Chern number and Berry curvature for Gaussian mixed states of fermions

Transitions between different phases of quantum matter are characterized either by a spontaneous breaking of symmetries or by changes of the topology of the many-body ground state [1, 2, 3]. Topologically different phases can be distinguished by invariants, which identify global properties of the system. The existence of these integer quantum numbers is also the origin of the robustness of characteristic features such as edge states and currents or quantized bulk transport [4, 5, 6, 7, 8]. Typical topological invariants are based on geometric phases such as the Berry or Zak phase [1] which characterize the parallel transport of the many-body ground-state upon cyclic changes of system parameters. They are thus applicable only to pure states. Several attempts have been made over the last years to extend the concept of topological invariants to mixed states of non-interacting fermions [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] with the aim of classifying finite-temperature or non-equilibrium steady states. Although some aspects of topology in open quantum systems can be captured by non-Hermitian Hamiltonians [21, 22, 23], a proper classification must account for both dissipation and fluctuations and requires the discussion of density matrices.

A possible generalization of Berry’s phase to density matrices has been given by Uhlmann [24]. Based on Uhlmanns construction a mixed-state topological invariant for one-dimensional (1D) systems was defined in [12] as the winding number of the Uhlmann phase upon cyclic parameter changes. For 1D lattice systems in a gapped ground state, the Uhlmann phase is identical to the Zak phase. In this limit its winding can thus be expressed as an integral of a Berry curvature over a 2D torus (lattice momentum and time) which is the well-known first Chern number. For mixed states the existence of a proper Berry connection is in general however not guaranteed. As shown in [16] the approach based on the Uhlmann phase fails when applied to two dimensions [13, 14]. The windings of the Uhlmann phases in $x$ and $y$ direction, $\int\mathrm{d}k_{x}\partial_{k_{x}}\phi^{U}_{y}(k_{x})$ and $\int\mathrm{d}k_{y}\partial_{k_{y}}\phi^{U}_{x}(k_{y})$, are for some parameters not the same, demonstrating that no proper Berry connection exists.

Recently, we have shown that a generalization of Resta’s many-body polarization [25] to mixed states, termed ensemble geometric phase (EGP) [17, 18], is an alternative way to define a topological invariant for Gaussian states of fermions in 1D. In the thermodynamic limit the EGP approaches the Zak phase of the lowest band of a single-particle Bloch Hamiltonian, termed fictitious Hamiltonian, which is defined by single-particle correlations and thus contains all properties of the Gaussian mixed state. The winding number of the EGP is a topological invariant characterizing this fictitious Hamiltonian. The EGP can be detected and a non-trivial topology has direct physical consequences such as quantized particle transport in an auxiliary system weakly coupled to the fermion chain [26, 27]. It can also be extended to 1D models with interactions including systems with fractional topological charges [28].

As is the case for the Uhlmann phase, the EGP depends on the choice of the momentum direction, when considering two-dimensional lattice models. However, as will be shown here, and in contrast to the Uhlmann case, the EGP in any spatial direction can be expressed as a closed-loop integral over the corresponding component of a single Berry connection. The latter describes the ground-state wave function of the fictitious Bloch Hamiltonian. Thus based on the EGP one can define a unique Chern number for mixed states in 2D. To illustrate this we discuss the asymmetric Qi-Wu-Zhang model, which is a two-dimensional, topologically non-trivial lattice model with two bands. Due to the asymmetry of the band structure the winding of the Uhlmann phase is different in $x$ and $y$ direction in a certain range of temperatures [16]. The EGP winding, on the other hand, is always the same in all directions.

To set the stage we start by shortly summarizing the topological classification of fermions in terms of the Berry curvature of Bloch states. To this end we consider insulating many-body states of non-interacting fermions on a two-dimensional lattice described by a number-conserving Hamiltonian. We set the lattice constant $a=1$, consider a total number of $N^{2}$ unit cells with periodic boundary conditions in $x$ and $y$ directions and use $\hbar=1$. The operators $\hat{c}_{\bm{j},\lambda},\hat{c}_{\bm{j},\lambda}^{\dagger}$ describe the annihilation and creation of a fermion in the $\bm{j}$th unit cell. $\bm{j}=(j_{x},j_{y})$ denotes the $x$ and $y$ coordinates of the unit cell, respectively, and the index $\lambda\in\{1,\dots,p\}$ labels a possible internal degree of freedom within a unit cell. Assuming translational invariance for simplicity, the Hamiltonian can be written in second quantization as

Here $\bm{k}=(k_{x},k_{y})$ is the lattice momentum and ${\sf h}$ is the $p\times p$ single-particle Hamiltonian matrix in momentum space. We assume that ${\sf h}$ has multiple bands, separated by finite gaps, and we consider an insulator, i.e assume that the chemical potential $\mu$ lies within a band gap.

If ${\sf h}$ breaks time-reversal symmetry, the topological properties of a gapped many-body state can be characterized by the Berry curvature ${\cal F}^{(n)}(\bm{k})$ of all occupied Bloch bands $n$

Here $|u_{n}(\bm{k})\rangle$ is the Bloch function of the $n$th band and $\bm{A}^{(n)}(\bm{k})$ is the Berry connection $A^{(n)}_{j}(\bm{k})=i\bigl{\langle}u_{n}(\bm{k})\bigr{|}\partial_{k_{j}}u_{n}(\bm{k})\bigr{\rangle}$. While the Berry curvature is not gauge invariant its integral over the two-dimensional torus of the Brillouin zone is. It furthermore defines an integer-valued topological invariant of the band, the first Chern number

The Chern number can also be related to the geometric phase picked up by a Block state $|u_{n}(\bm{k})\rangle$ upon parallel transport in momentum space through the Brillouin zone in either $k_{x}$ or $k_{y}$ direction. It can be written in terms of the winding of the Zak phases, defined as

or $\phi^{\textrm{Zak}}_{y}(k_{x})$ respectively. Note that we have dropped the band index for simplicity. Most importantly the two Zak phases in $x$ and $y$ direction can be expressed as integrals of the components of a single vector, the Berry connection $\bm{A}=i\langle u(\bm{k})|\bm{\nabla}_{\bm{k}}u(\bm{k})\rangle$. As a consequence the windings of the two phases

must be identical and equal to the Chern number

Topological invariants for pure states, such as the Chern number, characterize how a gapped many-body state $|\psi\rangle$ changes upon a parallel transport along a closed loop in parameter space. The parallel-transport requirement leads directly to the definition of the Berry phase, or for lattice systems to the Zak phase, eq.(4). The corresponding Berry connection transforms as a $U(1)$ gauge field according to $|u(\bm{k})\rangle\to e^{i\chi(\bm{k})}|u(\bm{k})\rangle$: $\bm{A}\to\bm{A}-\bm{\nabla}_{\bm{k}}\chi(\bm{k})$.

A generalization of geometric phases to density matrices $\rho$ has been introduced by Uhlmann [24], who pointed out that the decomposition $\rho=w\cdot w^{\dagger}$ of an $n\times n$ density matrix into matrices $w$ contains a gauge freedom $w\to w\,{\sf U}$, where ${\sf U}$ is a $U(n)$ unitary matrix. Since $\rho$ is positive semi-definite, $w$ can always be represented as

Let $\rho=\rho(\lambda)$ be a uniquely defined mixed state and let $\lambda\in\{0,\Lambda\}$ parametrize a closed loop in parameter space such that $\rho(0)=\rho(\Lambda)$. Requiring parallel transport of the density matrix in generalization of Berry’s construction, one can then define a $U(n)$ Uhlmann holonomy

Here ${\cal P}$ denotes path ordering and $\bm{A}_{\mathrm{U}}=i\partial_{\lambda}{\sf U}(\lambda){\sf U}^{\dagger}(\lambda)$ is the $U(n)$ Uhlmann connection. The $U(n)$ gauge freedom can be reduced to $U(1)$ by performing a trace which leads to the Uhlmann phase

For (pure) ground states of gapped fermionic models eq.(8) reduces to the well-known Zak phase, if $\lambda$ is identified with the lattice momentum $k$.

In [12], the winding of the Uhlmann phase (8) upon a cyclic change of an external parameter $t$ has been proposed as a topological invariant to classify one-dimensional lattice models in a mixed state $\rho$

Here the variable $\lambda$ entering the definition of the Uhlmann phase is the quasi momentum $k$ along the chain and the loop integral extends over the first Brillouin zone $(\lambda\to k\in[-\pi,\pi))$. While this defines a consistent invariant in 1D, its extension to two spatial dimensions as proposed in [13, 14] is problematic. As shown by Budich and Diehl in [16], the windings of the Uhlmann phase in $x$ or $y$ direction are in general not the same, i.e.

As an example they considered a finite-temperature state of a simple topological two-band model with asymmetric band structure

which is a modification of the Qi-Wu-Zhang model [29]. Here $\bm{\sigma}$ is the vector of Pauli matrices. Its band-spectrum is shown in Fig.3. While at temperatures $T=0$ or $T=\infty$ the windings of the Uhlmann phase in both directions are the same, i.e. $C_{y}^{\mathrm{U}}(T=0)=C_{x}^{\mathrm{U}}(T=0)=1$ and $C_{y}^{\mathrm{U}}(T=\infty)=C_{x}^{\mathrm{U}}(T=\infty)=0$, there is a range of temperatures where $C_{y}^{\mathrm{U}}(T)\neq C_{x}^{\mathrm{U}}(T)$. This shows that in contrast to the full $U(n)$ Uhlmann holonomy, there is no proper $U(1)$ gauge structure underlying the Uhlmann phase.

In the following we want to show that in contrast to the Uhlmann phase, the ensemble geometric phase, introduced for one-dimensional lattice models in [17, 18], can be used to define a proper Berry curvature and Chern number for mixed states of 2D bandstructures.

We have shown that the ensemble geometric phase which has been used to define topological winding numbers for mixed states of one-dimensional, Gaussian fermion systems can straightforwardly be extended to two spatial dimensions and defines a Chern number if there is translational invariance. Different from approaches based on other geometric phases for mixed states, such as the Uhlmann phase, this number is a true topological invariant as it is a two-dimensional integral over a proper Berry curvature. The latter is defined by the Bloch states of the fictitious Hamiltonian formed by the matrix of single-particle correlations in the Gaussian mixed state. Finite-temperature states of non-interacting fermion models, which are fully characterized by single-particle correlations are Gaussian, but Gaussian states can also emerge as non-equilibrium steady states of systems coupled to specific Markovian reservoirs. In the first case the mixed-state Berry curvature is identical to that of the original Bloch Hamiltonian counting all bands below the chemical potential as long as $T<\infty$. For genuine non-equilibrium states no such correspondence exists and the fictitious Hamiltonian has a meaning of its own. While the discussion in the present paper relies on the assumption of translational invariance, allowing a mapping to decoupled one-dimensional chains (see Fig.2), we anticipate the results to hold also in the presence of disorder and with interactions. This will be subject of further studies.