Geometric phases of Topological Systems under Quench Process

In recent years, with the development of quantum technology, the dynamics of the quench process of closed quantum systems has attracted a lot of attentions of both theoretical and experimental studies Bloch et al. (2008); Polkovnikov et al. (2011); Eisert et al. (2015). The quench processes of many quantum systems are frequently associated with the so-called dynamical quantum phase transition Heyl et al. (2013); Heyl (2018). These transitions can be detected by the zeros of the Loschmidt amplitudes or the divergences of the Loschmidt rate functions, which is in analogy to the non-analytical behaviors of thermodynamic phase transitions Heyl (2014, 2015); Hou et al. (2020). For quantum systems with nontrivial topology, other than the rate functions, one may also wonder about how the topological indices evolve under the quench dynamics. This is exactly the question we will try to address in this paper.

In one-dimensional systems, the geometric phases Chruscinski and Jamiolkowski (2004) provide the widely used indices to understand the topological properties of topological insulators and superconductors Qi and Zhang (2011); Hasan and Kane (2010). At zero temperature, the central notion is the Berry phase which is the accumulation of Berry connection along a closed loop in the parameter space Berry (1984); Zak (1989). In some sense, the Berry phase reflects the twist of the phase factor of the eigenstates as one travels around this loop Nakahara (2003). For one-dimensional systems, the Berry phase is usually quantized to the values of $0$ and $\pi$, which corresponds to the even and odd winding number of the mapping defined by the Hamiltonian.

At finite temperatures, the physics of mixed states is completely described by their density matrices. There are many attempts to generalize the Berry connection to the case of mixed states Sjoqvist et al. (2000); Budich and Diehl (2015); Bardyn et al. (2018). In this paper, we will follow the method proposed by Uhlmann Uhlmann (1986, 1991, 1993) long time ago, which is closely parallel to the notion of Berry connection in the geometric sense. The key step in defining Uhlmann connection is to decompose the density matrices into amplitudes and then set up a parallel condition that is strong enough to uniquely determine the arbitrary phase factors of the amplitudes. Similar to the Berry phase, the Uhlmann phase is also the accumulation of Uhlmann connection along a closed loop in the parameter space. The Uhlmann phase has been successfully used as a topological index to understand many topological systems Viyuela et al. (2014a, b); Huang and Arovas (2014); Mera et al. (2017); He et al. (2018); Zhang et al. (2021) and spin systems Morachis Galindo et al. (2021); Hou et al. (2021) at finite temperature. In terms of the Uhlamnn phase, it is usually found that there is a critical temperature $T_{c}$ which separates the topologically non-trivial phase from the trivial one.

In this paper, we will take the Creutz ladder Creutz (1999) as our example to investigate the time evolution of the Berry and Uhlmann phases under the quench process. For two-band models, the time evolution can be visualized as a closed loop on a unit sphere whose shape is deformed as time flows. From this point of view, we will show that the Berry phase will keep constant values after the quench, given the condition that the symmetry of the model is preserved. On the other hand, the Uhlmann phase will predict topological transitions at certain points of time after the quench. We will also try to compare the behavior of Uhlmann phase with that of Loschmidt rate functions.

The rest of this paper is organized as follows. In section II, we provide a geometric description of the quench process for generic two-band models. Then in section III, we briefly review the concepts of Berry and Uhlmann phases which is used as topological indices. The detailed calculations and numerical results of these two types of geometric phases under the quench process will be presented in section IV. At last, a simple summary will be given in section V.

In this paper, we will be mainly concerned with the quench process of two-band models. Since the topological properties are only determined by the eigenstates, we can always shift the average energy of the two bands to be zero, which makes the Hamiltonian traceless. Then the most generic two-band traceless Hamiltonian can be expanded by Pauli matrices $\sigma_{i}$ as

Clearly, the above Hamiltonian can be represented by this 3D vector $R_{i}$ whose components are functions of lattice momentum $\mathbf{k}$.

Now the quench process can be described as follows. First, suppose the initial Hamiltonian is given by $H_{0}=R^{0}_{i}(\mathbf{k})\sigma_{i}$. Then we make a sudden change of the parameters, thus the system will evolve under another Hamiltonian $H_{1}=R^{1}_{i}(\mathbf{k})\sigma_{i}$ where $R^{1}_{i}\neq R^{0}_{i}$. Here we assume both $H_{0}$ and $H_{1}$ are time independent, thus the time evolution can be computed analytically.

At zero temperature, we assume that the initial state $|\psi_{0}\rangle$ is an eigenstate of $H_{0}$. After the quench process, it evolves into

Since $H_{1}$ does not commute with $H_{0}$, the resulting $|\psi(t)\rangle$ is usually a superposition of the two eigenstates of $H_{0}$. In the next section, we will consider the time evolving of its Berry phase under the quench process.

At a finite temperature $T$, we assume that the initial state is a thermal equilibrium state with the following density matrix

for a fixed momentum $\mathbf{k}$.

Then the system is evolved under $H_{1}=\sum_{i}R^{1}_{i}\sigma_{i}$ where $R^{1}_{i}\neq R^{0}_{i}$. In the quench process, at some later time $t$, the density matrix can be computed as follows

Therefore, the time evolution is equivalent to making a unitary transformation of the initial Hamiltonian $H_{0}$. According to the well-known relation between the spinor and vector representation of $SU(2)$ group, we have the following identity

Here $r_{a}$ and $n_{a}$ are 3D vectors and $n_{a}$ has unit length. The angle $\theta$ is a real number. The matrix $R_{ab}$ rotates the vector $r_{a}$ around the axes along $n_{a}$ with angle $\theta$. To ease the notation, we assume that the repeated indices imply summations.

Making use of the above identity, the unitary transformation by $e^{-iH_{1}t}$ is equivalent to a rotation of the vector $R^{0}_{i}$

with $R^{1}=\sqrt{R^{1}_{a}R^{1}_{a}}$ and $\hat{R}^{1}_{a}=R^{1}_{a}/R^{1}$. Thus, the effect of the quench process is to rotate the initial vector $R^{0}_{a}$ around the axes along $\hat{R}^{1}_{a}$ with the angle $2R^{1}t$ to obtain a new vector $R_{a}(t)$. With this new vector $R_{a}(t)$, $\rho(t)$ can be expressed as

Here the partition function $Z$ is the same as the initial state. Based on these density matrices, Uhlmann connections can be computed for any later time $t$.

We want to consider the behavior of geometric phases under quench processes. At zero temperature, the most widely used geometric phases are the Berry phases. Let us consider a given energy band of a lattice system. Then the eigenstate of this band $|u_{n}(\mathbf{k})\rangle$ depends on the parameter of lattice momentum $\mathbf{k}$. For this band, the Berry connection is defined as

Here the Berry connection measures the phase difference of the neighboring wave functions in the $\mathbf{k}$ space. The Berry phase is defined to be the phase accumulated as one travels around a closed loop in the $\mathbf{k}$ space, which can be written as

Here $C$ denotes a closed loop in $\mathbf{k}$ space. Although $A_{\mu}$ is not gauge invariant, it is easy to see that the above defined $\Phi$ is gauge invariant and thus is also an observable.

At finite temperatures, the above notion of Berry connection can be generalized to the so-called Uhlmann connection Uhlmann (1986, 1991, 1993). For a mixed state, the physical properties are determined by the density matrix $\rho$ instead of wave functions. It is convenient to make a spectral decomposition of density matrices as $\rho=\sum_{i}p_{i}|u_{i}\rangle\langle u_{i}|$. Here $|u_{i}\rangle$ for $i=1,\cdots,n$ are eigenstates and $p_{i}$ is the Boltzmann weight for systems at thermal equilibrium. The concept of Uhlmann connection is based on the amplitude decomposition of the density matrix as

Here $w$ can be thought of as the counterpart of the wave function for the mixed states. Note that for a fixed $\rho$, $w$ can be multiplied by an arbitrary unitary matrix $U$ but still reproduce the same $\rho$. This matrix $U$ represents a $U(n)$ phase factor, which is very similar to the $U(1)$ phase factor of a wave function.

In order to determine a preferred phase factor $U$, Uhlmann Uhlmann (1986) proposed the following parallel condition:

Where $C>0$ means that $C$ is a Hermitian and positive definite matrix. With this condition, the relative phase factor between $w_{1}$ and $w_{2}$ is uniquely determined. Let us see how to derive this as follows.

Consider two different density matrices and their amplitudes $w_{1}=\sqrt{\rho_{1}}U_{1}$ and $w_{2}=\sqrt{\rho_{2}}U_{2}$. The self multiplication of the parallel condition gives

After taking the square root, one finds that

Substitute $C$ back to the parallel condition, we find a relative phase factor as

This result can be considered as a finite version of the Uhlmann connection. The above formula requires the density matrix $\rho$ to be a full-rank matrix, since the inverse of $\rho$ must be inserted.

To simplify the above result, one can also work with an infinitesimal Uhlmann connection. To this end, we can set $\rho_{1}=\rho(\mathbf{k})$ and $\rho_{2}=\rho(\mathbf{k}+\Delta\mathbf{k})$ in Eq.(15), then the infinitesimal Uhlmann connection can be defined as

where $\partial_{\mu}=\frac{\partial}{\partial k_{\mu}}$. After some straightforward calculations, the explicit result of the Uhlmann connection can be written as

We would like to mention that, if one makes another gauge choice, then $A^{U}$ will transform like an ordinary $U(n)$ non-Abelian gauge field. Similar to the Berry connection, we can construct a gauge invariant Wilson loop based on the Uhlmann connection as

Here $C$ represents a closed loop in the $\mathbf{k}$ space and $\mathcal{P}$ represents the path ordering. Now suppose we pick a starting point at the loop $C$. The amplitude at this point is given by $w_{1}=\sqrt{\rho_{0}}$. If we parallel transport $w_{1}$ along the loop $C$ and back to starting point, the amplitude becomes $w_{2}=\sqrt{\rho_{0}}V$ which is usually different from $w_{1}$. Now we can define the Uhlmann phase as the phase angle of the overlap between $w_{1}$ and $w_{2}$, which is given as follows

This index has already been used to study several 1D topological systems at finite temperatures Viyuela et al. (2014a).

As a specific example, we consider the quench process of the Creutz ladder, which is a typical one-dimensional topological model. In real space, the Hamiltonian of this model can be written as

Here $a$ and $b$ are fermion annihilation operators for A and B sublattices. $m$ is the hopping constant inside each unit cell. $w$ is the hopping constant between neighboring unit cells and we set $w=1$ as an energy unit. The phase angle $\theta$ represents the magnetic flux applied to each unit cell.

If we transfer the above Hamiltonian to momentum space and ignore the term proportional to identity matrix, we find that

In this form, it is easy to see that $\sigma_{2}H\sigma_{2}=-H$. Therefore, the Creutz ladder has chiral symmetry and belongs to the AIII symmetry class Chiu et al. (2016). As a comparison example, we also introduce a modified Creutz ladder as

with a constant $\gamma\neq 0$. The $\gamma\sigma_{2}$ term breaks the chrial symmetry of the original model.

At $T=0$, the topological properties of Creutz ladder can be easily understood by the vector $R_{i}(k)$. As $k$ increases from $0$ to $2\pi$, $R_{i}(k)$ sweeps an ellipse on the $xz$ plane. If $|m|<1$, this ellipse encloses the coordinate origin, which means the winding number is one. On the other hand, if $|m|>1$, then the coordinate origin is outside the ellipse and the winding number is zero. Therefore, $m=\pm 1$ is the boundary between the topological and trivial phases.

In this paper, we have shown that the Berry phase of 1D topological models is constant under the quench process if the symmetry of the model is preserved. The time evolution after the quench can be visualized as a closed loop on a unit sphere deforming its shape with time. It is found that the value of Berry phase can be converted to the area of the surface bounded by this closed loop. Based on this observation, we have analytically shown that the symmetry of the model dictates that the Berry phase must be a constant. At finite $T$, we found that the Uhlamnn phase is always quantized to two possible values after the quench when the symmetry of the model is intact. As time flows, the system jumps between the topological non-trivial and trivial phases according to sudden changes of the Uhlamnn phase. If a term that breaks the chiral symmetry is included, then both the Berry and Uhlmann phases deviate from the quantized values. But the topological meaning of these geometric phases is not clear in this case.