Non-Abelian Geometric Phases Carried by the Spin Fluctuation Tensor
H. M. Bharath






School of Physics, Georgia Institute of Technology




The expectation values of the first and second moments of the quantum mechanical spin operator can be used to define a spin vector and spin fluctuation tensor, respectively. The former is a vector inside the unit ball in three space, while the latter is represented by an ellipsoid in three space. They are both experimentally accessible in many physical systems. By considering transport of the spin vector along loops in the unit ball it is shown that the spin fluctuation tensor picks up geometric phase information. For the physically important case of spin one, the geometric phase is formulated in terms of an SO(3) operator. Loops defined in the unit ball fall into two classes: those which do not pass through the origin and those which pass through the origin. The former class of loops subtend a well defined solid angle at the origin while the latter do not and the corresponding geometric phase is non-Abelian. To deal with both classes, a notion of generalized solid angle is introduced, which helps to clarify the interpretation of the geometric phase information. The experimental systems that can be used to observe this geometric phase are also discussed.


Non-Abelian Geometric Phases Carried by the Spin Fluctuation Tensor
    Bharath H. M.
    December 30, 2023
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§ INTRODUCTION

Berry's geometric phase <cit.>, has attracted renewed interest in recent years due to applications in, for example, phase transitions with topological order parameters <cit.> and fault tolerant quantum computation <cit.>. Although Berry's phase was defined for adiabatic transport of a quantum system along a loop in the parameter space of the system's Hamiltonian, it was later realized that it is a kinematic property, which does not depend on the dynamics of the system <cit.>. In <cit.>, it is shown that a quantum system transported along a closed loops picks up a geometric phase irrespective of how the transport was induced. In <cit.>, geometric phase has been generalized to open paths in the space of mixed states by defining an SU(N) operator corresponding to every path in the space of N× N density matrices, with no reference to the question of how the transport along the path is induced. In other words, Berry's phase depends only on the path in the parameter space along which the quantum state is transported, and not on the dynamical equation governing the transport or the rate of transport. This insight has resulted in a kinematic formulation <cit.>, <cit.>, which we shall also adopt in this paper. The space of ground state eigenvectors of a non-degenerate Hamiltonian has a line bundle structure over its parameter space. Geometrically, Berry's phase can be viewed as the holonomy of Berry's connection form on this line bundle <cit.>. When the Hamiltonian is degenerate, Berry's phase generalizes to a non-Abelian Wilczek-Zee phase, which can also be formulated as a holonomy <cit.>. In general, geometric phase can be defined as a holonomy element of a connection form on a fiber bundle structure imposed on the space of quantum states <cit.>, <cit.>, <cit.>.

The geometric phase is essentially the geometric information stored in the overall phase of the wave-function of a quantum mechanical system. In this paper, we show that such geometric information may be extracted from second and higher order spin moments of a quantum spin system, which we formulate as a non-Abelian geometric phase. We restrict our analysis to pure quantum states, i.e., quantum states that can be represented by a vector in a finite-dimensional Hilbert space. Vectors in a finite dimensional Hilbert space may be regarded as states of a quantum spin system. Corresponding to every pure state, one can define a spin vector in real space as s⃗= (⟨ S_x⟩, ⟨ S_y⟩, ⟨ S_z⟩ )^T, where S_i are the Hilbert space spin operators and ⟨ S_i⟩ are their expectation values with respect to the given pure state.  For a spin-1/2 system, the real space spin vector has unit length and therefore lies on the unit sphere; known as the Bloch sphere. For a spin-1 system, the length of the real space spin vector lies in the interval [0,1] and therefore s⃗ lies in the closed unit ball, known as the Bloch ball (𝔹):

    𝔹 = {s⃗∈ℝ^3 :  |s⃗ | ≤ 1}

A measure of the quantum fluctuations of the spin in the quantum state is given by the covariance matrix, a rank-2 tensor:

    𝐓=
    (
    [              ⟨ S_x^2⟩-⟨ S_x⟩^2  1/2⟨{S_x,S_y}⟩ -⟨ S_x⟩⟨ S_y ⟩  1/2⟨{S_x,S_z}⟩ -⟨ S_x⟩⟨ S_z ⟩; 1/2⟨{S_x,S_y}⟩ - ⟨ S_x⟩⟨ S_y ⟩             ⟨ S_y^2⟩ -⟨ S_y⟩^2 1/2⟨{S_z,S_y}⟩ - ⟨ S_z⟩⟨ S_y ⟩; 1/2⟨{S_x,S_z}⟩ - ⟨ S_x⟩⟨ S_z ⟩  1/2⟨{S_z,S_y}⟩ -⟨ S_z⟩⟨ S_y ⟩            ⟨ S_z^2⟩ - ⟨ S_z⟩^2;                                ])

Here, {S_i, S_j}=S_iS_j + S_jS_i is the anticommutator of S_i and S_j. Hereafter, we refer to this covariance matrix as the spin fluctuation tensor.


When the real space spin vector is transported along a loop in 𝔹, the geometric phase information is encoded in the spin fluctuation tensor. To see this, we will introduce a geometric picture of the spin fluctuation tensor. The latter is a symmetric, positive semi-definite matrix with three non-negative eigenvalues and orthogonal eigenvectors. It may be represented by an ellipsoid whose principle axes have lengths given by the square-roots of the eigenvalues and whose orientation is determined by the eigenvectors. The pair (s⃗, 𝐓) can be visualized by a vector in 𝔹 with an ellipsoid representing the spin quantum fluctuations centered at its tip (FIG1(a)). Let us consider a loop inside 𝔹 along which the spin vector is transported. Analogous to the parallel transport of tangent vectors on a sphere, one can introduce a notion of parallel transport of the ellipsoids along the loop, where each of the ellipsoid's axes is parallel transported (Of course, the center of the Bloch ball poses a certain non-triviality which we address in this paper).  Upon circumscribing the loop the ellipsoid will, in general, return in a different orientation, capturing the geometric phase of the loop (FIG1(b)). In this paper, we rigorously formulate this geometric phase as an element of the group SO(3) and provide a geometric interpretation for the same.





The key step in formulating our geometric phase is to rigorously define parallel transport of the ellipsoids. In the sequel we show, however, that the parallel transport cannot be defined using the standard theory of connections on a fiber bundle. Hereafter, we restrict ourselves to spin-1 systems. The quantum state of a spin-1 system is represented by a non-zero vector ψ=(z_-1,z_0, z_+1)^T in the 3-dimensional complex Hilbert space ℂ^3; here superscript T denotes matrix transpose. The physical properties of the spin system are invariant under a multiplication of this vector by a non-zero complex scalar, i.e., ψ and λψ are physically equivalent state vectors for λ∈ℂ-{0}; this defines an equivalence class under the equivalence relation ψ∼λψ. The quotient space under this equivalence is a four dimensional manifold.  Topologically, the manifold is the complex projective plane (ℂℙ^2), defined as the space of all lines in a 3-dimensional complex vector space passing through the origin:

    ℂℙ^2 = {ψ∈ℂ^3-{0⃗}:    ψ∼λψ for λ∈ℂ-{0}}

Equivalently, ℂℙ^2 is the space of all 1-dimensional subspaces of ℂ^3. Each 1-dimensional subspace of ℂ^3 represents an equivalence class. The spin expectation values can be written as ⟨ S_i⟩ = ⟨ψ, S_iψ⟩/⟨ψ, ψ⟩, where ⟨· , ·⟩ is the standard inner product on ℂ^3. We may define a map ϕ : ℂℙ^2 →𝔹 that takes every equivalence class of ℂ^3 to its real space spin vector: ϕ(ψ)= s⃗. In terms of coordinates, for a vector ψ = (z_-1, z_0, z_+1)^T representing the equivalence class {λψ: λ∈ℂ-{0}}, the image under this map is:

    ϕ(([ z_-1;  z_0; z_+1;      ]))=1/|z_-1|^2+|z_0|^2+|z_+1|^2([ √(2)Re(z_-1z_0^*+z_0z_+1^*); √(2)Im(z_-1z_0^*+z_0z_+1^*);           |z_+1|^2-|z_-1|^2;                             ]) = s⃗∈𝔹

We note that the map ϕ is independent of the choice of the representative in any equivalence class and therefore, it is well defined. In other words, ϕ(ψ)=ϕ(λψ) for λ∈ℂ-{0}. 

The components of the spin fluctuation tensor can also be written in terms of the coordinates of ψ. Together, the spin vector and the spin fluctuation tensor contain all the information about the spin-1 quantum state. Indeed, every spin-1 state is uniquely represented by the pair (s⃗, 𝐓). Defining a parallel transport of ellipsoids along a loop in 𝔹 is tantamount to defining a horizontal lift of loops in 𝔹 to ℂℙ^2. The map ϕ:ℂℙ^2→𝔹 does not, however, have a fiber bundle structure. Any fiber bundle over 𝔹 is necessarily a product bundle as 𝔹 is a contractible space. The space ℂℙ^2, being 4-dimensional, is not a product bundle over 𝔹 because it has non-trivial second homology. Any 4-dimensional product bundle over 𝔹, being homotopic to the 1-dimensional fiber itself, would have a trivial second homology. Therefore, this geometric phase cannot be formulated as a holonomy of loops in 𝔹, in general. Circumventing this difficulty is the first of the two problems that we address in this paper.

The interpretation of this geometric phase poses a separate problem. Berry's phase associated with a loop on the Bloch sphere, ∂𝔹, is generally interpreted as the solid angle enclosed by the loop <cit.>. The definition of solid angles easily extends to loops in the Bloch ball, provided they do not pass through the center. A convenient way to determine this solid angle is to radially project the loop onto the boundary ∂𝔹 where it subtends the same solid angle as the original loop (FIG2 (a)). We refer to such loops as non-singular loops. Loops in 𝔹 that pass through the center break into discontinuous pieces when projected to the boundary of 𝔹; their solid angle cannot be defined by projection and therefore we refer to them as singular loops (FIG2 (b)). It can be seen intuitively that for non-singular loops, the geometric phase is a rotation of the ellipsoid by an angle equal to the solid angle subtended by the loop, about the vector s⃗, at the base point of the loop. However, interpretation of the geometric phase of singular loops is non-trivial. It is the second problem we address in this paper. We restate these two problems as:



  (i) Given that ℂℙ^2 is not a fiber bundle over 𝔹, can we still define a curve in ℂℙ^2 to be a horizontal lift of a loop in 𝔹 and formulate a definition of geometric phase?

  (ii) What is the interpretation of this geometric phase? In particular, can we attach a meaning to “solid angles" for singular loops?

Pivotal to our solution of (i) is the idea that horizontal lifts in every known version of geometric phase minimize a certain metric in the fiber bundle <cit.>. In sec 2A, we provide an outline of our solution to (i). As for the interpretation of geometric phases of singular loops, the term is justified by noting that the difficulty in defining solid angles for such a loop can not be solved by perturbing it and taking a limit <cit.>. It requires a more detailed construction and a generalization of the notion of solid angles, which we provide in the next two sections and outline in sec 2B. While sec 2 as a whole outlines all of our results, the details of definitions and proofs of the theorems therein are provided in sec 3. Along with a few examples, we address the question of how to observe this geometric phase experimentally in sec 4. 






§ OUTLINE OF RESULTS

 We state our solutions to (i) and (ii) in sec 2a and sec 2b respectively.


 §.§ Definition of Horizontal Lift And Geometric Phase


In definitions 1 and 2 below, we answer (i) by invoking the important role played by metrics in the theory of geometric phase <cit.>, <cit.>, <cit.>. In the definition of Berry's phase and Uhlmann's phase, horizontal lifts are constructed using Berry's connection form <cit.> and Uhlmann's connection form <cit.>, respectively. It has been noted that in both of these cases, the horizontal lift can also be defined as the lift with minimal length in the respective fiber bundles <cit.>. For a general Ehresmann connection <cit.>, if the horizontal subspace of the tangent space of a fiber bundle is defined as the orthogonal complement of the vertical subspace under a Riemannian metric, the resulting horizontal lift of a loop always minimizes the length among all lifts of the loop. While ℂℙ^2 is not a fiber bundle over 𝔹, it has a standard, natural (i.e., maximally symmetric) metric — the Fubini-Study metric (s_FS) <cit.>. It is essentially the “angle" between two quantum state vectors in the Hilbert space:

    s_FS(ψ_1, ψ_2)= cos^-1(|⟨ψ_1, ψ_2⟩|/√(⟨ψ_1, ψ_1 ⟩⟨ψ_2, ψ_2 ⟩))

We note that this definition extends to ℂℙ^2, when we employ any Hilbert space representatives for the equivalence classes corresponding to the points in ℂℙ^2, i.e., it is invaraint under scalar multiplications: s_FS(ψ_1, ψ_2)=s_FS(λ_1ψ_1, λ_2ψ_2) where λ_1, λ_2 ∈ℂ-{0}. We define a horizontal lift for loops in 𝔹 using this metric.

Definition 1 (Horizontal Lift): A continuous path γ̃:[0,1]→ℂℙ^2  is called a horizontal lift of a loop γ:[0,1]→𝔹 iff ϕ∘γ̃ = γ and γ̃ minimizes the Fubini-Study length in ℂℙ^2.

In sec 3, we show that the earlier described intuitive notion of parallel transport of the ellipsoids along a loop in 𝔹 is equivalent to the above definition of a horizontal lift of the loop. We show that, corresponding to every choice of γ̃(0) satisfying ϕ(γ̃(0))=γ(0), there is a unique horizontal lift of γ.  In eqn_u, we provide explicit equations to compute the horizontal lift of a given loop and a given initial point of the lift. Before proceeding to define a geometric phase using this horizontal lift, we note that not every loop in 𝔹 has a well-defined horizontal lift in ℂℙ^2. The relevant regularity conditions on the loop are summarized in theorem 1.

Theorem 1 (Existence criteria for horizontal lifts): A continuous, piece-wise differentiable loop γ: [0,1]→𝔹 has a horizontal lift if it is differentiable at every t∈ [0,1] where γ(t)=0⃗∈𝔹.

This theorem essentially states that a loop in 𝔹 has a horizontal lift if it has no “kinks" while passing through the center of 𝔹. We refer to the loops satisfying the condition mentioned in this theorem as liftable loops. Clearly, any piece-wise differentiable loop not passing through the center of 𝔹 is liftable. FIG3 shows two examples of liftable loops and one example of a loop that is not liftable. FIG3 (b) is an important example of a loop that appears to have a kink at the center of 𝔹, but is nevertheless liftable. The apparent non-differentiability at the center is removable. If we choose the center as the starting and the ending points of the loop, i.e., γ(0)=γ(1)=0⃗∈𝔹, the loop satisfies all conditions mentioned in the theorem. However, the loop in FIG3 (c) is not liftable. There are multiple points of non-differentiability at the center, and so this loop does not satisfy the conditions mentioned in the above theorem. Therefore, a loop is liftable, if there is at least one parametrization under which it is differentiable at every visit to the center.

We now define geometric phase using the horizontal lift defined above. For a given loop γ and a horizontal lift γ̃, the end points γ̃(0) and γ̃(1) are in ℂℙ^2 and therefore, there is an operator U∈ SU(3) such that γ̃(1)=Uγ̃(0). This is because, SU(3) acts transitively on ℂℙ^2. The operator is not unique — there are infinitely many such operators. Through its irreducible representation in SU(3), SO(3) can be regarded as a subgroup of SU(3). We denote the representation as 𝒟 : SO(3)→ SU(3). In sec 3, we show that there is an SO(3) choice for the operator U, i.e., there is an operator R∈ SO(3) with a representation 𝒟(R)∈ SU(3) such that γ̃(1)=𝒟(R)γ̃(0). However, this operator is still not unique — it has a two fold ambiguity. We clear up this ambiguity and provide a more rigorous definition in sec 3. We also show that this operator is independent of the choice of γ̃(0), and so it is well-defined for γ. We define this SO(3) operator as the geometric phase of γ.


Definition 2 (Geometric Phase): If γ is a liftable loop in 𝔹, its geometric phase is the operator R∈ SO(3) such that, γ̃(1)=𝒟(R)γ̃(0) holds for every horizontal lift γ̃ of γ, where 𝒟(R) ∈ SU(3) is the representation of R in SU(3).

In sec 3, eqn_R, we provide an explicit way of computing the geometric phase of a given loop. Going back to the earlier described geometric picture of representing a quantum state by a spin vector and an ellipsoid centered at its tip, the end points γ̃(0) and γ̃(1) are two quantum states with the same spin vector but different ellipsoids; i.e., we can represent them as γ̃(0)≡(s⃗, 𝐓_1) and γ̃(1)≡(s⃗, 𝐓_2). The geometric phase of γ, we show, is precisely the rotation R which rotates the ellipsoid 𝐓_1 to 𝐓_2, i.e., 𝐓_2 = R𝐓_1R^T.

In the axis-angle representation, a right-hand rotation about a unit vector n̂∈ℝ^3 by an angle θ∈ [0,2π) is represented by R_n̂(θ). For non-singular loops, the geometric phase is R=R_γ(0)(Ω), a rotation about the spin vector γ(0) by an angle Ω, equal to the solid angle enclosed by γ (FIG1 (b)).  To see this, we need the following simple facts about the ellipsoids, which follow from the map phi. One of the eigenvectors of 𝐓 coincides with s⃗ with an eigenvalue 1-|s⃗|^2. Therefore, the ellipsoid is always oriented with one axis parallel to s⃗ (see Appendix for a detailed derivation). The other two eigenvalues are 1/2(1±√(1-|s⃗|^2)) and that leaves only one degree of freedom for the ellipsoid when the spin vector is fixed, namely, rotation about the spin vector (FIG1 (a)).  Therefore, if γ(t)≠ 0 throughout the loop, the geometric phase is necessarily a rotation about the vector γ(0). The parallel transport of the ellipsoid is reminiscent of the parallel transport of a tangent line to S^2 along a loop and thus the holonomy is the solid angle of the loop. Therefore, the angle of rotation of the ellipsoid is also this solid angle.

The above interpretation, however, does not work for singular loops. We provide a generalization of the above interpretation in the following section.





 §.§ Interpretation of Geometric Phase


To answer point (ii), we define a generalized solid angle for all loops inside 𝔹 in definition 3 below.  The ideas is to first project a loop in the Bloch ball onto the real projective plane (ℝℙ^2) and to define a solid angle for this projection. We begin with the definition of the projection.

We recall that ℝℙ^2 is the space of all lines through the origin of ℝ^3. Equivalently, it is the space obtained from the 2-sphere S^2 by identifying diametrically opposite points. We use the following notation for points in ℝℙ^2:

Notation: The projection of a unit vector n̂∈ S^2, to ℝℙ^2 is the equivalence class {+n̂, -n̂} and will be denoted by ±n̂.

Every loop in S^2 can be projected to a loop in ℝℙ^2.  As described earlier, the solid angle of a non-singular loop can be pictured by radially projecting it to the boundary of 𝔹, which is S^2 (FIG2 (a)). A singular loop can also be projected to S^2 after removing the point(s)at the center. The projected path will, however, be discontinuous (FIG2 (b)). Every time the loop crosses the center of 𝔹, the projected path makes a discontinuous jump across the diameter of S^2, parallel to the tangent of the loop at the center. This holds for all liftable loops. The discontinuity can be removed by identifying diametrically opposite points on S^2 and in doing so, we obtain an ℝℙ^2. Thus, every liftable loop γ in 𝔹 can be projected to a continuous path α: [0,1] →ℝℙ^2:

    α (t) =
    ±γ (t)/|γ (t)|    γ(t)≠ 0
    ±γ̇ (t)/|γ̇ (t)|    γ(t)= 0

Here, γ̇=dγ/dt. Note that the projection is in general an open path.


We will next define a solid angle for paths in ℝℙ^2, as an appropriate U(1) holonomy. Indeed the relevant U(1) bundle over ℝℙ^2 is isomorphic to the lens space L(4,1). We recall that the lens space L(4,1) is a quotient of the 3-sphere S^3 by the discrete group Z_4 action (z_1,z_2) ↦ (iz_1,iz_2), where S^3 is represented as the set of all normalized vectors in ℂ^2, i.e.,

    S^3 = {(z_1, z_2)∈ℂ^2:    |z_1|^2+|z_2|^2=1},

and Z_4= {1, i, -1, -i}.
Thus L(4,1) is obtained by identifying the orbits of Z_4 in S^3,

    L(4,1)=S^3/(z_1,z_2)∼ (iz_1, iz_2)

S^3 is a 4-sheet covering space of L(4,1).

The lens space L(4,1) is a U(1) bundle over both ℝℙ^2 and S^2 (we will show this explicitly in sec 3). In fact, this is the only lens space that is a U(1) bundle over ℝℙ^2 <cit.>. The solid angle of a loop in S^2 can be defined as the U(1) holonomy of its lift in L(4,1). Similarly, we define the solid angle of a loop in ℝℙ^2 as the U(1) holonomy of its lift in L(4,1). An important property of this solid angle is that it is preserved under the projection map from S^2 to ℝℙ^2 — the solid angle of a loop in S^2 is equal to the solid angle of its projection in ℝℙ^2. We prove this in lemma 3 in sec 3b.  The appropriate generalization of a holonomy to open paths is a vertical displacement <cit.>. The vertical displacement of the horizontal lift of a path in ℝℙ^2 is a map from the fiber above the initial point of the path to the fiber above the final point of the path. Noting that SO(3)≈ L(2,1) is a double cover of L(4,1) and it acts transitively on L(4,1), the vertical displacement can be represented by an SO(3) action on L(4,1), i.e., by an operator V∈ SO(3). We provide the details in sec 3.

We now define the generalized solid angle of a loop in 𝔹.

Definition 3 (Generalized Solid Angle): Let γ be a liftable loop in 𝔹 and let α be its projection in ℝℙ^2. If α̃ is a horizontal lift of α in L(4,1) with a vertical displacement V∈ SO(3), and k̂ is any unit vector normal to both α(0) and α(1), the generalized solid angle (Ω) of the loop γ is given by Ω = cos^-1(k̂· Vk̂).

In sec 3c, we show that the expression Ω = cos^-1(k̂· Vk̂) is the correct holonomy of α when it is closed, and a meaningful definition of the solid angle of α, also when it is open.  Furthermore, we also show that it is equal to the standard solid angle of γ when it is non-singular. Hence we refer to this angle as the generalized solid angle of γ. The following theorem establishes the connection between the generalized solid angle and geometric phase:

Theorem 2: If γ is a liftable loop in 𝔹 and α is its projection in ℝℙ^2, then the geometric phase of γ is equal to the vertical displacement of α.

Thus, the geometric phase of any loop inside 𝔹 can be interpreted in terms of the generalized solid angle of its projection into ℝℙ^2. This interpretation builds on the m=0 geometric phases introduced in <cit.>. In the following section, we fill in the details of definitions 1, 2,  3  and provide proofs of theorem 1 and theorem 2. Before proceeding, we make a few remarks contrasting our geometric phase with Berry's phase. Unlike Berry's phase, our geometric phase does not arise naturally from the dynamics of the system. For any liftable loop inside 𝔹, our geometric phase is well defined, regardless of how the physical system is transported along the loop. Therefore, our geometric phase is similar to the mixed state geometric phase introduced in <cit.> and the non-adiabatic geometric phases introduced in
<cit.>. Both of these formulations have been observed experimentally <cit.>. In sec 4, we briefly address the question of how to observe our geometric phase experimentally.  




§ FORMULATION AND PROOFS OF THEOREM 1 AND THEOREM 2


The basic idea behind the proof of theorem 1 is that although ϕ: ℂℙ^2→𝔹 does not have a fiber bundle structure, it is closely related to a fiber bundle. In fact, it can be constructed as a quotient of a fiber bundle. 𝔹 can be constructed as a quotient space of S^2×[0,1], by collapsing the sphere S^2×{0} to a point.  We show in lemma 2(a) below that ℂℙ^2 can also be constructed as a quotient space of L(4,1)×[0,1] by collapsing L(4,1)×{0} and L(4,1)×{1} to an ℝℙ^2 and an S^2 respectively. L(4,1)× [0,1] is an S^1 bundle over S^2×[0,1], because L(4,1) is an S^1 bundle over S^2. Thus, ℂℙ^2 →𝔹 can be constructed from the fiber bundle L(4,1)× [0,1]→ S^2 ×[0,1]. Before proceeding to state and prove lemma 2, we develop a geometrical construction of L(4,1). We show, in lemma 1, that L(4,1) is the space of all tangent lines to a unit sphere.

Lemma 1: L(4,1) is homeomorphic to the space of all tangent lines to a unit sphere and it is an S^1 bundle over both S^2 and ℝℙ^2.

Proof: A tangent line (ℓ) to a sphere is uniquely represented by the pair ℓ = (v̂, ±û) (FIG4(a)) of orthogonal unit vectors, v̂ representing the point of tangency of ℓ and û representing the direction of ℓ. Here, -û and +û represent the same tangent line and therefore, we use a “±" sign before û, as a short hand for the equivalence class {+û, -û}. We show that the space of all tangent lines to a sphere, i.e, {ℓ = (v̂, ±û):   v̂·û=0} is homeomorphic to L(4,1) by explicitly constructing a 4-sheeted covering map from S^3 to this space and showing that this space is also obtained as a quotient of S^3 under a Z_4 action (L41).

Noting that SU(2) is topologically homeomorphic to S^3 and SO(3) acts transitively on the space of tangent lines to a sphere, we construct a composition of the following two maps:

    SU(2) SO(3) {ℓ = (v̂, ±û):   v̂·û=0}

 f is the standard double cover from SU(2) to SO(3) i.e., f : e^in̂·σ⃗θ/2↦ R_n̂(θ)∈ SO(3), where n̂ is a unit vector in ℝ^3 and  σ⃗=(σ_x, σ_y, σ_z) are the Pauli matrices:

    σ_x=([ 0 1; 1 0;   ]),   σ_y=([  0 -i;  i  0;    ]),   σ_z=([  1  0;  0 -1;    ])

The map g is constructed from the action of SO(3) on the space of tangent lines to a sphere. Fixing a tangent line ℓ_0 = (ẑ, ±x̂) (FIG4(a)), we obtain:

    g : R_n̂(θ) ↦ R_n̂(θ)ℓ_0 = (R_n̂(θ)ẑ, ± R_n̂(θ)x̂)

We now show that g ∘ f : SU(2)→{(v̂, ±û):  v̂·û=0} is the required 4-sheet covering map. The action of SO(3) on a tangent line to a sphere has a Z_2 stabilizer. For instance, the stabilizer of ℓ_0 is {1,R_ẑ(π)}. Therefore, g is a double covering map. For an arbitrary tangent line ℓ, the pre-image set under g contains two points in SO(3). If ℓ = R_n̂(θ)ℓ_0, for some n̂ and θ, then its pre-image set is g^-1(ℓ)={R_n̂(θ), R_n̂(θ)R_ẑ(π)}. Further, f^-1∘ g^-1(ℓ) is a set of 4 elements in SU(2) given by:

    f^-1∘ g^-1(ℓ)= e^in̂·σ⃗θ/2{1, iσ_z, -1, -iσ_z}

Thus, the pre-image set is generated by a Z_4 action and therefore, g∘ f is the required covering map and L(4,1)≈{ℓ = (v̂, ±û):   v̂·û=0}. We can now define the bundle maps π_1 : L(4,1)→ S^2 and π_2 : L(4,1)→ℝℙ^2:

    π_1: (v̂, ±û)    ↦v̂∈ S^2
    π_2 :  (v̂, ±û)   ↦±û∈ℝℙ^2

π_1 takes every tangent line to its point of tangency, and π_2 takes every tangent line to a parallel line through the center, which is an element of ℝℙ^2. It is straight forward  to verify that they are both S^1 bundle maps ▪.

A natural metric on L(4,1) is induced by the round metric (i.e., the standard Cartesian metric) on S^3. This metric, at a point ℓ = (v̂, ±û)∈ L(4,1) is:

    ds^2 = dv̂· dv̂ + dû· dû-(v̂· dû)^2

The first term (dv̂· dv̂) corresponds to the distance covered by the point of contact on S^2. The term dû· dû-(v̂· dû)^2 corresponds to the angle of rotation of the tangent line about its point of contact.

Using a similar argument, it can be shown that the lens space L(2,1) is the space of all unit tangent vectors to a unit sphere, i.e.,  L(2,1) ≈{(v̂, û):  û·v̂=0} (FIG4 (b)).


Lemma 2:


  (a) ℂℙ^2 can be constructed from the stack L(4,1)× [0,1] by collapsing L(4,1)×{0} to an ℝℙ^2 and L(4,1)×{1} to an S^2 using the respective bundle maps π_1 and π_2. That is,

    ℂℙ^2 = L(4,1)× [0,1]/π

where π=1 on L(4,1)×(0,1), π=π_1 on L(4,1)×{1} and  π=π_2 on L(4,1)×{0}

  (b) Writing 𝔹^∘-{0} = S^2 ×(0,1), where 𝔹^∘ is the interior of 𝔹, the restriction of ϕ to L(4,1)×(0,1) is,

    ϕ = π_1× 1 : L(4,1)× (0,1) → S^2× (0,1).


  (c) ℂℙ^2 is the space of all chords to a unit sphere and ϕ maps each chord to its center.


Proof:
We begin with a proof of (a). Let us consider the pre-image sets of ϕ:

    ϕ^-1(s⃗)=ℝℙ^0  if  |s⃗|=1 
    ℝℙ^1  if  0<|s⃗|<1 
    ℝℙ^2  if  |s⃗|=0

This can be shown using the explicit algebraic expression of ϕ,
the map phi. However, it is more illuminating to use the earlier described geometric picture of representing a point in ℂℙ^2 as a vector and an ellipsoid, i.e., (s⃗, 𝐓) (FIG1 (a)). The lengths of the axes of the ellipsoid are 1-|s⃗|^2, 1/2(1±√(1-|s⃗|^2)) (see Appendix). Therefore, its dimensions depend only on the length of the spin vector. When |s⃗|≠ 0, one of its axes is parallel to s⃗. For a given spin vector with 0<|s⃗|<1, the ellipsoid has one degree of freedom — rotation about s⃗, which produces the set of all quantum states with spin vector s⃗. This set is an ℝℙ^1, because the ellipsoid has a two fold symmetry when rotated about s⃗.

On the boundary of 𝔹, when |s⃗|=1, the lengths of the two transverse axes of the ellipsoid are equal and the length of the third axis is zero. Therefore, the ellipsoid degenerates into a disk perpendicular to s⃗. It has no degrees of freedom; it is the only quantum state with the given spin vector. Thus, the pre-image set of this spin vector is just a point i.e., ℝℙ^0.

Finally, when |s⃗|=0, the ellipsoid again degenerates to a disk at the center of 𝔹. This time, however, it has two degrees of freedom. The pre-image set ϕ^-1(0⃗) is the space of all orientations of a disk in ℝ^3 centered at the origin. This is indeed ℝℙ^2.

It follows, now, that the pre-image set of the boundary of 𝔹, i.e., ϕ^-1({s⃗:    |s⃗|=1}) is a sphere in ℂℙ^2. For a shell of radius 0<r<1, the pre-image set is a lens space L(4,1):

    ϕ^-1({s⃗:    |s⃗|=r})= L(4,1)    0<r<1

To show this, we use lemma 1 and construct a bijective map from the pre-image of the shell to L(4,1). Consider the map (s⃗, 𝐓)↦ (v̂, ±û) where v̂=s⃗/r and û is the eigenvector of 𝐓 normal to s⃗, with the smaller eigenvalue. Indeed, there is a one-one correspondence between the orientations of an ellipsoid at s⃗ and tangent lines at s⃗ to a sphere of radius |s⃗|. Thus, it follows from lemma 1 that the pre-image of a shell is homeomorphic to L(4,1).

We can now construct ℂℙ^2 using the pre-image sets:

    ϕ^-1({s⃗:    |s⃗|=1})   = S^2
    ϕ^-1({s⃗:    0<|s⃗|<1})   = L(4,1)×(0,1)
    ϕ^-1(0⃗)   = ℝℙ^2

ℂℙ^2 is therefore obtained by attaching an ℝℙ^2 and an S^2 to either ends of L(4,1)×(0,1). The attaching maps are easily seen to be π_1 and π_2, using the geometric picture. Thus, ℂℙ^2 is obtained from L(4,1)× [0,1] by collapsing L(4,1)×{0} to an ℝℙ^2 and L(4,1)×{1} to an S^2 using the respective bundle maps.

(b) follows trivially from the above construction of pre-image sets. The geometrical construction of ℂℙ^2 claimed in (c) can be shown as follows. The chords passing through the center of a unit sphere form an ℝℙ^2. The chords at some distance r∈(0,1) from the center form an L(4,1) and the chords at a distance 1 from the center degenerate to points on a sphere, forming a sphere. Thus, the space of all chords to a unit sphere has the same structure as ℂℙ^2 and is homemorphic to it. ▪.


Lemma 2(c) is also a consequence of Majorana constellation <cit.> which has been used very fruitfully to understand geometric phases <cit.>. States of a spin-1 system can be considered as symmetric states of a two coupled spin-1/2 systems. A spin-1/2 state is a point on a Bloch sphere (i.e., ℂℙ^1) and therefore, a spin-1 state is an unordered pair of points on the Bloch sphere (see ref. <cit.> for a detailed description of this representation). This is equivalent to a chord[This picture has a generalization. ℂℙ^n is a unordered product of n ℂℙ^1's. It is the space of all unordered set of n points on a unit sphere. That is,  ℂℙ^n = ℂℙ^1×⋯×ℂℙ^1/∼ where (r_1,⋯ r_i,⋯ r_j,⋯ r_n)∼(r_1,⋯ r_j,⋯ r_i,⋯ r_n) for r_i ∈ℂℙ^1. This is known as Majorana constellation.]. ϕ maps each chord to its center.

We can represent a chord as (r, v̂, ±û), where rv̂ is the center of the chord and û is its direction. This corresponds to a quantum state whose spin vector is rv̂ and the ellipsoid is oriented such that the eigenvector normal to s⃗ with the smaller eigenvalue is parallel to û. It is straightforward to construct this quantum state ψ∈ℂ^3. For instance, written in the standard basis,
 
    (r, ẑ, ±x̂)↦ψ = (√(1-r/2), 0, √(1+r/2)) ∈ℂ^3

Quantum states corresponding to any chord can be obtained by preforming rotations on both sides of the above equation. Conversely, the chord corresponding to given quantum state can be obtained from its spin vector and fluctuation tensor, (s⃗, 𝐓) — it is the chord centered at s⃗ and oriented parallel to the largest axis of 𝐓 perpendicular to s⃗.


The Fubini-Study metric on ℂℙ^2 can be applied to the space of all chords to a unit sphere.  At (r, v̂, ±û), the metric is:

    ds_FS^2 = 1/2(1-√(1-r^2))dv̂· dv̂+ √(1-r^2)(û· dv̂)^2+ (1-r^2)(dû· dû-(v̂· dû)^2)+1/4(1-r^2)dr^2

This follows from FS. We now proceed to prove theorem 1.



 §.§ Proof of theorem 1

Without loss of generality, we may assume that γ̇(t)≠ 0 whenever it is well-defined. Therefore, γ^-1(0⃗) is a zero dimensional compact manifold, i.e., it is a finite set of points. Adding the end points 0  and 1 to this finite set, we obtain a set of points,  γ^-1(0⃗)∪{0,1}={a_0,⋯ a_n+1} where, a_i<a_i+1, a_0=0 and a_n+1=1. This set divides the loop into n+1 pieces, γ_j:[a_j-1,a_j]→𝔹 for j=1,2,⋯ n+1. Each piece γ_j may start and end at the center of 𝔹, but lies away from the center otherwise. That is, its interior lies away from the center, γ((a_j-1,a_j))⊂ S^2×(0,1]. The closure of this path in S^2× [0,1] has a horizontal lift in L(4,1)× [0,1], defined using the standard theory of connections <cit.>,  because this space has a circle bundle structure over S^2×[0,1]. We denote this horizontal lift by γ̃_j: [a_j-1,a_j]→ L(4,1)×[0,1]. This path can be projected to ℂℙ^2 by composing it with π, as shown in lemma 2(a). The idea behind this proof is to show that these  projected paths can be attached continuously under the assumptions of the theorem, and the resulting path in ℂℙ^2 is a lift of γ that minimizes the Fubini-Study length.

Within (a_j-1, a_j), we may write γ_j(t) = (γ_j(t)/|γ_j(t)|, |γ_j(t)|)∈ S^2×(0,1] where the two components represent the coordinates in S^2 and (0,1] respectively, i.e., γ_j(t)/|γ_j(t)|∈ S^2 and |γ_j(t)|∈ (0,1]. Let us define the closure of the first component as β_j: [a_j-1,a_j]→ S^2:

    β_j(t) = γ_j(t)/|γ_j(t)|   a_j-1<t<a_j
    lim_t'→ a_kγ_j(t')/|γ_j(t')|   t=a_k,   k =j, j-1

Note that β_j are indeed the closures of the discontinuous radial projections shown in FIG2 (b). Let β̃_j denote a horizontal lift of β_j in L(4,1). We define paths γ̃_j:[a_j-1,a_j]→ L(4,1)×[0,1] as:
 
    γ̃_j(t)= (β̃_j(t), |γ_j(t)|)

We next show that after projecting these paths to ℂℙ^2, i.e., π∘γ̃_j can be attached continuously at all a_j for j=1, 2⋯ n. Note that γ(a_j)=0⃗ for j=1,2⋯ n. The end points of the two neighboring paths, γ̃_j and γ̃_j+1 at a_j, projected to ℂℙ^2 are given by:

    π∘γ̃_j(a_j)=π∘ (β̃_j(a_j), 0)    ≡π_2∘β̃_j(a_j)∈ℝℙ^2 = ϕ^-1(0⃗)
    π∘γ̃_j+1(a_j)=π∘ (β̃_j+1(a_j), 0)    ≡π_2∘β̃_j+1(a_j)∈ℝℙ^2 =ϕ^-1(0⃗)

It suffices to show that the first point of the lift, β̃_j+1(a_j), can be chosen such that the above two points coincide in ℂℙ^2. We begin with a simple observation; since γ is liftable, it is differentiable at a_j and it follows that [We have used lim_t→ a_k^±γ_j(t)/|γ_j(t)| =  lim_t→ a_k^±γ_j(t)-γ_j(a_k)/|γ_j(t)-γ_j(a_k)|= ±γ̇_j(a_k)/|γ̇_j(a_k)|]:

    β_j(a_j) = lim_t→ a_jγ_j(t)/|γ_j(t)| = γ̇(a_j)/|γ̇(a_j)|
    β_j+1(a_j) = lim_t→ a_jγ_j+1(t)/|γ_j+1(t)| = -γ̇(a_j)/|γ̇(a_j)|
 Let β̃_j(a_j)=(β_j(a_j), ±û)∈ L(4,1) for some û normal to β_j(a_j), following lemma 1. We may choose

    β̃_j+1(a_j)= (β_j+1(a_j), ±û) ∈π_1^-1(β_j+1(a_j))

This is a valid choice because û is normal to β_j+1(a_j) (this follows from β_j+1(a_j)= -β_j(a_j)). It now follows that π_2∘β_j(a_j) = π_2 ∘β_j+1(a_j)= ±û∈ℝℙ^2 and therefore, γ̃_j and γ̃_j+1 can be attached continuously .

It remains to show that the lift γ̃ obtained by attaching π∘γ̃_j  minimizes the Fubini-Study metric. It suffices to show this for the interior of each segment π∘γ̃_j, which is contained in L(4,1)× (0,1). Consider γ̃_j(t)=(r(t), v̂(t), ±û(t)) as a path in the set of all chords to a unit sphere, following lemma 2(c) and using the notation (r, v̂, ±û) for a chord with center at rv̂ and in direction û. It follows from the construction of γ̃_j that:

    r(t)     = |γ_j(t)| 
    
    (v̂(t), ±û(t))    = β̃_j(t) ∈ L(4,1)   and
    v̂(t)    = β_j(t)

r(t) and v̂(t) are determined by |γ_j(t)| and β_j(t) respectively. The key observation is that the horizontal lift β̃_j minimizes the length under the induced round metric on L(4,1) (Metric on L41) among all lifts of β_j <cit.>, <cit.>. That is,  û(t) is chosen so as to minimize the length of β̃_j in L(4,1). From Metric on L41, it follows that û̇·û̇ = (v̂·û̇)^2 (here, û̇=dû/dt). This is the condition for minimizing the length. From Fubini-Study, it follows that the same condition minimizes the Fubini-Stuidy length of γ̃_j in L(4,1)×(0,1). Thus, γ̃ is a horizontal lift of γ. ▪

We next demonstrate that the horizontal lift defined by minimizing the Fubini-Study metric is equivalent to the intuitive notion of parallel transport of ellipsoids inside 𝔹. It is easier to use chords in 𝔹, following lemma 2(c), instead of ellipsoids. Let ψ be a quantum state vector represented by the chord (r, v̂, ±û). Its spin vector is s⃗=rv̂. We show that for every infinitesimal change ds⃗ of the spin vector, the corresponding intuitive parallel transport of the chord also minimizes the Fubini-Study metric. ds⃗ is a 3-dimensional vector and it can be written as a superposition of v̂, û and v̂×û. It suffices to consider the three cases where ds⃗ is parallel to the above mentioned three vectors separately. Let us begin with the case ds⃗=|ds⃗|v̂ (FIG5 (a)). Intuitively, the chord should be moved radially, parallel to itself, i.e., after parallel transport, the new chord will be (r+|ds⃗|, v̂, ±û). From Fubini-Study, it follows that this is consistent with the minimization of the Fubini-Study metric. When ds⃗ is perpendicular to both v̂ and û, intuitively, the chord should be moved to (r, v̂+ds⃗/r, ±û)— consistent with minimization of Fubini-Study metric (FIG5 (b)). Finally, when ds⃗ is parallel to û, the chord should be parallel transported like a tangent line to the shell of radius r. Using straightforward geometry, it follows that the new chord is (r, v̂+ds⃗/r, ±(û-|ds⃗|/rv̂)) (FIG5 (c)). This satisfies the correct minimization condition for the Fubini-Study metric, dû· dû=(v̂· dû)^2.


Geometric phase was defined in sec 2 as the operator R∈ SO(3) such that γ̃(1)=𝒟(R)γ̃(0) holds for all lifts γ̃ of γ. However, this operator is not unique — it has a two fold ambiguity because γ̃(0) has a non-trivial stabilizer in SO(3). For instance, when |s⃗|≠ 0, R_s⃗(π)γ̃(0)=γ̃(0). We now use the details of the construction of γ̃ to clear this ambiguity and provide a rigorous definition of R.

Corresponding to each segment β_j in S^2, we define a vertical displacement R_j ∈ SO(3) such that its lift satisfies β̃_j(a_j)= R_j β̃_j(a_j-1). Here, β̃_j is considered a path in the space of tangent lines to a sphere and R_j acts on the tangent lines as a rotation. To define R_j uniquely, we note that SO(3)≈ L(2,1) is a double cover of L(4,1). As remarked earlier, L(2,1) is the space of all unit tangent vectors to a unit sphere. β̃_j can be lifted to L(2,1), and the end points of this lift will define a unique R_j ∈ SO(3). For example, if β̃_j(t) = (v̂(t), ±û(t)), we may assume without loss of generality, that (v̂(t), û(t)) represents a continuous path in the space of all unit tangent vectors, i.e., in L(2,1). Indeed, this is a lift of β̃_j in L(2,1). The only other lift is (v̂(t), -û(t)). Both of these lifts define the same, unique vertical displacement R_j∈ SO(3) with

    R_j v̂(a_j-1)= v̂(a_j)   and   R_j û(a_j-1)= û(a_j)

Noting that L(4,1) is a U(1) bundle over S^2, it is straightforward to show that this operator is independent of the choice of the first point, β̃_j(a_j-1) of the lift <cit.>. We now define the geometric phase as

    R= R_n+1R_n⋯ R_1

It follows that 𝒟(R)γ̃(0)=γ̃(1).

We end this section with an explicit formula to compute the horizontal lift in ℂℙ^2 and the geometric phase of a given loop in 𝔹. It suffices to compute β̃_j and R_j for each piece γ_j of the loop. Assuming that β_j = v̂(t) for t∈ [a_j, a_j+1], we are to find a û(t) such that (v̂(t), ±û(t)) is a horizontal lift of β_j with a given initial point û(0). Using the minimization condition for Metric on L41 and û(t)·v̂(t)=0, it follows that û(t) is the solution to the differential equation with the given initial value:

    d/dtû(t) = - (dv̂(t)/dt·û(t))v̂(t)

To find the geometric phase, we introduce X:[a_j,a_j+1]→ SO(3) satisfying û(t)=X(t)û(a_j), v̂(t)=X(t)v̂(a_j) and X(a_j)=1. The geometric phase will then be R_j= X(a_j+1). It is straightforward to see that X(t) is the solution to the following initial value problem:

    d/dtX(t)= (dv̂(t)/dtv̂(t)^T-v̂(t)dv̂(t)/dt^T)X
       X(a_j) = 1

The above two equations, along with CP2-chord provide a complete set of equations to compute the horizontal lift and the geometric phase for any loop in 𝔹. 

Before proving theorem 2, we make a few remarks regarding the points on the boundary of 𝔹. The pre-image set of these points is trivial (pre-image sets). This implies that the corresponding quantum states can not carry any geometric phase information. Nevertheless, the definition of horizontal lifts and geometric phase given above are valid even for loops that visit the boundary of 𝔹. 

To understand what the horizontals lift and geometric phases of the class of loops that visit the boundary of 𝔹 mean, we note that such loops can be pushed to the interior of  𝔹 through infinitesimal perturbations. It is straightforward to see that the horizontal lift (geometric phase) of such loops is indeed equal to the limit of the horizontal lift (geometric phase) of the perturbed loops. Therefore, although no geometric phase can be extracted physically from this particular class of loops, for the purpose of theoretical completeness, it is possible to consistently define a geometric phase for them.



 §.§ Proof of Theorem 2


As shown in lemma 1, L(4,1) admits two S^1 bundle structures, namely, π_1: L(4,1)→ S^2 and π_2 : L(4,1)→ℝℙ^2.  Accordingly, loops in S^2  and loops in ℝℙ^2 both have well-defined solid angles in terms of the respective U(1) holonomies. The natural projection from S^2 to ℝℙ^2 preserves the solid angle.  This is the core ingredient in the interpretation of the geometric phase and the proof of theorem 2. We prove this fact in lemma 3 and then proceed to prove theorem 2. We denote the natural projection map from S^2 to ℝℙ^2 by p.

Lemma 3: Let β be a piece-wise differentiable path in S^2 and p∘β be its projection in ℝℙ^2. The vertical displacements of the horizontal lifts of β and p∘β in L(4,1) are equal.

Proof: Let β(t)=v̂(t) and let β̃(t)= (v̂(t), ±û(t)) be its horizontal lift in L(4,1). The projection of β in ℝℙ^2 is p∘β = ±v̂(t). We first show that the path obtained by interchanging the two vectors û and v̂ in β̃, i.e., (û(t), ±v̂(t)), is a horizontal lift of p∘β in L(4,1).

From the condition û(t)·v̂(t)=0, it follows that û̇(t)·v̂(t)+û(t)·v̇̂̇(t)=0. Therefore, the paths (v̂(t), ±û(t)) and (û(t), ±v̂(t)) have the same length in L(4,1)(see Metric on L41).  Further, (û(t), ±v̂(t)) is a lift of p∘β because, π_2∘ (û(t), ±v̂(t))= ±v̂(t) = p∘β(t). We show, by contradiction, that it is indeed a horizontal lift. If it is not a horizontal lift, let (û'(t), ±v̂(t)) be the unique horizontal lift with the initial value û'(0)=û(0). It must have a shorter length than (û(t), ±v̂(t)). It follows now that (v̂(t), ±û'(t)) is a lift of β with a length shorter than β̃(t)=(v̂(t), ±û(t)), and they have the same initial point i.e., (v̂(0), ±û'(0)) = (v̂(0), ±û(0)). This contradicts with the hypothesis that β̃ is a horizontal lift.

Thus, p̃∘̃β̃=(û(t), ±v̂(t)) is a horizontal lift of p∘β. Let us now consider lifts of β̃ and p̃∘̃β̃ in L(2,1) i.e., (v̂(t), û(t)) and (û(t), v̂(t)) respectively. It is straightforward to see that the vertical displacements are identical and is given by
the unique SO(3) operator V which satisfies Vv̂(0)=v̂(1) and Vû(0)=û(1). ▪

We now return to prove theorem 2. Although the pieces β_j in S^2 cannot be attached continuously, their projections in ℝℙ^2 can be attached continuously:

    p∘β_j(a_j)= ±γ̇(a_j)/|γ̇(a_j)|= p∘β_j+1(a_j)

This follows from beta. Indeed, the path obtained by attaching the segments p∘β_j in ℝℙ^2 is α, the projection of γ defined in Projection to rp2. From lemma 3, it follows that the vertical displacements of β_j and p∘β_j are equal. Thus, the vertical displacement of α is given by

    V=R_n+1R_n ⋯ R_1

Where, R_j is the vertical displacement of β_j. This is equal to the geometric phase of γ, defined in def of gp.▪



 §.§ Generalized Solid Angle

The notion of generalized solid angle was introduced through definition 3 in sec 2. In the following, we show that this generalized solid angle reduces to the standard solid angle for non-singular loops. Furthermore, we discuss the reasons why this definition is a meaningful generalization of solid angles for singular loops. In particular, we discuss the case when the projected path α is open in ℝℙ^2.

When γ is non-singular, its projection α is necessarily closed. We consider the following three cases separately — (i) γ is non singular, (ii) γ is singular and α is closed and (iii)  γ is singular and α is an open path.

For a non-singular loop, by definition |γ(t)|≠ 0 throughout. Therefore it comprises of only one piece, i.e., a_0=0 and a_1=1. The corresponding projected path in S^2, β = γ/|γ| is closed. From lemma 3 and the definition of the geometric phase given by def of gp, it follows that the geometric phase (R) of γ is a rotation about β (0) (or equivalently, about α (0)) by an angle equal to the solid angle of γ. This angle is obtained by the expression cos^-1(k̂· Rk̂) for some unit vector k̂ normal to α(0). Thus, the generalized solid angle is consistent with the standard solid angle for non-singular loops.


For a singular loop, the standard solid angle is not well-defined. However, if the projection α is closed, i.e., α(0)=α(1), the geometric phase (i.e., the vertical displacement of α) is still a rotation about α(0) — it maps the fiber above α(0) in L(4,1) to itself. Therefore, the angle of rotation about α(0) is well-defined and is the natural extension of solid angles to this case.

Finally, we consider the case where α is open. FIG3(b) shows one such example of a loop γ, whose projection is open in ℝℙ^2. That is, γ(0)=γ(1)=0⃗ but ±γ̇(0)/|γ̇(0)|=α(0)≠α(1)=±γ̇(1)/|γ̇(1)|. Solid angles are well-defined for open paths in S^2 by closing them using a geodesic in <cit.>, <cit.> (see also ref. <cit.> for an alternative formulation). We adopt a similar technique to define solid angles for open paths in ℝℙ^2. The geometric phase (R) maps the fiber above α(0) to the fiber above α(1) in L(4,1). Indeed, it can be written uniquely as a product of two rotations, one that takes α(0) to α(1) and another that rotates about α(1):

    R = R_α(1)(Ω_2)R_k̂(Ω_1)

where k̂ is a vector normal to α(0) and α(1) and Ω_1 is the angle between α(0) and α(1). The natural definition of solid angle for such a path is Ω_2, which is given by cos^-1(k̂· Rk̂).






§ EXAMPLES AND EXPERIMENTAL CONSIDERATIONS

In this section, we discuss two examples and make a few remarks on how this geometric phase can be observed experimentally.  To begin, we present the general procedure to determine the Horizontal lift and geometric phase for any given loop, in five steps:


  1. Verify if the loop is liftable. If it is not liftable, the horizontal lift and the geometric phase are not well defined.

  2. Identify the “zeros" of the loop, i.e., the set {a_0, a_1, ⋯ , a_n+1}= γ^-1(0⃗) at which the loop visits the center of 𝔹. Because the loop is liftable, it will be differentiable at each a_i.

  3. Divide the loop into the segments γ_i:[a_i-1, a_i]→𝔹 and determine their projections β_i(t), as defined in the proof of theorem-1.

  4. Solveeqn_u for β̃_i(t) (and hence obtain γ̃_i(t)) and eqn_R for X_i(t) and hence obtain R_i.

  5. By concatenating γ̃_i, obtain the horizontal lift γ̃, i.e., the lift that minimizes the Fubini -Study length. The geometric phase is, using (<ref>), R=R_n+1R_n⋯ R_1 and the generalized solid angle is given by cos^-1(k· Rk̂) for some unit vector k̂ normal to both β_1(0) and β_n(1). 



We now illustrate this procedure to compute the generalized solid angle of the singular loop shown in FIG2(b), as a first example. This is also the loop “a" in FIG7. In Cartesian coordinates, this loop is (using the notation γ_a(t)=(x(t), y(t), z(t))): 

    γ_a(t)=(sin (2π t)sin (π/6)cos(2π t), -sin (2π t)sin (π/6)sin(2π t), sin (2π t)cos (π/6))
 In spherical polar coordinates, this loop can be represented as r(t)=sin(2π t) (allowing negative values), θ(t)=π/6 and ϕ(t)=-2π t. We now follow the procedure outlined above to compute its horizontal lift, geometric phase and generalized solid angle:


  1. This loop is liftable, because γ_a(t) is differentiable everywhere.

  2. This loop has zeros at t=0, 1/2  & 1, i.e, Thus, n=1 and a_0=0, a_1=1/2, a_2=1.

  3. The segments γ_1 and γ_2 are given by restricting the loop to [0,1/2] and [1/2, 1] respectively. The corresponding projections β_i, spherical polar coordinates, are given by:

    β_1(t) =    (1/2cos(
    2π t), -1/2sin(2π t), √(3)/2),  t∈ (0,1/2)
    β_2(t)=   (-1/2cos(
    2π t), 1/2sin(2π t),-√(3)/2),   t∈ (1/2,1)

FIG6 (a) shows these projections.

  4. eqn_u and eqn_R both have analytical solutions. In particular, the solutions toeqn_R are:

    X_1(t)   =R_β_1(t)(-√(3)π t)R_z(2π t):    t ∈ [0, 1/2] 
     
    X_2(t)   =R_β_2(t)(√(3)π (t-1/2))R_z(2π (t-1/2)):    t ∈ [1/2, 1]

This indeed implies that eqn_u also has an analytic solution. Using the notation from the proof of theorem-1, β̃_i(t)=(β_i(t), ± u_i(t)) for i=1,2, it is straightforward to see that u_1(t)=X_1(t)u_1(0) and u_2(t)=X_2(t)u_1(1/2). The horizontal lifts depend on the choice of the initial point, u_1(0). 

  5. Thus, R_1=R_β_1(1/2)(-√(3)π /2)R_z(π)= R_z(π)R_β_1(0)(-√(3)π /2) and R_2= R_β_2(1)(√(3)π/2)R_z(π). Noting that R_z(2π)=R_β_1(0)(2π) and β_1(0)=-β_2(1), it follows that the geometric phase is R_a=R_2R_1 = R_β_1(0)((2-√(3))π). Following GSA, the generalized solid angle is Ω =π (2-√(3)). The most straightforward way to obtain the concatenated lift γ̃, starting from an initial point γ̃(0) is using the equation γ̃(t)= (|γ_a(t)|, β̃(t)), where we are representing a quantum state by a chord.

The projection of this loop to ℝℙ^2 is α(t)=± (1/2cos(
2π t), -1/2sin(2π t), √(3)/2). If every point in ℝℙ^2 is represented by a diameter of S^2, this loop represents a cone (see FIG6(b)). The generalized solid angle of this loop is indeed equal to the solid angle of the cone, as shown by theorem 2. 

The geometric phase of this loop, R_a=R_β_1(0)((2-√(3))π) is a rotation about the axis, (1/2, 0, √(3)/2) by an angle (2-√(3))π. The axis is the tangent to the loop at t=0. This already indicates that geometric phase of two loops based at the center can be non-commuting. If, for instance, we choose another loop congruent to this one but oriented differently (see FIG7(b)), its geometric phase will also be a rotation by an angle (2-√(3))π, but about a different axis because the loop would start with a different tangent. For instance, FIG7(b) shows a loop γ_b, congruent to γ_a, obtaned by rotating the latter about the z-axis by π/2. Its geometric phase is R_b=R_z(π/2)R_aR_z(-π/2). Although the two geometric phases do not commute, the product of the geometric phases is not equal to the geometric phase of the product of the loops, because the latter is not liftable. 

We provide another example, illustrating the non-Abelian nature of this geometric phase, where the geometric phases of two liftable loops do not commute and the concatenations of the loops are liftable.  These loops are also experimentally implemented easily, as we argue below. Let us consider the loops γ_c and γ_d shown in FIG7(c) and FIG7(d) respectively. It suffices to work out one of them in detail. γ_c can be parametrized as:

    γ_c(t)=
    (0, 0, 2t):  0≤ t≤ 1/4 
    
    (0, 1/2sin (2π(t-1/4)), 1/2cos (2π(t-1/4))):   1/4 ≤ t≤ 1/2
    
    ( 1/2sin (2π(t-1/2)), 1/2cos (2π(t-1/2)), 0):   1/2 ≤ t≤ 3/4
    
    (2(1-t), 0, 0):  3/4≤ t≤ 1

We now follow the procedure to compute this loop's horizontal lift and geometric phase:


  1. Despite its appearance, this loop does not have a non-differentiable point at the center; it visits zero only twice, at t=0 and t=1, under the given parametrization. And it is differentiable at both these times. Therefore, this loop is liftable.

  2. The set γ_c^-1(0⃗) is quite obviously {0,1}. 

  3. There is only one segment, and its projection β (t) is given by:

    β(t)=
    (0, 0, 1):  0≤ t≤ 1/4 
    
    (0,  sin (2π(t-1/4)), cos (2π(t-1/4))):   1/4 ≤ t≤ 1/2
    
    (  sin (2π(t-1/2)),  cos (2π(t-1/2)), 0):   1/2 ≤ t≤ 3/4
    
    (1, 0, 0):  3/4≤ t≤ 1
 

  4. The solution to eqn_R are:

    X(t)=
    1:  0≤ t≤ 1/4 
    
    R_x(-2π(t-1/4)):   1/4 ≤ t≤ 1/2
    
    R_z(-2π(t-1/2))R_x(-π/2):   1/2 ≤ t≤ 3/4
    
    R_z(-π/2)R_x(-π/2):  3/4≤ t≤ 1

In the axis-angle representation. The lift β̃ is easy to obtain using this solution. 

  5. The geometric phase is R_c=R_z(-π/2)R_x(-π/2). The lift can be obtained, again, by using the equation γ̃(t)= (|γ_c(t)|, β̃(t)).

Similarly, for γ_d, the geometric phase is R_d = R_x(π/2)R_z(-π/2). Thus, their geometric phases do not commute. They both have a generalized solid angle of π/2. 




In order to experimentally observe this geometric phase, the physical spin -1 system must satisfy the following criteria: (i) the quantum state vector of the system must be controllable and, (ii) a complete state tomography, i.e., a measurement of all the components of the spin vector and the spin fluctuation tensor must be possible.  Physical systems that satisfy both of these criteria include trapped atoms, ions, nitrogen-vacancy centers, superconducting qubits and experiments using nuclear-magnetic resonance. In the following, we briefly describe how the loops shown in FIG7 can be induced and their geometric phase can be measured using trapped atoms. We consider a Bose-Einstein condensate of trapped rubidium atoms, which is a spin-1 system. The spin eigenstates corresponding to eigenvalues 0 and ± 1 are magnetically sensitive. Therefore, using a strong spatial gradient in the magnetic field, a condensate can be spatially separated into three clouds, corresponding to the eigenstates 0 and ± 1. This is also known as Stern-Gerlach separation. The population of atoms in each cloud  can be estimated by measuring the intensity of fluorescent light emitted by each cloud. These populations are proportional to |z_0|^2 and |z_±|^2 where (z_-1, z_0, z_+1) is the quantum state vector of the system. The elements of the spin vector and the spin fluctuation tensor can be extracted from such measurements done in different basis. This technique has been demonstrated in <cit.>. Further, the loops shown in FIG7 can be induced in this system using rotating magnetic fields and microwave pulses. The curved parts of this loop can be induced by rotating the spin vector using a radio-frequency magnetic field and the linear parts can be induced using microwave transitions. These techniques have also been demonstrated recently <cit.>. The geometric phase of these loops can be measured by starting with two condensates prepared in the same quantum state and inducing the loop on one of them and then comparing the measured spin fluctuation tensor for both of them. For instance, the ellipsoid at the center of the Bloch sphere degenerates into a disk and has two independent parameters. Therefore, in an experiment where a loop that starts and ends at the center is induced, geometric phase can be observed by measuring this pair of parameters of the spin fluctuation tensor. Such an experiment has been recently done in a system of ultracold Rubidium atoms <cit.>. 



§ CONCLUSIONS

To conclude, we have shown that the geometrical properties of a loop traversed by the spin vector inside the Bloch ball can be extracted from the spin fluctuation tensor of a spin-1 quantum state. This property crucially depends on the Fubini-Study metric on ℂℙ^2, and it reflects the deep synchrony between the geometry of real space and the geometry of the abstract space of quantum states.

For a loop inside the Bloch ball we call a path in the space of quantum states, a lift, if it projects down to the given loop upon evaluating the expectation value of the spin vector at each point and a horizontal lift, if it also minimizes its length under the Fubini-Study metric. Not every loop inside the Bloch ball has a well-defined horizontal lift. A loop is liftable, i.e., has a well-defined horizontal lift if it has at least one parameterization under which it is differentiable at every visit to the center. We have defined a geometric phase corresponding to each loop in a Bloch ball in the form of an SO(3) operator.

Among liftable loops,  those that actually visit the center, which we call singular loops, are particularly interesting because of two non-trivial properties — first, their geometric phase is non-Abelian and second, they don’t have a well-defined solid angle and therefore prompt a generalization of the same. We have introduced the notion of generalized solid angles, defined for both singular and non-singular loops, which reduces to the standard solid angle for non-singular loops, and use it to provide an interpretation of the geometric phase of liftable-singular loops.

On the experimental side, we have discussed the possibility of observing this geometric phase in a system of a Bose-Einstein condensate of rubidium atoms. We have argued that in this system, it is possible to use the existing experimental techniques to induce the loops shown in FIG7 and the geometric phase can be observed by comparing the tomographies performed on the quantum state before and after inducing the loop. 

Although we have considered a spin-1 system, our analysis can be generalized to any spin system. A spin-S system has independent moment tensors up to order 2S. A natural extension of our work is to explore the geometric information carried by these higher order tensors.

We end with a remark on the possible theoretical applications of our geometric phase. One of the recent theoretical applications has been in characterizing topological phases of matter. Berry's phase along a loop in the parameter space of a Hamiltonian is given by the integral of the Berry curvature evaluated over the region enclosed by the loop <cit.>. The total integral of the Berry curvature over the entire parameter space (usually the momentum space in condensed matter systems) is a topological invariant of the parameter space known as the Chern number. A topological phase transition is characterized by a “sudden change" of the the Chern number. Recent explorations <cit.>, <cit.>, have shown that mixed state generalizations of Berry's phase <cit.>, <cit.> can also be used to characterize topological phase transitions.  The geometric phase introduced in this paper could also be used to characterize topological states of 1-dimensional quantum systems. In particular the nature of singular loops is akin to critical points of quantum phase transitions, an example of which can be seen in <cit.>.









§ ACKNOWLEDGMENTS

We thank T. A. B. Kennedy for discussions and providing extensive feedback during the preparation of the manuscript. We thank M. S. Chapman, Carlos Sa de Melo and Matthew Boguslawski for stimulating insights and discussions. We also thank John Etnyre for help with the formulation.  Finally, we acknowledge support from the National Science Foundation, grant no. NSF PHYS-1506294.



§ APPENDIX A: EIGENVALUES OF THE SPIN FLUCTUATION TENSOR

In this section, we derive the expressions for the quantum state vector with a given spin vector and the eigenvalues of its spin fluctuation tensor. The system has an SO(3) symmetry, i.e., if R∈ SO(3) and 𝒟(R)∈ SU(3) is its representation, under the transformation ψ→𝒟(R)ψ, the spin vector transforms as s⃗→ Rs⃗ and the spin fluctuation tensor transforms as 𝐓→ R𝐓R^T.  Therefore, for the purpose of derivation of the eigenvalues of 𝐓, without loss of generality we may assume that s⃗=(0,0,|s⃗|)^T. Any normalized quantum state vector ψ=(z_-1, z_0, z_+1)^T, (i.e., ⟨ψ, ψ⟩=1) with this spin vector must satisfy:

    z_-1z_0^*+z_0z_+1^* = 0 
    
    |z_+1|^2-|z_-1|^2 = |s⃗|

 This follows from the map phi. When 0<|s⃗|<1, the solutions to the above equation are
 
    ψ = ( [ √(1-|s⃗|/2) e^-iθ;                 0;  √(1+|s⃗|/2) e^iθ;                   ])

Each θ∈ [0,π) produces a distinct quantum state with spin vector equal to s⃗. From tensor, the corresponding spin fluctuation tensor can me computed:

    𝐓= ([ 1/2+√(1-|s⃗|^2)/2cos 2θ     √(1-|s⃗|^2)/2sin 2θ                       0;     √(1-|s⃗|^2)/2sin 2θ 1/2-√(1-|s⃗|^2)/2cos 2θ                       0;                       0                       0                1-|s⃗|^2;                         ])

One of the eigenvectors of this matrix is s⃗=(0, 0, |s⃗|)^T with an eigenvalue 1-|s⃗|^2. The other two eigenvalues are easily seen to be 1±√(1-|s⃗|^2)/2.



§ APPENDIX B: THE CONNECTION FORM OF THIS GEOMETRIC PHASE

The well known parallel transport of tangent vectors on a sphere is formulated using the affine connection. However, the geometric phase introduced in this paper can not be formulated using affine connection because its features are compatible only with Ehresmann connection. In order to put our geometric phase in perspective with the other well known examples and to bridge the gap between affine connection and Ehresmann connection, in this section, we show how the former naturally generalizes to the latter and that minimizing the length is a concise way of defining parallel transports. 

We begin with an overview of affine connection. Let M be an n-manifold and TM be its tangent bundle. At a point q∈ M, with local coordinates (x^1,x^2, ⋯ , x^n), the tangent plane, T_qM is an n dimensional vector space spanned by e_μ= ∂/∂ x^μ, μ = 1,2,⋯, n. Central to an affine connection is the covariant derivative, which comes from “differentiating" the basis vectors, ∂/∂ x^νe_μ=Γ^σ_μνe_σ. In the more formal language, the covariant derivative, denoted by “D" is defined as De_μ = Γ^σ_μνe_σ⊗ dx^ν. 

The basic problem, that the affine connection is designed to solve is to define a parallel transport γ̃, i.e., a horizontal lift, in TM for a path γ in M. This can be restated as follows: for a point q∈ M on γ with a given a lift (q, v)∈ TM on γ̃ (where v=v^μe_μ∈ T_qM) and the local tangent vector y=y^μe_μ of γ, along which q is moved, how do we change the coordinates v^μ in order to maintain the vector v parallel to itself? In other words, we are to decide the local tangent vector t of γ̃, that moves the point (q,v)∈ TM such that q is moved along y and v remains parallel to itself. This tangent vector is in the 2n dimensional tangent plane of TM at (p,v), i.e., t∈ T_(q,v)(TM) . This space is spanned by the basis vectors (e_1, e_2, ⋯, e_n, f_1, f_2, ⋯ , f_n), where f_μ=∂/∂ v^μ. Quite obviously, t=y^μe_μ+z^νf_ν, for a suitable choice of coefficients z^ν such that tDv=0. This condition is the parallel transport criterion. Using Dv= D(v^μe_μ)=dv^μ e_μ+ v^μ De_μ, we get z^μe_μ+ Γ^σ_μνe_σ y^νv^μ =0. For convenience, the connection matrix is defined as ω_ν^μ=Γ^μ_νσy^σ, in terms of which, z^μ=-ω^μ_νy^ν <cit.>.

In order to move towards Ehresmann connection, we rewrite the affine connection in a coordinate-free form. Two observations are crucial. First, in the 2n dimensional tangent space T_q, v(TM), the n dimensional subspace spanned by f_μ represent changes to the tangent v alone and is therefore known as the vertical subspace of T_(q,v)(TM).  And second, the special vectors t∈ T_(q,v)(TM) that satisfy tDv=0 also form an n dimensional subspace spanned by {e_μ-ω_μ^νf_ν:  μ= 1, 2, ⋯, n}. This space complements the vertical subspace and is known as horizontal subspace of T_(q,v)(TM).  It is straightforward to see that all that the affine connection does is to identify this horizontal subspace at each point of TM. Indeed, any n dimensional subspace of T_(q,v)(TM) that complements the vertical subspace uniquely defines the elements ω_μ^ν of the connection matrix. This follows from the fact that the vector e_μ can be written uniquely as a sum of two vectors, one in the horizontal subspace and one in the vertical subspace. Thus, in the coordinate-free form, a connection is a specification of a horizontal subspace at each point of TM, that complements the vertical subspace. Indeed, it is an n dimensional distribution over TM.

This definition extends to any fiber bundle and is known as Ehresmann connection. Apart from being coordinate-free, it also has two other advantages over the affine and the Levi-Civita connections. The covariant derivative, uipon which the affine and Levi-Civita connections are based, is rooted in differentiating vector fields over a manifold (i.e., sections of the vector bundle). In a fiber bundle where the fiber has non-trivial topology,  in general there are no sections. The Ehresmann connection is therefore the most natural way of defining horizontal lifts. Furthermore, the Ehresmann connection generalizes naturally to structures that are not bundles. The dimension of the horizontal space can be non-uniform. Such a connection, however, can not be traced back to a connection matrix and therefore, regarding a connection as a distribution becomes inevitable. 

If a fiber bundle has a natural Riemannian metric, the horizontal subspace can be defined as the orthogonal complement of the vertical subspace. The resulting horizontal lifts mimimize the path length. The tangent vector to the horizontal lift is always confined to the local horizontal subspace and because the vertical subspace is orthogonal to it, any added component would only make the lift longer. We illustrate with two examples, that by minimizing the length of a lift, we recover the standard connection form:




  * Berry phase for spin-1/2 systems: Let n̂(t) be a loop on the Bloch sphere and ψ(t) be one of its lifts in ℂ^2. We assume that ψ(t) is normalized. All other lifts of n̂(t) with the same initial point are of the form e^ix(t)ψ(t), where x(t) is a real scalar with x(0)=0. The length of these lifts (s) is given, under the Euclidean metric, by s=∫√(||ψ̇(t)||^2 +|ẋ|^2+ iẋ(⟨ψ̇, ψ⟩ - ⟨ψ, ψ̇⟩)) dt. The first order functional derivative of s w.r.t to x(t) is ⟨ψ̇, ψ⟩ - ⟨ψ, ψ̇⟩ and it should vanish if ψ is a horizontal lift, according to the new definition. Together, with ||ψ(t)||=1, we obtain the correct parallel transport criterion: ⟨ψ̇, ψ⟩=0, i.e., the Berry connection form <cit.>. The same argument applies for Aharonov-Anandan phase <cit.>.


  * Wilczek-Zee phase <cit.>: We follow the generalization of Wilczek-Zee phase in <cit.>, and show that the connection form thereof can be derived by minimizing the length. Let us assume that the Hilbert space can be decomposed into a direct sum of two subspaces with dimensions n and m respectively (denoted by V_n and V_m). V_n is the time dependent n dimensional eigenspace of the Hamiltonian. We are to define parallel transport of a vector ψ∈ V_n(0) in time, i.e., we are to define a path ψ(t)∈ V_n(t) with ψ(0)=ψ and minimal length. Given any lift, ψ(t), we can construct other such lifts by  a transformation U(t)ψ(t), where U(t) is a unitary operator with all of its non-trivial eigenvectors in V_n(t). The length of such a path is s=∫√(||ψ̇(t)||^2 + ⟨U̇ψ|U̇^†ψ⟩ + ⟨ U^†U̇ψ|ψ̇⟩ + ⟨ψ̇ |U^†U̇ψ⟩) dt. Again, setting U^†U̇=iH(t) for some Hermitian operator H(t), the first derivative of s is ⟨ H(t)ψ̇|ψ⟩ - ⟨ψ |H(t)ψ̇⟩. H(t) is an arbitrary Hermitian with zero eigenvalues in V_m(t). Therefore, s is stationary iff ψ̇ is orthogonal to V_n(t), indeed the same condition as equation (7) in ref. <cit.>. 
 


In our system, at most of the points in ℂℙ^2, the vertical subspace is one dimensional and the horizontal subspace, defined by the Fubini-Study metric is three dimensional. However, at some points, i.e., within the pre-image of the center of the Bloch ball, the vertical and the horizontal subspaces are both two dimensional. This feature adds all the non-trivialities to the system and therefore, we have to use Ehresmann connection in this problem. 



ieeetr
