,N$ across the Brillouin zone (Sec. 3a). In the state eigenbasis $\{|u_{k,s}⟩\}$ (where $s=0,1,\dots ,n-1$ indexes purity bands), matrices $e^{-G}$ and $e^{-G_k}$ are diagonal and can be seen as “weight factors.” In the conventional case depicted in (c) [where states $|u_{k,s}⟩$ represented by circles are connected by operators $(\dots )$], a single “selection” by weight factors occurs (indicated by a dashed line). For the EGP, in contrast, a thermodynamically large number of times $N$ of selections come into play. As a result, connections to states $|u_{k,s}⟩$ with $s>0$ are efficiently damped, and the EGP exhibits the $U(1)$ geometric properties of $|u_{k,0}⟩$ alone (up to corrections that vanish in the limit $N\to \infty$).

Figure 2

Scaling of finite-size corrections $|{\phi }_E-{\phi }_Z|$ in the finite-temperature Rice-Mele model, for different temperatures $T=0.1$, 1, 10, and 100, with $t_1-t_2=-1/4$ and $\mathrm{\Delta }=\sqrt{3}/4$ in Eq. (33). Solid lines are fitting curves of approximately $N^{-2}$, which provide a good description of data points in the limit of large system sizes $N$. Inset: Difference $[1/(2\pi )]\mathrm{\Delta }{\phi }_E$ accumulated per closed cycle in $(t_1-t_2,\mathrm{\Delta })$ space encircling the purity-gap-closing point $t_2=t_1$, $\mathrm{\Delta }=0$, as a function of inverse temperature $\beta$. The same plot is found for $N=8$, 16, 32, showing that the quantization of $[1/(2\pi )]\mathrm{\Delta }{\phi }_E$ is independent of system size. As discussed in the main text, a topological transition occurs at infinite temperature ($\beta =0$), where the purity gap globally closes, highlighting the key role of the latter.

Figure 3

Scaling of finite-size corrections $|{\phi }_E-{\phi }_Z|$ in the nonequilibrium analog of the Rice-Mele model introduced in Ref. [17], with parameters $\epsilon =-\sqrt{3}/4$ and $\lambda =-1/4$ in Eq. (36). As in the thermal case (Fig. 2), the expected scaling behavior approximately $N^{-2}$ is verified. Inset (a): Difference $[1/(2\pi )]\mathrm{\Delta }{\phi }_E$ accumulated per full cycle in $(\epsilon ,\lambda )$ parameter space, as a function of a parameter $\delta$ controlling whether or not the purity-gap-closing point at $\epsilon =\lambda =0$ is encircled [as illustrated in inset (b), with the gapless point shown in red]. As in the thermal case, quantization is observed irrespective of system size $N$, and a topological phase transition occurs at $\delta =1$, where the purity-gap-closing point becomes encircled.

Figure 4

(a) Schematic setup for the interferometric measurement of the EGP. Photons are injected into one input, and the intensity difference between the two outputs is detected. In the lower arm of the interferometer, each photon interacts with the 1D chain of fermions corresponding to the probed system. (b) Internal fermionic level scheme. On each site $i$, the photonic mode $\widehat{c}$ couples ground and excited levels (with annihilation operators ${\widehat{a}}_i$ and ${\widehat{b}}_i$, respectively) in an off-resonant way (large detuning $\mathrm{\Delta }$).

Figure 5

Dynamical measurement of the EGP ${\phi }_E$ as a function of the parameter $\varphi (t)/(2\pi )=t/T_{\varphi }$, completing a closed cycle in parameter space over the time period $T_{\varphi }$. The system examined here is the Rice-Mele model in thermal equilibrium discussed in Sec. 4. Inset: Total EGP difference $[1/(2\pi )]\mathrm{\Delta }{\phi }_E$ accumulated over one cycle, as a function of $T_{\varphi }$.