Phase space formulation of the Abelian and non-Abelian quantum geometric tensor

Consider a quantum system defined by a Hamiltonian $\hat{\bf{H}}(\hat{\bf{q}},\hat{\bf{p}};x)$, where $\hat{\bf{q}}=\{\hat{\bf{q}}_{a}\}$ and $\hat{\bf{p}}=\{\hat{\bf{p}}_{a}\}$ ($a,b,\dots\!=1,\dots,N$) are respectively the position and momentum operators, and $x(\tau)=\{x^{i}\}$ ($i,j,\dots\!=1,\dots,m$) is a set of $m$ real adiabatic parameters, i.e., slowly varying functions of time $\tau$. To simplify notation, $x(\tau)$ will be denoted as $x$. If $\hat{\bf{H}}$ has at least one orthonormal eigenvector $\lvert n(x)\rangle$ with nondegenerate eigenvalue $E_{n}(x)$, then the Abelian quantum geometric tensor defined in the parameter space of the system is given by

where $\partial_{i}=\partial/\partial x^{i}$. By construction, Eq (1) is invariant under the phase transformation [3]

where $\alpha_{n}$ is an arbitrary real function of the parameters, meaning that we can choose the basis $\lvert n\rangle$ up to a $U(1)$ gauge transformation (see also Ref. [29]). The relevance of this tensor lies in the fact that it provides the fundamental structures underlying the parameter space.

The (symmetric) real part of Eq. (1) gives the quantum metric tensor [3]

which defines the line element as $dl^{2}=g^{(n)}_{ij}\delta x^{i}\delta x^{j}$ and provides a distance between the two neighbor states $\lvert n(x)\rangle$ and $\lvert n(x+\delta x)\rangle$ over the parameter space. Actually, the metric (3) stems from the leading term of the fidelity $f=|\langle n(x)|n(x+\delta x)\rangle|$, a measurement of the “similarity” between the states $\lvert n(x)\rangle$ and $\lvert n(x+\delta x)\rangle$.

The (antisymmetric) imaginary part of Eq. (1) yields the Berry curvature [30]

which defines the 2-form $F^{(n)}=\frac{1}{2}F^{(n)}_{ij}\delta x^{i}\wedge\delta x^{j}$ in the parameter space and is associated with the (Abelian) Berry connection

It is worthy of noticing that $A^{(n)}_{i}(x)$ is real since $\langle n|\partial_{i}n\rangle$ is purely imaginary and that, under the gauge transformation (2), it changes as

which is the transformation law for a genuine $U(1)$ gauge connection. The integral of the 1-form $A^{(n)}=A^{(n)}_{i}\delta x^{i}$ over a closed path $C$ in the parameter space or the flux of the 2-form $F^{(n)}$ through a surface with boundary $C$ yields the gauge invariant Berry phase [30].

On the other hand, if the Hamiltonian $\hat{\bf{H}}$ has a set of $g_{n}$ orthonormal eigenvectors $\lvert n_{I}(x)\rangle$ ($I,J,\dots\!=1,2,\dots,g_{n}$) associated with the eigenvalue $E_{n}(x)$, i.e., $\hat{\bf{H}}(x)\lvert n_{I}(x)\rangle=E_{n}(x)\lvert n_{I}(x)\rangle$, then the non-Abelian quantum geometric tensor defined in the parameter space of the system is given by [24]

It is easy to see that in the nondegenerate case $g_{n}=1$, the non-Abelian tensor (7) reduces to the Abelian one (1). Furthermore, under the unitary transformation

where $U_{IJ}(x)$ are the entries of a parameter-dependent unitary $g_{n}\times g_{n}$ matrix $U(x)$, the tensor (7) transforms covariantly

where ‘^∗’ stands for complex conjugation.

By analogy with Eqs. (3) and (4), the corresponding (symmetric) non-Abelian quantum metric tensor and the (antisymmetric) non-Abelian Berry curvature or Wilczek-Zee curvature are given by

and

respectively. Moreover, the associated non-Abelian Berry connection or Wilczek-Zee connection is defined by [25, 31]

This connection provides the entries of a $g_{n}\times g_{n}$ Hermitian matrix for each $i$ and, under the transformation (8), transforms as a proper gauge potential

Given an operator $\hat{\bf{Q}}(\hat{\bf{q}},\hat{\bf{p}};x)$, the Weyl correspondence associates a quantum phase space function $\mathcal{Q}(q,p;x)$ defined as [23]

which is known as the Weyl transform of $\hat{\bf{Q}}$. Here, the variables $q=\{q_{a}\}$ and $p=\{p_{a}\}$ are respectively the eigenvalues of the operators $\hat{\bf{q}}$ and $\hat{\bf{p}}$, and we have written ${\rm d}^{N}\!y\equiv{\rm d}y_{1}\dotsi{\rm d}y_{N}$, and $p\cdot y\equiv\sum_{a=1}^{N}p_{a}y_{a}$. An important property of the Weyl correspondence is that it allows to write the trace of the product of two operators, $\hat{\bf{Q}}$ and $\hat{\bf{O}}$, as

where $\mathcal{Q}$ and $\mathcal{O}$ are the Weyl transform of these operators.

The Wigner function $W_{n}(q,p;x)$, which is the main tool of this formalism, is defined as the function corresponding to the density operator $\hat{\rho}(x)$. More precisely, for a pure state, $\hat{\rho}_{n}(x)=\lvert n(x)\rangle\langle n(x)\rvert$, it is given by [19]

where $\psi_{n}(q+\tfrac{y}{2};x)=\langle q+\tfrac{y}{2}|n(x)\rangle$ and $\psi^{*}_{n}(q-\tfrac{y}{2};x)=\langle n(x)|q-\tfrac{y}{2}\rangle$. Thus, the Wigner function provides a phase space representation of the quantum state $\lvert n(x)\rangle$.

From Eq. (15) with $\hat{\bf{Q}}\rightarrow\hat{\rho}_{n}$, it is straightforward to see that the expectation value of an operator $\hat{\bf{O}}$ can be written as

This expression is reminiscent of the average of the function $\mathcal{O}$ (to which $\hat{\bf{O}}$ corresponds) with respect to the phase space “quasiprobability” distribution $W_{n}$, which is real but not always positive. Notice that setting $\hat{\bf{O}}$ equal to the identity operator $\hat{\mathds{1}}$ in Eq. (17) and using the fact that the Weyl transform of $\hat{\mathds{1}}$ is $1$, we have

which shows that $W_{n}$ is normalized in phase space. For later purposes, we also write down the following expression derived by taking $\hat{\bf{O}}\rightarrow\hat{\rho}_{n}$ in Eq. (17):

For more details on the Wigner function, we refer the reader to Ref. [32].

Let us first consider the Abelian Berry connection. We start by writing Eq. (5) as

where $\hat{\bf{A}}^{(n)}_{i}$ is a quantum operator defined by

Note that $\hat{\bf{A}}^{(n)}_{i}$ is non-Hermitian, and then its Weyl transform

is a complex function in phase space variables. Actually, by using Eqs. (16) and (22), it is not hard to show that

Having Eq. (20), we can now apply Eq. (15) to relate the phase space function $\mathcal{A}_{i}^{(n)}$ with the Berry connection. Indeed, taking $\hat{\bf{Q}}\rightarrow\hat{\mathds{1}}$ and $\hat{\bf{O}}\rightarrow\hat{\bf{A}}^{(n)}_{i}$ in Eq. (15) and using Eq. (20), we have

This equation provides an expression of the Berry connection in the Wigner-function formalism. It can be checked that, under the gauge transformation (2), the function $\mathcal{A}_{i}^{(n)}$ transforms according to

from which, together with Eq (18), it follows that the connection (24) satisfies the transformation law (6), as expected. Notice that the transformation law for $\mathcal{A}_{i}^{(n)}$ is similar to the one of a $U(1)$ gauge connection, but not identical because of the presence of $W_{n}$ in Eq. (25). This is a consequence of the fact that, while the gauge transformation (2) is restricted to the parameter space, $\mathcal{A}_{i}^{(n)}(q,p;x)$ is a mixed structure of the parameter space and the physical phase space. Of course, it is possible to consider a less restrictive and more general gauge transformation (see Ref. [29], for instance). It is also worth noting that only the real part of the phase space function $\mathcal{A}_{i}^{(n)}$ contributes to the Berry connection (24), which is a consequence of Eq. (23) and the fact that

To recast the Abelian quantum metric tensor into the Wigner-function formalism, here we follow the same procedure as in the case of the Berry connection. We observe that the quantum geometric tensor (1) can be written as

where $\hat{\bf{Q}}^{(n)}_{ij}$ is an operator defined by

with $\hat{\bf{A}}^{(n)}_{i}$ given by Eq.  (21). Therefore, noting that $\hat{\bf{A}}^{(n)\dagger}_{i}-\hat{\bf{A}}^{(n)}_{i}=-\rmi\partial_{i}\hat{\rho}_{n}$, from Eq. (15) with $\hat{\bf{Q}}\rightarrow-\rmi\partial_{i}\hat{\rho}_{n}$ and $\hat{\bf{O}}\rightarrow\hat{\bf{A}}^{(n)}_{j}$ it follows that

which, after using Eq. (23), becomes

This is a formulation of the quantum metric tensor within the phase space formalism. Notice that we only need the knowledge of $\mathcal{A}_{i}^{(n)}$ in order to compute the quantum geometric tensor (30). This shows that the new phase space functions $\mathcal{A}_{i}^{(n)}$, introduced here for calculating the Berry connection, actually encode all the relevant information embodied in the parameter space. On the other hand, it can be checked, by using Eq. (25) together with

which follows from Eq. (19), that Eq. (30) is gauge invariant under the transformation (2).

Separating Eq. (30) into its real and imaginary parts, we obtain the expression for Abelian quantum metric tensor

which, after using Eq. (23), can be written as

and the expression for the Abelian Berry curvature tensor

Some remarks are in order. First, note that in contrast to the standard approach to determine the Abelian quantum metric tensor (namely, Eq. (3)), the phase space approach based on Eq. (33) does not require the knowledge of wave functions. This is because the Wigner functions can also be obtained as the solutions of suitable functional equations in phase space, bypassing the use of wave functions [16]. Second, note that the expression (34) for the Abelian Berry curvature is different from the one conjectured in Ref. [13], although both expressions must yield identical results. In fact, while Eq. (34) only involves the phase space function $\mathcal{A}_{i}^{(n)}$, the expression proposed in [13], which uses Eq. (17) and takes elements of the curvature associated with the classical Hannay angle [33], requires the knowledge of variables $q=\{q_{a}\}$ and $p=\{p_{a}\}$ in terms of the angle-action variables. In this regard, it is worth recalling that the angle-action variables are restricted to classical integrable systems, and hence the expression of Ref. [13] can be applied only to those quantum systems whose classical counterpart is integrable, in contrast to Eq (34) which does not have that limitation. Furthermore, we would also like to point out that Eqs. (24) and (32)–(34) for the Abelian geometrical structures have not, to our knowledge, been reported in the literature. They are new and can be useful not only to gain more insight in the nature of the parameter space of quantum systems, but also for practical applications in a variety of quantum models. Third, notice that in Eq. (20) we could also consider the operator $\hat{\bf{B}}^{(n)}_{i}:=(\beta_{1}\hat{\bf{A}}^{(n)}_{i}+\beta_{2}\hat{\bf{A}}^{(n)\dagger}_{i})/(\beta_{1}+\beta_{2})$ with $\beta_{1}$ and $\beta_{2}$ being arbitrary complex numbers, because of $A_{i}^{(n)}={\rm Tr}\,\hat{\bf{B}}^{(n)}_{i}$. Nevertheless, a more complicated choice could be inconvenient for practical applications and does not give any further insight into the nature of the parameter space.

Here, we extend the phase space description of the parameter space to include the non-Abelian geometrical structures. Following the procedure considered for the Abelian case, we begin by expressing the Wilczek-Zee connection as

where now the associated quantum operator takes the form

The Weyl transform for this operator is given by

and, as in the Abelian case, it gives rise to a complex function in phase space variables, since $\mathcal{A}^{(n)}_{iIJ}$ is a non-Hermitian operator.

By combining Eq. (35) and Eq. (15) with $\hat{\bf{Q}}\rightarrow\hat{\mathds{1}}$ and $\hat{\bf{O}}\rightarrow\hat{\bf{A}}^{(n)}_{iIJ}$, it is direct to see that the expression for the Wilczek-Zee connection in phase space formalism is

which is the natural generalization of the (Abelian) Berry connection (24). It is worth noting that Eq. (38) gives the entries of a $g_{n}\times g_{n}$ Hermitian matrix for each $i$, as it should be. This can be proved as follows. From Eq. (37) we have

where $W_{nIJ}$ are the non-diagonal Wigner functions

Then, plugging Eq. (39) into Eq. (38) and using the fact that

it is straightforward to see that $A^{(n)}_{iIJ}=A^{(n)*}_{iJI}$.

Let us now prove that the connection (38) obeys the gauge transformation law (13) for the Wilczek-Zee connection. Under the unitary transformation (8), the phase space function $\mathcal{A}_{iIJ}^{(n)}$ changes as

which generalizes Eq. (25). Note that while the second term on the right side of Eq. (25) is purely imaginary, the corresponding term in Eq. (42) is complex in general. Now, taking into account Eq (42), Eq (38) immediately implies

which, after using Eq. (41), becomes the transformation law (13).

We now turn to the phase space formulation of the non-Abelian quantum geometric tensor. We find that Eq. (7) can be written as

where the operator $\hat{\bf{Q}}^{(n)}_{ijIJ}$ is defined as

with $\hat{\mathfrak{P}}_{n}:=\sum_{I=1}^{g_{n}}\lvert n_{I}\rangle\langle n_{I}\rvert$ the projection operator. Then, taking $\hat{\bf{Q}}\rightarrow-\rmi\,\partial_{i}\hat{\mathfrak{P}}_{n}$ and $\hat{\bf{O}}\rightarrow\hat{\bf{A}}^{(n)}_{jIJ}$ in Eq. (15), it is clear that Eq (44) is equivalent to

Let us write this equation directly in terms of the phase space function $\mathcal{A}_{iIJ}^{(n)}$. Using Eqs. (37) and (40), it can be demonstrated that

which is the analog of Eq. (23). Thus, substituting Eq. (47) into Eq. (46), we get

which is an expression for the non-Abelian quantum geometric tensor in the phase space formalism. This equation tells us that, as in the Abelian case, all we need to know in order to compute the non-Abelian quantum geometric tensor is the phase space function $\mathcal{A}_{iIJ}^{(n)}$.

Now we are in a position to find the expressions for the non-Abelian quantum metric tensor and Wilczek-Zee curvature. From Eqs. (10) and (48), the non-Abelian quantum metric tensor turns out to be

which, with the help of Eqs. (39) and (47), takes the form

whereas from Eqs. (11) and (48), the Wilczek-Zee curvature is recasted as

We remark here that, as in the Abelian case, the expression (51) for the Wilczek-Zee curvature is different from the one proposed in Ref. [13]. In fact, whereas Eq. (51) only depends on the phase space function $\mathcal{A}_{iIJ}^{(n)}$, the expression of Ref. [13] requires the non-diagonal Wigner functions and the variables $q=\{q_{a}\}$ and $p=\{p_{a}\}$ in terms of the angle-action variables, meaning that it is restricted to quantum systems that have a classical integrable counterpar. It is also worth mentioning that the phase space formulation obtained in this subsection for the non-Abelian geometrical structures is new and can be used to deal with a wide variety of degenerate quantum systems.

The following sections are devoted to exemplify and apply this phase space approach to the parameter space.