A unified theory of quantum holonomies

Annals of Physics 324 (2009) 1340–1359

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A unified theory of quantum holonomies Atushi Tanaka a,*, Taksu Cheon b a b

Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan Laboratory of Physics, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan

a r t i c l e

i n f o

Article history: Received 23 February 2009 Accepted 14 March 2009 Available online 24 March 2009

PACS: 03.65.Vf 03.65.Ca 42.50.Dv Keywords: Geometric phase Exotic hononomies Gauge theory

a b s t r a c t A periodic change of slow environmental parameters of a quantum system induces quantum holonomy. The phase holonomy is a well-known example. Another is a more exotic kind that exhibits eigenvalue and eigenspace holonomies. We introduce a theoretical formulation that describes the phase and eigenspace holonomies on an equal footing. The key concept of the theory is a gauge connection for an ordered basis, which is conceptually distinct from Mead–Truhlar–Berry’s connection and its Wilczek–Zee extension. A gauge invariant treatment of eigenspace holonomy based on Fujikawa’s formalism is developed. Example of adiabatic quantum holonomy, including the exotic kind with spectral degeneracy, are shown. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction Consider a quantum system in a stationary state. Let us adiabatically change a parameter of the system along a closed path where the spectral degeneracy is assumed to be absent. We ask the destination of the state after a change of the parameter along the path. This question is frequently raised in discussions of the Berry phase [1]. An answer, which is widely shared since Berry’s work, is that a discrepancy remains in the phase of the state vector, even after the dynamical phase is excluded. Indeed this is correct in a huge amount of examples [2,3]. However, it is shown that this answer is not universal in a recent report of exotic anholonomies [4] in which the initial and the final states are orthogonal in spite of the absence of the spectral degeneracy. In other words, the eigenspace

* Corresponding author. Fax: +81 426 77 2483. E-mail addresses: [email protected] (A. Tanaka), [email protected] (T. Cheon). 0003-4916/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2009.03.006

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associated with the adiabatic cyclic evolution exhibits discrepancy, or anholonomy. Furthermore, the eigenspace discrepancy induces another discrepancy in the corresponding eigenenergy. For the phase discrepancy, an established interpretation in terms of differential geometry allows us to call it the phase holonomy [5]. This interpretation naturally invites its non-Abelian extension, which has been subsequently discovered by Wilczek and Zee in systems with spectral degeneracies [6]. Contrary to this, any successful association of the eigenspace discrepancy with the concept of holonomy has not been known. The aim of this paper is to demonstrate that an interpretation of the eigenspace discrepancy in terms of holonomy is indeed possible. To achieve this, we introduce a framework that treats the phase and the eigenspace holonomies in a unified manner in Section 2. The key concept is a nonAbelian gauge connection that is associated with a parameterized basis [7], and the identification of the place where the gauge connection resides in the time evolution. This is achieved through a fully gauge invariant extension of Fujikawa’s formulation that has been introduced for the phase holonomy [8,9]. Our approach is illustrated by the analysis of adiabatic quantum holonomies of three examples. First, Berry’s Hamiltonian with spin-12 is revisited in Section 3. The role of parallel transport [10,5], which accompanies the multiple-valuedness of a parameterized basis, in our formulation will be emphasized. The second example, shown in Section 4, exhibits exotic holonomies without spectral degeneracy. The last example, shown in Section 5, is the simplest examples of the exotic holonomies in the presence of degeneracy, i.e., the eigenspace holonomy á la Wilczek and Zee. Section 6 provides a summary and an outlook. A brief, partial report of the present result can be found in Ref. [11]. 2. A gauge theory for a parameterized basis Two building blocks of our theory, a gauge connection that is associated with a parameterized basis [7], and Fujikawa formalism, originally conceived for the phase holonomy, are presented in order to introduce our approach to quantum holonomies. 2.1. A gauge connection In the presence of the quantum holonomy, basis vectors are, in general, multiple-valued as functions of a parameter. In order to cope with such multiple-valuedness, we introduce a gauge connection for a parameterized basis. This has been introduced by Filipp and Sjöqvist [7] to examine Manini–Pistolesi off-diagonal geometric phase [12]. As is explained below, this gauge connection is different from Mead–Truhlar–Berry’s [13,1] and Wilczek–Zee’s gauge connections [6], which describe solely the phase holonomy. N1 For N-dimensional Hilbert space H, let fjnn ðsÞign¼0 be a complete orthogonal normalized system that is smoothly depends on a parameter s. The parametric dependence induces a gauge connection AðsÞ, which is a N  N Hermite matrix and whose ðn; mÞ-th element is

Anm ðsÞ  ihnn ðsÞj

o jn ðsÞi: os m

ð1Þ

By definition, AðsÞ is non-Abelian. For given AðsÞ, the basis vector jnn ðsÞi obeys the following differential equation

i

X o Anm ðsÞjnn ðsÞi jnm ðsÞi ¼ os n

ð2Þ

and we may solve this equation with an ‘‘initial condition” at s ¼ s0 . The dynamical variable of the equation of motion (2) is an ordered sequence of basis vectors, also called a frame,

f ðsÞ  ½ jn0 ðsÞi; jn1 ðsÞi; . . . ; jnN1 ðsÞi :

ð3Þ

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Its conjugation

2

hn0 ðsÞj 6 hn ðsÞj 1 6 f ðsÞy ¼ 6 4 ...

3 7 7 7 5

ð4Þ

hnN1 ðsÞj is also useful. For example, the resolution of unity by fjnn ðsÞigN1 n¼0 is expressed as

n o ^H ; f ðsÞ f ðsÞy ¼ 1

ð5Þ

^ H is the identical operator for H, and the gauge connection AðsÞ is written as where 1

n oo AðsÞ ¼ i f ðsÞy f ðsÞ: os

ð6Þ

Now we have the equation of motion for f ðsÞ

i

o f ðsÞ ¼ f ðsÞAðsÞ: os

ð7Þ

Its formal solution is

f ðs00 Þ ¼ f ðs0 Þ exp i !

Z

!

s00

AðsÞds ;

ð8Þ

s0

where exp! is the anti-ordered exponential for the contour integration by s [7]. Note that we need to specify the integration path to deal with the multiple-valuedness of f ðsÞ, in general. Our designation ‘‘gauge connection” for the whole AðsÞ is intended to clarify the difference from two famous gauge connections for the phase holonomy. When we choose fjnn ðsÞigN1 n¼0 as an adiabatic basis of b a non-degenerate Hamiltonian HðsÞ with an adiabatic parameter s, the elements of the gauge connection AðsÞ have a well-known interpretation: a diagonal element Ann ðsÞ is Mead–Truhlar–Berry’s Abelian gauge connection for a single adiabatic state jnn ðsÞi. The off-diagonal elements are nonadiabatic transition matrix elements, and, constitute ‘‘the field strength” corresponding to Mead–Truhlar–Berry’s Abelian gauge connection [1]. This also applies to a degenerate Hamiltonian with Wilczek–Zee’s non-Abelian gauge connection, which describes the change in an eigenspace. On the other hand, AðsÞ that contains all elements fAmn ðsÞg is defined with respect to the change of the frame f ðsÞ, instead of each eigenspace [7]. As to be shown below, AðsÞ plays the central role in the quantum holonomy. 2.2. An extended Fujikawa formalism Fujikawa has introduced a formulation to examine the quantum holonomy accompanying time evolution that involves a change of a parameter [8,9]. We will focus on the unitary time evolution for pure state in the following. As a building block of the time evolution, we examine a parameterized quantum map, whose stroboscopic, unit time evolution from jw0 i to jw00 i is described by 0 b jw00 i ¼ UðsÞjw i;

ð9Þ

b where UðsÞ is a unitary operator with a parameter s. This is because periodically driven systems, b whose Floquet operator is UðsÞ, are our primary examples, and our approach is immediately applicable b to a Hamiltonian time evolution, where UðsÞ correspond to an infinitesimal time evolution operator. Let us vary s along a path C in the parameter space during L iterations of the quantum map (9), where s0 and s00 are the initial and finial points, respectively. Accordingly we examine the whole time evolution operator

  b L1 Þ Uðs b fsl gL1  Uðs b L2 Þ    Uðs b 0 Þ; U l¼0

ð10Þ

where sl is the value of s at l-th step. Although the present formulation is applicable to investigate nonadiabatic settings, our primary interest here is an adiabatic behavior induced by the limiting proce-

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b dure L ! 1. The ‘‘f ðsÞ-representation” of the building block UðsÞ of the whole evolution is a N  N unitary matrix

n o b ZðsÞ  f ðsÞy UðsÞf ðsÞ:

ð11Þ

b In other words, we have UðsÞ ¼ f ðsÞZðsÞff ðsÞy g. In order to deal with the change of s from sl to slþ1 , we have

n o n o b l Þ ¼ f ðslþ1 Þ f ðslþ1 Þy  Uðs b l Þ ¼ f ðslþ1 ÞZ F ðslþ1 ; sl Þ f ðsl Þy ; Uðs

ð12Þ

where an effective time evolution matrix Z F ðslþ1 ; sl Þ incorporates the unit dynamical evolution and the parametric change of s along a part of path C

Z F ðslþ1 ; sl Þ  exp i

Z

!

slþ1

AðsÞds Zðsl Þ;

ð13Þ

sl

where exp

is the ordered exponential. Hence the whole time evolution is expressed as

   n o b fsl gL1 ¼ f ðs00 ÞBd fsl gL U f ðs0 Þy l¼0 l¼0

ð14Þ

where we have an ‘‘effective” time evolution operator

  Bd fsl gLl¼0  Z F ðsL ; sL1 ÞZ F ðsL1 ; sL2 Þ . . . Z F ðs1 ; s0 Þ:

ð15Þ

Finally, we have ‘‘the f ðs0 Þ-representation” of the whole time evolution operator

   n o b fsl gL1 ¼ f ðs0 ÞWðCÞB d fsl gL U f ðs0 Þy ; l¼0 l¼0

ð16Þ

 Z  WðCÞ  exp i AðsÞds :

ð17Þ

where

!

C

Since the above definitions are exact, our formulation is invariant against a basis transformation with N  N unitary matrix GðsÞ

f ðsÞ # f ðsÞGðsÞ

ð18Þ

once we incorporate the following transformations

AðsÞ # GðsÞy AðsÞGðsÞ þ iGðsÞy 0 y

00

oGðsÞ ; os

WðCÞ # Gðs Þ WðCÞGðs Þ;   Bd ðfsl gLl¼0 Þ # Gðs00 Þy B d fsl gLl¼0 Gðs0 Þ:

ð19aÞ ð19bÞ ð19cÞ

This is Fujikawa’s hidden local gauge invariance [9] in a generalized form. The strategy of Fujikawa formalism is to extract a geometric information from the whole time evolution operator via the expression (16) with an appropriate restriction of GðsÞ, as is shown below. Let us examine the case that the change of s is slow enough so that we may employ the adiabatic approximation [14]. Accordingly it is suitable to choose f ðsÞ as an adiabatic basis, i.e., each basis vector b to make ZðsÞ a diagonal matrix, whose non-zero elements are the jnn ðsÞi is an eigenvector of UðsÞ, b b corresponding to an eigenvector jnn ðsÞi, i.e., eigenvalues of UðsÞ. Let zn ðsÞ be the eigenvalue of UðsÞ

b UðsÞjn n ðsÞi ¼ zn ðsÞjnn ðsÞi;

ð20Þ

where we assume that there is no degeneracy in eigenvalue. Note that zn ðsÞ is unimodular due to the b unitarity of UðsÞ. Now the gauge transformation GðsÞ is restricted to Uð1ÞN times a permutation matrix, which correspond to the freedoms to choose the phases of basis vectors, and assign the quantum

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numbers, respectively. The permutation matrix is required to deal with the eigenspace holonomy, as is shown below. In terms of Z F ðslþ1 ; sl Þ, the adiabatic approximation is the diagonal approximation [9]

Z F ðslþ1 ; sl Þ ’

Z DF ðslþ1 ; sl Þ

 exp i

Z

slþ1

! D

A ðsÞds Zðsl Þ;

ð21Þ

sl

where AD ðsÞ is the diagonal part of the gauge connection AðsÞ, i.e., ADmn ðsÞ ¼ dmn Amm ðsÞ. Namely, ADmm ðsÞ is Mead–Truhlar–Berry’s gauge connection for the m-th state jnm ðsÞi [13,1]. The corresponding adiabatic approximation of Bd ðfsl gLl¼0 Þ (15) is

  Bad fsl gLl¼0  Z DF ðsL ; sL1 ÞZ DF ðsL1 ; sL2 Þ . . . Z DF ðs1 ; s0 Þ;

ð22Þ

which are decomposed into two parts:

    Bad fsl gLl¼0 ¼ BðCÞD fsl gLl¼0 ;

ð23aÞ

 Z  BðCÞ  exp i AD ðsÞds

ð23bÞ

where

C

is the geometric part, and, L1   Y D fsl gLl¼0  Zðsl Þ

ð23cÞ

l¼0

contains dynamical phases. We retain the path-ordered exponential in BðCÞ to make it applicable to the cases with the presence of spectral degeneracies, as shown below. To sum up, the adiabatic approximation of the whole time evolution is

   n o b fsl gL1 ’ f ðs0 ÞMðCÞD fsl gL f ðs0 Þy ; U l¼0 l¼0

ð24Þ

where

 Z   Z  MðCÞ  WðCÞBðCÞ ¼ exp i AðsÞds exp i A D ðsÞds ; !

C

ð25Þ

C

is the geometric part determined by the path C, and two gauge connections AðsÞ and AD ðsÞ. For a cyclic path C, we call MðCÞ (Eq. (25)) a holonomy matrix, which describes the adiabatic change of state vector, starting from an eigenstate at s ¼ s0 , along the closed path C. An explanation why WðCÞ is required to describe the eigenspace holonomy is the following. Let us assume that f ðsÞ is single-valued. This implies that WðCÞ is the N  N identical matrix. Consequently, MðCÞ ¼ BðCÞ is always diagonal and thus cannot describe the eigenspace holonomy, though the singlevaluedness assumption does not prevent the conventional approach from describing the phase holonomy. On the other hand, in the presence of the eigenspace holonomy, a factor of MðCÞ need to be a permutation matrix. Since BðCÞ, which is always a diagonal matrix according to its definition, cannot be such a factor, the permutation matrix is need to be supplied by WðCÞ. This is consistent with the fact that the presence of the eigenspace anholonomy implies the multiple-valuedness of f ðsÞ. Furthermore, when we employ the parallel transport condition AD ðsÞ ¼ 0 [10,5,7], the holonomy matrix takes extremely simple form:

 Z  M p:t: ðCÞ ¼ exp i AðsÞds ; !

ð26Þ

C

which is determined only by WðCÞ. In other words, all the adiabatic quantum holonomies can be summarized as a holonomy in the ordered basis f ðsÞ (Eq. (3)). For the phase holonomy, this observation is already reported by Fujikawa [8,9]. In this sense, the parallel transport condition offers a privileged gauge.

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We explain the consequence of the gauge transformation (18) under the adiabatic approximation, where GðsÞ is restricted to be a product of a permutation matrix and a diagonal unitary matrix. The invariance of the adiabatic time evolution operator, which appears at the right hand side of Eq. (24), is assured due to the following

AD ðsÞ#GðsÞy AD ðsÞGðsÞ þ iGðsÞy 0 y

oGðsÞ ; os

00

ð27aÞ

WðCÞ#Gðs Þ WðCÞGðs Þ;

ð27bÞ

00 y

ð27cÞ

0

BðCÞ#Gðs Þ BðCÞGðs Þ: Hence we obtain the manifest covariance of MðCÞ

MðCÞ#Gðs0 Þy MðCÞGðs0 Þ:

ð28Þ

Eq. (25) provides a correct expression of the holonomy matrix MðCÞ not only for maps (9) but also for flows, i.e., Hamiltonian systems and periodically driven systems. For a system whose time evolub tion is generated by a nearly static Hamiltonian HðsÞ, we will derive Eq. (25) in Appendix A with a suitb sÞ, where able discretization of time. For a periodically driven system described by a Hamiltonian Hðt; b b Hðt; sÞ ¼ Hðt þ T; sÞ is assumed, a Floquet operator

  Z i T b Hðt; sÞdt exp  h 0

ð29Þ

b is the unitary operator UðsÞ to provide a stroboscopic description of the system. Hence this system is reduced to a quantum map. An extension of our formulation to the case that the presence of spectrum degeneracy whose degree is independent with s along a closed path C is shown. The resultant expression for the holonomy matrix (25) remains the same. This is achieved by a suitable extension of gauge connections AðsÞ and b we have a normalized orthogAD ðsÞ. For the eigenspace corresponding to the eigenvalue zn ðsÞ of UðsÞ, onal vectors jnnm ðsÞi with an index m for the eigenspace, where

b UðsÞjn nm ðsÞi ¼ zn ðsÞjnnm ðsÞi

ð30Þ

and hnn00 m00 ðsÞjnn0 m0 ðsÞi ¼ dn00 n0 dm00 m0 . The gauge connection for the parameterized basis is

An00 m00 ;n0 m0 ðsÞ  ihnn00 m00 ðsÞj

o jn 0 0 ðsÞi: os n m

ð31Þ

It is straightforward to see that AD ðsÞ that appears in Eq. (25) is non-Abelian:

ADn00 m00 ;n0 m0 ðsÞ  dn00 n0 An00 m00 ;n0 m0 ðsÞ; where

ADnm00 ;nm0 ðsÞ

ð32Þ

is Wilcek-Zee’s gauge connection for the n-th eigenspace [6].

3. An example of the phase holonomy Our formulation is applied to Berry’s simplest example of the adiabatic phase holonomy [1]. Let us suppose that a spin-12 is under static magnetic field B. With a suitable choice of units, the spin is described by a Hamiltonian

b HðBÞ ¼Bb r; P

ð33Þ

^ j ej is the Pauli operator for the spin, and, ej (j ¼ x; y; z) is the unit vector for j-axis. ^ ¼ j¼x;y;z r where r b The spectrum of HðBÞ is fBg, where B  kBk. To investigate the adiabatic holonomy, we have to exclude the degeneracy point B ¼ 0. The unit vector n  B=B is parameterized with the spherical coordinate, i.e., n ¼ ex cos u sin h þ ey sin u sin h þ ez cos h. Let jn ðBÞi be a normalized eigenvector of b HðBÞ, corresponding to the eigenvalue B:

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h h jnþ ðBÞi ¼ eiu=2 cos j "i þ eiu=2 sin j #i 2 2 h h jn ðBÞi ¼ eiu=2 sin j "i þ eiu=2 cos j #i: 2 2

ð34aÞ ð34bÞ

Note that jn ðBÞi are multiple-valued as functions of ðh; uÞ. To assure them single-valued, the range of ðh; uÞ needs to be restricted within an open set U M  fðh; uÞ j 0 < h < p and 0 < u < 2pg, for example. The corresponding frame f ðBÞ  ½ jnþ ðBÞi; jn ðBÞi  is

"

f ðBÞ ¼ ½ j "i; j #i 

eiu=2 cos 2h eiu=2 sin 2h eiu=2 sin 2h

#

eiu=2 cos 2h

ð35Þ

:

Gauge connections Ax ðBÞ  iff ðBÞy gof ðBÞ=ox (x ¼ h; u; B) for parametric changes of B are

1 ry ; 2 1 Au ðBÞ ¼ ðrz cos h  rx sin hÞ; 2 AB ðBÞ ¼ 0; Ah ðBÞ ¼

ð36aÞ ð36bÞ ð36cÞ

where we employ 2  2 complex matrices

rx 



0

1

1 0

 ;

ry 



0 i i

0

 ;

rz 



1 0 0 1

 :

ð37Þ

Accordingly, the Mead–Truhlar–Berry gauge connections are

ADh ðBÞ ¼ 0;

ADu ðBÞ ¼

1 rz cos h; 2

ADB ðBÞ ¼ 0:

ð38Þ

As is well known, the strength of the magnetic field B plays no particular role in the computation of the phase holonomy once B is kept non-zero. Evaluations of the holonomy matrix MðCÞ for typical closed loops in the parameter space are shown. First, we examine a loop C in which f ðBÞ is single-valued, e.g., C U M . Consequently WðCÞ is the 2  2 identical matrix. This is a conventional wisdom to obtain a formula of the phase holonomy, where an appropriate gauge for f ðBÞ (or, equivalently, jn ðBÞi) against the loop is chosen to avoid the multiple-valuedness of f ðBÞ [1]. Hence all the holonomies reside in BðCÞ ¼ MðCÞ. The evaluation of the contour integral in BðCÞ is straightforward to obtain the classic result in the matrix MðCÞ:

  i MðCÞ ¼ exp  XðCÞrz ; 2

ð39Þ

where XðCÞ is the solid angle for C [1]. Second, we examine a meridian great circle C h in which h moves 0 to 2p with u kept fixed. The parametric change along such a circle can induce a change of the sign of an eigenvector [15]. In our example, f ðBÞ cannot be single-valued on C h :

f ðBÞjh#0 ¼ f ðBÞjh"2p :

ð40Þ

Although the conventional strategy mentioned above is to avoid such multi-valuedness, we insist the present choice of the gauge (35) to show an alternative way to reproduce the conventional result. Thanks to the present choice of the gauge, the Mead–Truhlar–Berry gauge connection satisfies the parallel transport condition ADh ¼ 0 and BðC h Þ ¼ 1. This enable us to employ Eq. (26) to obtain the holonomy matrix:

 I MðC h Þ ¼ exp i !

Ch

 1 ry dh ¼ eipry ¼ 1; 2

ð41Þ

which is consistent with Eq. (39). This is an example that our formulation properly deals the multiplevaluedness of f ðBÞ. On the other hand, if we choose an appropriate gauge to make f ðBÞ single-valued on the circuit C h , WðC h Þ is trivial to put all the nontrivial holonomy in BðC h Þ, as is stated above.

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Finally, let us consider a circle of latitude C u , where h is kept fixed and u is increased from 0 to 2p. It is straightforward to obtain BðC u Þ ¼ expðiprz cos hÞ and WðC u Þ ¼ 1. The latter indicates again the sign change in the parametric dependence along C u . These elements are combined to reproduce a well-known result

MðC u Þ ¼ exp fipð1  cos hÞrz g:

ð42Þ

We remark that, in all the examples above, the holonomy matrices M are diagonal, so that the eigenspace holonomy is absent. Accordingly the eigenvalue anholonomy is also absent. 4. An example of the exotic holonomies in quantum map spin-12 In Berry’s Hamiltonian (Eq. (33)), the strength of the magnetic field B plays no role in quantum holonomies. One reason is that the corresponding gauge connection AB ðBÞ vanishes. Another reason is that it is impossible to make any loop in the parameter space by an increment of B, with n being kept fixed. To make a loop for a strength parameter, we may examine the following quantum map for a spin-12

^Þg; exp fikða þ bn  r

ð43Þ

where n is a normalized real vector, a and b are real constants to be specified later, and k is the strength. The periodicity of the quantum map with respect to the increment of k implies that there is a loop in the parameter space of k. In particular, if we choose a ¼ q=2 and b ¼ ð2  q=2Þ with an integer q, Eq. (43) is periodic as a function of k, with a primitive period 2p. Accordingly, the parameter space of k is identified with S1 and it might be suitable to investigate quantum holonomies for a periodic variation of k. However, such a loop does not allow us to study adiabatic holonomies, since there remains a spectral degeneracy along the loop at k ¼ 0 ðmod 2pÞ. A simple way to lift the degeneracy at k ¼ 0 is to concatenate two quantum maps:

      q 2q p 2p ^ ^ ; exp il þ mr exp ik þ nr 2 2 2 2

ð44Þ

where q and p are integers, m and n are normalized vectors in R3 , and, l and k are strengths. Due to the periodicity in l and k, the parameter space of ðl; kÞ is a two-dimensional torus S1  S1 . Both m and n specify points on a sphere S2 . In the following, we fix m ¼ ez and parameterize n by spherical variables h and u as n ¼ ex cos u sin h þ ey sin u sin h þ ez cos h. If we change m with keeping m  n fixed, it induces only the Berry phase. To facilitate the following analysis, we examine the symmetric version of the quantum map (44)

      p 2p b  exp i l q þ 2  q m  r ^ ^ U exp ik þ nr 2 2 2 2 2    l q 2q ^ :  exp i þ mr 2 2 2

ð45Þ

For brevity, we omit the parameters in the following. A possible implementation of the quantum map (45) is available by a periodically driven system that is described by the following Hamiltonian

   X q 2q p 2p b ^ þk ^ HðtÞ l dðt  jÞ; þ mr þ nr 2 2 2 2 j2Z

ð46Þ

b The magnitudes of the magwhere the Floquet operator for a unit time interval 1=2 6 t < 1=2 is U. netic fields of the unperturbed system and the perturbation are

1 ð2  qÞl; 2 1 Bk  ð2  pÞk; 2

Bl 

respectively. It is straightforward to show

ð47Þ

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 b  eiðlqþkpÞ=2 cos D  i^ U r  ~l ; 2

ð48Þ

where

 D  2 cos1 cos Bl cos Bk  cos h sin Bl sin Bk ; ~l  sin Bl cos Bk þ cos h cos Bl sin Bk m þ ðsin Bk Þfn  ðn  mÞmg:

ð49aÞ ð49bÞ

2 It is also easy to see k~lk2 ¼ sin ðD=2Þ. Hence l  ~l= sinðD=2Þ is a unit vector. Now we have

   b  exp i lq þ kp þ D r U ^l ; 2 2

ð50Þ

and its eigenvalues are

   lq þ kp D : z  exp i  2 2

ð51Þ

Corresponding quasienergies are E  ðlq þ kp  DÞ=2, which is defined up to modulus 2p. In order to study the adiabatic holonomies, we need to identify the spectral degeneracies, whose condition is eiD ¼ 1, in the parameter space. It is useful to see D as a function of Bk

D ¼ 2 cos1 ðA cos ðBk þ a~ÞÞ

ð52Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ~ is an ‘‘initial phase” that is independent with k. We choose the where A  1  sin h sin Bl and a branch of cos1 A in ½0; p. If A < 1, D oscillates within the range ½2 cos1 A; 2p  2 cos1 A ð0; 2pÞ as a function of Bk , and encounters no spectral degeneracy. On the other hand, the condition A ¼ 1 (i.e., sin h sin Bl ¼ 0) implies the presence of the spectral degeneracy. From the similar argument for Bl , we will encounter spectral degeneracies if sin h sin Bl sin Bk ¼ 0. Let us examine the case sin h ¼ 0. Since this implies cos h ¼ 1, we have



 D ¼ 2 cos1 cos Bk  Bl :

ð53Þ

Accordingly the degeneracy points draw lines in ðBl ; Bk Þ-plane as



Bk  Bl ðBl ; Bk Þ

2Z

p

ð54Þ

corresponding to the condition cos h ¼ 1. On the other hand, if we assume sin Bl sin Bk ¼ 0, we have

 D ¼ 2 cos1 cos Bl cos Bk :

ð55Þ

Hence another condition for the spectral degeneracy is j cos Bl cos Bk j ¼ 1, i.e., the degeneracy points are at lattice points:





 Bl Bk Bl ; Bk

; 2 Z ; p p

ð56Þ

for all h. Summarizing above, we show the location of the spectral degeneracies in terms of ðl; mÞ, whose space is the two-dimensional torus. The degeneracy lines are specified by ðl; k; hÞ as



kð2  pÞ þ lð2  qÞ cos h h ðl; k; hÞ

2 Z; 2 Z : 2p p

ð57Þ

In addition to this, we have isolated degeneracy points as



l¼

 2pk 2pl ; k¼ jk; l 2 Z : 2q 2p

Except these degeneracy points, it is legitimate to introduce a zenith angle H of l, s.t.,

ð58Þ

A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

sin Bl cos Bk þ cos h cos Bl sin Bk ; sinðD=2Þ sin h sin Bk sin H ¼ : sinðD=2Þ

1349

cos H ¼

ð59Þ

b corresponding to the eigenvalues z : It is straightforward to obtain the eigenvectors jn i of U,

jnþ i ¼ eiu=2 cos

H 2

jn i ¼ eiu=2 sin Let f  ½ jnþ i;

j "i þ eiu=2 sin

H

H

j "i þ eiu=2 cos

2

j #i;

2

H 2

j #i:

ð60aÞ ð60bÞ

jn i  be a frame. The gauge connections (6) for f are

1 oH ry ; 2 oh 1 Au ¼ ðrz cos H  rx sin HÞ; 2 1 oH Ak ¼ r y ; 2 ok 1 oH ; Al ¼ ry 2 ol

Ah ¼

ð61aÞ ð61bÞ ð61cÞ ð61dÞ

and the corresponding Mead–Truhlar–Berry gauge connections are AhD ¼ 0, ADu ¼ 12 rz cos H, ADk ¼ 0, and ADl ¼ 0. With these gauge connections, we will examine the holonomy matrices of typical loops in the parameter space. First, we examine the meridian great circle C h in which h moves 0 to 2p with other parameters are kept fixed. It is straightforward to see BðC h Þ ¼ 1, due to the parallel transport condition ADh ¼ 0. Hence all the holonomies reside in WðC h Þ ¼ MðC h Þ:

 I WðC h Þ ¼ exp i !

Ch

   1 oH 1 p ry dh ¼ exp i ry Hj2h¼0 ; 2 2 oh

ð62Þ

p where Hj2h¼0 , the change of H along C h , is determined by the image of C h in the sphere ðH; uÞ. If the p ¼ 2p, where  correspond to the direction of the path. Both image is a closed circle, we have Hj2h¼0 cases provides a Longuet-Higgins type phase change MðC h Þ ¼ e ipry ¼ 1. On the other hand, if the imp ¼ 0, which implies MðC h Þ ¼ 1. The folage is closed self-retracing curve along an ark, we have Hj2h¼0 lowing index

r

  Bk þ Bl

p

 

Bk  Bl

p

 ;

ð63Þ

where ½x is the maximum integer not greater than x, determines which is the case, as shown in Appendix B:

( p j2h¼0

H

¼

2pð1Þr=2

for r is even

0

for r is odd

:

ð64Þ

Hence we obtain

MðC h Þ ¼ ð1Þ1þr :

ð65Þ

Next, we examine a circle of latitude C u , where u is increased from 0 to 2p and the other parameters are kept fixed. We have BðC u Þ ¼ expðiprz cos HÞ and WðC u Þ ¼ 1. Accordingly, we obtain

MðC u Þ ¼ exp ðipð1  cos HÞrz Þ:

ð66Þ

So far, the holonomy matrices MðCÞ are diagonal, so that neither C h nor C u incorporates the exotic holonomies. The following is the first example of the exotic holonomies in this paper.

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A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

Let us examine a closed loop C k , in which k is increased from 0 to 2p, being kept fixed other parameters. To avoid degeneracies along C k , we choose lð2  qÞ=ð2pÞ R Z and h=p R Z. When we increase k from k ¼ k0 to k ¼ k0 þ 2p, we have



Djk¼k0 þ2p ¼

Djk¼k0 2p  Djk¼k0

for even p for odd p

ð67Þ

;

i.e., an anholonomy in D occurs. Accordingly we have an eigenvalue holonomy

z jk¼k0 þ2p ¼



z jk¼k0

for even p

z jk¼k0

for odd p

ð68Þ

;

which implies the presence of the eigenspace holonomy. We proceed to evaluate the holonomy matrix MðC k Þ. Because of the parallel transport condition ADk ¼ 0, we have MðC k Þ ¼ WðC k Þ. On the other hand, we have

 I WðC k Þ ¼ exp i Ck

   1 oH 1 p ry dk ¼ exp i ry Hjk¼2 : k¼0 2 2 ok

ð69Þ

Hence we need to examine the zenith angle H. From Eq. (59) and sinðD=2Þjk¼k0 þ2p ¼ sinðD=2Þjk¼k0 , we p ¼ pð2  pÞ mod 2p. Since this is not suffice to determine have eiH jk¼2p ¼ ð1Þð2pÞ :eiH jk¼0 , i.e., Hj2k¼0 p . the precise value of WðC k Þ, we need keep track of H against the increment of k to obtain Hj2k¼0 Now the parameter space perpendicular to C k is ðh; u; lÞ 2 S2  S1 , which is divided into subspaces p is constant, by the spectral degeneracies l ¼ 2pk=ð2  qÞ (k is integer). Since, in each subspace, Hj2k¼0 it is suffice to evaluate it at a representative point. Let us choose a point h ¼ p=2 and l ¼ pð2k þ 1Þ=ð2  qÞ, where the spectral gap takes a constant value D ¼ p, from Eq. (49a). Accordingly we have eiH ¼ ð1Þk expfið1Þk kð2  pÞ=2g; from Eq. (59). Hence we obtain p Hj2k¼0 ¼ ð1Þk pð2  pÞ;

ð70Þ

which also holds for 2pk=ð2  qÞ < l < 2pðk þ 1Þ=ð2  qÞ; i.e., k ¼ ½lð2  qÞ=ð2pÞ. Hence we have

#   " ð1Þk sin p2p  cos p2p ; 1 k : MðC k Þ ¼ exp i ð1Þ pð2  pÞry ¼ 2 ð1Þk sin p2p ;  cos p2p

ð71Þ

In particular, if p is odd, the off-diagonal elements of MðC k Þ remains, so that the eigenspace holonomy exhibits. This is consistent with the emergence of the eigenvalue holonomy for odd p. Now the similar analysis of quantum holonomies for the circuit C l , where l is increased from 0 to 2p, is trivial. Hence we show only the holonomy matrix

" MðC l Þ ¼

 cos q2p ;

ð1Þk sin q2p

ð1Þk sin q2p ;

 cos q2p

# ;

ð72Þ

where k ¼ ½kð2  pÞ=ð2pÞ. We conclude that odd q along C l implies the exotic holonomies. 5. Example 3: the exotic holonomies a lá Wilczek–Zee A simple example of the eigenspace holonomy accompanying spectral degeneracy is shown. Extending Mead’s study [16,17] on non-Abelian adiabatic phase holonomy [6], we introduce a quantum map with Kramer’s degeneracy. 5.1. Quantum map for spin-32 with Kramers’ degeneracy To introduce our model, we review the time-reversal invariance structure in an atom with oddnumber of electrons [16,17]. For a comprehensive explanation, we refer Avron et al. [17]. Let bJ be the total angular momentum of our system and jJ; Mi the standard basis vector for bJ , i.e., pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bJ 2 jJ; Mi ¼ JðJ þ 1ÞjJ; Mi, bJ  ez jJ; Mi ¼ MjJ; Mi, and bJ  ðex  iey ÞjJ; Mi ¼ JðJ þ 1Þ  MðM  1ÞjJ; M  1i.

A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

1351

b  expðipbJ y Þ K b 0 , where K b 0 is The standard time-reversal operator for bJ is an anti-unitary operator K the complex conjugate operation in the jJ; Mi-representation. We examine the fermion case b , its spectrum b 2 ¼ 1, which implies that J is a half-integer. If an Hermite operator commutes with K K exhibits Kramer’s degeneracy. The same is true for unitary operators. We focus on the case J ¼ 32, and introduce basis vectors as follows:

 

3 3

b ðje1 iÞ ¼ 3 ;  3 ; je1 i 

; ; jKe1 i  K

2 2 2 2

 

3

3 1 1 b ðje2 iÞ ¼ ; : je2 i 

;  ; jKe2 i  K

2 2 2 2

ð73Þ

Our physical observables are spanned by the following time-reversal invariant operators

s^0  je1 ihe1 j þ jKe1 ihKe1 j  je2 ihe2 j  jKe2 ihKe2 j; s^1  je1 ihKe2 j þ jKe1 ið1Þhe2 j þ h:c:; s^2  je1 iðiÞhKe2 j þ jKe1 iðiÞhe2 j þ h:c:; s^3  je1 ihe2 j þ jKe1 ihKe2 j þ h:c:; s^4  je1 iðiÞhe2 j þ jKe1 iihKe2 j þ h:c:;

ð74Þ

which are traceless and form a Clifford algebra

s^a s^b þ s^b s^a ¼ 2dab :

ð75Þ

Several properties of sa are shown in Appendix C. We introduce an extension of the quantum map for spin-12 (Eq. (45))

( !)    4 p 2p X b  exp i l q þ 2  q s ^0 ^a exp ik U na s þ 2 2 2 a¼0 2 2    l q 2q^  exp i s0 ; þ 2 2 2

ð76Þ

P 2 2 where ðna Þ4a¼0 is a unit vector in R5 , i.e., a na ¼ 1, and ðq; pÞ 2 Z . The quantum map (76) can be implemented by a periodically pulsed driven system in a similar way shown in the previous section for the quantum map (45). Since the unitary operator (76) is 2p-periodic both in l and k, the parameter space of ðl; kÞ forms a two-dimensional torus S1  S1 . The unit vector ðna Þ is parameterized by spherical variables:

 n0 ¼ cos h;

n1 n2



 ¼

cos v sin v

In Appendix D, we show

b ¼ eiðlqþkpÞ=2 cos D  i U 2

X



sin g sin h;

! ~la s ^a ;



n3 n4



 ¼

cos u sin u



cos g sin h:

ð77Þ

ð78Þ

a

where D is defined in Eq. (49a),

~l ¼ sin B cos B þ cos h cos B sin B ; 0 l k l k

ð79aÞ

~l  n sin B a a k

ð79bÞ

and P for a – 0, where the definitions of Bl and Bk are shown in Eq. (47). Since 4a¼0~l2a ¼ sinðD=2Þ2 , we normalize ~la :

la 

1 ~la : sinðD=2Þ

Accordingly we have

ð80Þ

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A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

(

lq þ kp D X ^ la sa þ

b ¼ exp i U

2

2

!) ð81Þ

:

a

b are The eigenvalues of U

   lq þ kp D z  exp i  : 2 2

ð82Þ

Namely, the spectrum is completely same with the example shown in Section 4. To examine eigenvectors, we parametrize la with the zenith angle H, which is already introduced for the quantum map spin-12 in Eq. (59). We have

l0 ¼ cos H;



l1 l2



 ¼

cos v



sin v

sin g sin H;



l3 l4



 ¼

cos u sin u



cos g sin H:

ð83Þ

b are From Appendix E, the eigenvectors of U

jnþ i ¼ jd1 i cos

H 2

þ jd2 i sin

H 2

;

H

H

jKnþ i ¼ jKd1 i cos þ jKd2 i sin ; 2 2  H H þ jd2 i cos ; jn i ¼ jd1 i  sin 2 2   H H jKn i ¼ jKd1 i  cos þ jKd2 i cos ; 2 2

ð84Þ

where

  g g þ jKe1 i eþiðuþvÞ=2 sin ; jd1 i  je1 i eiðuþvÞ=2 cos 2 2   g g þ jKe2 i eiðuvÞ=2 sin ; jd2 i  je2 i eiðuvÞ=2 cos 2 2   g g jKd1 i ¼ je1 i þeiðuþvÞ=2 sin þ jKe1 i eþiðuþvÞ=2 cos ; 2 2   g g þiðuvÞ=2 iðuvÞ=2 sin cos : þ jKe2 i e jKd2 i ¼ je2 i e 2 2

ð85Þ

Note that we put each basis vector before its complex coefficient above to prevent a confusion due to the presence of anti-Hermite operation K. To conclude this subsection, we introduce a frame comb [18]: posed by the eigenvectors U

f  ½ jnþ i; jKnþ i; jn i; jKn i :

ð86Þ

5.2. Analysis of adiabatic holonomies We examine the adiabatic holonomies of the quantum map. Note that D and H depend only on l, k, h and is the same ones for the quantum map spin-12 (see, Eqs. (45) and (59)). Hence the degeneracy points in the parameter space and the holonomy in the eigenvalues are the completely the same. We will focus on the eigenspace holonomy in the following. It is straightforward to obtain the gauge connection from the eigenvectors. Since H depends on l, k, h, the corresponding gauge connections are defined through the derivative of f by H:

Ax ¼ if

y

 o 1 0 f ¼ ox 2 þiI2

 iI2 oH ; ox 0

ð87Þ

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A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

for x ¼ h; l; k, where I2 is the 2  2 unit matrix. The corresponding Wilczek–Zee gauge connections vanish to satisfy the parallel transport condition, i.e., ADh ¼ ADl ¼ ADk ¼ 0. For other gauge connections, l, k, and h-dependences are introduced through H:

  1 ry cos H ry sin H ; 2 ry sin H ry cos H     1 rz cos H rz sin H 1 rx 0 cos g þ sin g; Au ¼ 2 rz sin H rz cos H 2 0 rx     1 rz 0 1 rx cos H rx sin H cos g þ sin g; Av ¼ 2 0 rz 2 rx sin H rx cos H

Ag ¼

ð88aÞ ð88bÞ ð88cÞ

and

  1 ry 0 cos H; 2 0 ry     0 1 rz 1 rx 0 ADu ¼ cos H cos g þ sin g; 2 0 rz 2 0 rx     0 1 rz 0 1 rx cos g þ cos H sin g: AvD ¼ 2 0 rz 2 0 rx

ADg ¼

ð89aÞ ð89bÞ ð89cÞ

It is straightforward to evaluate the holonomy matrix MðC a Þ (25) for a closed loop C a where

a(=l; k; h; g; u; v) is increased from 0 to 2p, once we take care the ‘‘anholonomy” in H along C a , which is also clarified in Section 4. For l; k; h, we have   0 i MðC a Þ ¼ exp  2 þiI2

iI2



H

0

j2ap¼0



2 ¼4

I2 cos

Hj2ap¼0

þI2 sin

2

Hj2ap¼0 2

I2 sin I2 cos

Hj2ap¼0

2 Hj2p

3 5;

ð90Þ

a¼0

2

where

( p Hj2h¼0 ¼

2pð1Þr=2

for r is even

0

for r is odd

ð91aÞ

;

p Hj2k¼0 ¼ ð1Þ½lð2qÞ=ð2pÞ pð2  pÞ;

jl2p¼0

H

¼ ð1Þ

½kð2pÞ=ð2pÞ

ð91bÞ

pð2  qÞ;

ð91cÞ

and r is defined in Eq. (63). Hence MðC h Þ exhibits only Herzberg and Longuet-Higgins’ sign change [15]

MðC h Þ ¼ ð1Þ1þr I4 :

ð92Þ

Also, the same kind of sign change appears along C l (C k ) with even p (q), i.e., MðC l Þ ¼ ð1Þ1þp=2 I4 (MðC k Þ ¼ ð1Þ1þq=2 I4 ). On the other hand, a mixture of the eigenspace holonomy and the Herzberg and Longuet-Higgins’ sign change occurs along C l with odd p

MðC l Þ ¼ ð1Þ½lð2qÞ=ð2pÞþðp1Þ=2



0 I2

 I2 ; 0

0

I2

I2

0

ð93Þ

and, along C k with odd q

MðC k Þ ¼ ð1Þ½kð2pÞ=ð2pÞþðq1Þ=2



 ;

ð94Þ

which do not incorporate mixing within the degenerate eigenspaces. Other holonomy matrices MðC a Þ (a ¼ g; u; v) describes genuine Wilczek–Zee’s phase holonomies:

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A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

" MðC g Þ ¼  MðC u Þ ¼ MðC v Þ ¼



#

 exp iry Xg =2 0

 ; 0 exp iry Xg =2 exp ½iðrz cos g1 þ rx sin g1 ÞX1 =2 0 exp ½iðrz cos g2 þ rx sin g2 ÞX2 =2 0

ð95aÞ 0



; exp ½iðrz cos g1  rx sin g1 ÞX1 =2  0 ; exp ½iðrz cos g2  rx sin g2 ÞX2 =2

ð95bÞ ð95cÞ

where Xg  2pð1  cos HÞ, Xj  2pð1  bj Þ, gj  i lnfðcos H cos g þ i sin gÞ=bj g for j ¼ 1; 2, b1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ðsin H cos gÞ2 and b2  1  ðsin H sin gÞ2 .

6. Summary and outlook We have introduced a framework that is capable of describing the eigenspace holonomy and the phase holonomy in a unified manner. Several examples have been shown. In hindsight, it might seem rather odd that, in two decades since the first discovery of Berry phase, the full, gauge invariant formulation for quantum holonomy has not been conceived prior to this work. It should probably be attributed to the lack of the incentive to improve on the original expression of Berry; Nothing other than the phase anholonomy has been anticipated for the adiabatic cyclic variation of parameters for regular Hamiltonian system. As a result, the gauge invariant formulation must have seemed a redundant luxury. However, if we once recognize the possibility of eigenstates’ exchange without level crossing for cyclic parameter variation, both for singular systems and for time-periodic systems, it becomes imperative to treat the choice of basis frame explicitly within the formalism, which has naturally lead us to arrive at the full, gauge invariant formulae. In a simple minded view, it is the periodicity of quasienergy, which is a result of the time-periodicity of the system, that enables the eigenvalue holonomy in a natural manner. For a Hamiltonian system, with the energy defined on entire real number, an energy eigenstate after cyclic variation of parameter cannot reach another eigenstate of different energy in a usual way, since the crossing of levels is prohibited for adiabatic variation. The only possible exception appears to be singular Hamiltonian systems, for which the highest and the ground eigenenergy diverge [4]. In this work, we have focused on the most elementary setting of quantum holonomy, i.e., the adiabatic excursion of pure quantum eigenstates along a closed path in the parameter space. A vast amount of studies on the phase holonomy naturally suggests possible directions of extension of the present result. We mention only few of them. A straightforward extension is to examine noncyclic path [19,12]. Also, loosening of the assumption of adiabaticity and resulting extension into e.g., Aharonov–Anandan’s nonadiabatic settings [20], are expected to be straightforward thanks to the generality of Fujikawa’s formulation [9], which is the basis of our theory. It seems timely as well as interesting to examine the eigenspace holonomy in dissipative systems, for which we will need to horn appropriate techniques to treat of the eigenspace holonomy in mixed states [21]. Finally, we mention a question that is raised from the main result Eq. (25), which supplies a complete prescription to quantify the adiabatic quantum holonomy. How this helps to understand the exotic holonomies intuitively? Is it possible to find any underlying object or concept that governs them, for example, in the manner of diabolic point for the case of Berry phase? Herzberg and Longuet-Higgins has shown that the phase holonomy along a closed loop C implies the presence of spectral degeneracy in a surface S enclosed by C in the parameter space [15] (see also, Refs. [10,22]). Is there any counterpart of the argument of Herzberg and Longuet-Higgins for the exotic holonomies? This does not seem likely, for the case of exotic holonomy, at the first glance, since there is no room to make S from C in all the examples shown in this paper. We now believe that an affirmative answer is to be found in the complex parameter plane, on which we shall focus our attention in a forthcoming publication.

A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

1355

Acknowledgement This work has been partially supported by the Grant-in-Aid for Scientific Research of Ministry of Education, Culture, Sports, Science and Technology, Japan under the Grant number 18540384. Appendix A. The gauge theory for Hamiltonian time evolution b We will derive the holonomy matrix (Eq. (25)) for a system described by Hamiltonian HðsÞ that depends on a time-dependent parameter s. Suppose that s is moved from s0 to s00 along a path C, during 0 6 t 6 T. The corresponding time evolution operator is

  Z i T b b f st g Uð Hðst Þdt : t2½0;T Þ  exp  h  0

ð96Þ

b l is We divide the time interval ½0; T into L parts. Let tl  ðl=LÞT. A short time evolution operator U accordingly introduced:

b l  exp  i U h

Z

t lþ1

! b Hðst Þdt :

ð97Þ

tl

When L is large enough, we have

 

 b l Þ þ O  2 ; b l ¼ exp  i Hðs U h where sl  12 ðstlþ1 þ stl Þ and

ð98Þ

  T=L. Now we introduce a s-dependent unitary operator

  b  : b s  exp  i HðsÞ U h

ð99Þ

Accordingly we have

  Y L1 b b fst g U t2½0;T ¼ l¼0 U sl þ OðÞ:

ð100Þ

Since we choose L so as to satisfy   1, our formulation explained in Section 2 is straightforwardly applicable. First, for a given f ðsÞ, the f ðsÞ-representation of UðsÞ is

  o

 in b ZðsÞ ¼ exp  f ðsÞy HðsÞf ðsÞ þ O 2 : h

ð101Þ

On the other hand, we have

exp i

Z

!

slþ1

AðsÞds

 ¼ exp ðiAðsl Þs_ Þ þ O 2 ;

ð102Þ

sl

where we assumed s_ ¼ Oð1Þ. Hence we have

exp i

Z sl

slþ1

!

 

 i AðsÞds Zðsl Þ ¼ exp  Fðsl ; s_ Þ þ O 2 ; h

ð103Þ

where we introduce Fujikawa’s Hamiltonian matrix [9]

n o b Fðs; s_ Þ  f ðsÞy HðsÞf ðsÞ  hAðsÞs_ :

ð104Þ

In the limit L ! 1, the effective time evolution operator for Fujikawa’s formulation [9] is

  Z i T Bd ðfst gÞ ¼ exp  Fðst ; s_ t Þdt : h 0

ð105Þ

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A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

b Next, let us examine the adiabatic change of s. Let f ðsÞ be an adiabatic basis for HðsÞ. Now

b ðsÞ HD ðsÞ  f ðsÞy HðsÞf

ð106Þ

b is a diagonal matrix whose non-zero elements are the eigenvalues of HðsÞ. Thanks to the adiabatic theorem, we employ the diagonal approximation for Bd ðfst gÞ:

  Z i T D Bd ðfst gÞ ’ Bad ðfst gÞ  exp  F ðs; s_ Þdt ; h  0

ð107Þ

where F D ðs; s_ Þ is defined as

F D ðs; s_ Þ  HD ðsÞ  hAD ðsÞs_ :

ð108Þ

Hence Bad ðfst gÞ is decomposed into geometric and dynamical factors:

Bad ðfst gÞ ¼ exp i

Z s0

s00

!

  Z i T D A ðsÞds exp  H ðsÞdt : h 0 D

ð109Þ

Now it is trivial to apply our formulation explained in the main text to obtain the holonomy matrix (Eq. (25)). Appendix B. A derivation of Eq. (65) To evaluate Eq. (62), we need to obtain the image of C h in the sphere ðH; uÞ. First, we remind that H p is a periodic function of h from Eqs. (49a) and (59). Hence Hj2h¼0 must be a multiple of 2p. Second, we remind that there is no spectrum degeneracy along the path C h if r R Z, where r  fkð2  pÞ lð2  qÞg=ð2pÞ. Namely, the non-degenerate regions are divided into squares by the lattice p takes a constant value. Accordingly, it is suffice to evaluate ðr þ ; r Þ 2 Z2 . Within each square, Hj2h¼0 them at representative points of the squares. Let us examine the case ðr þ  r Þ=2 is an integer k, which implies D ¼ 2 cos1 ðð1Þk cosðpðrþ þ r  Þ=2Þ, cos H ¼ ð1Þk cos h sinðpðrþ þ r  Þ=2Þ= sinðD=2Þ, and sin H ¼ sin h sinðpðrþ þ r  Þ= 2Þ= sinðD=2Þ. Accordingly we have

eiH ¼ ð1Þ½kþðrþ þr Þ=2 expfið1Þk hg;

ð110Þ

which implies the image of C h is also a great meridian loop in the sphere ðH; uÞ. Thus we have p Hj2h¼0 ¼ ð1Þk 2p:

ð111Þ

This result is also valid for the case that ð½rþ   ½r  Þ=2 is an integer k. Let us examine the case ðrþ þ r Þ=2 is an integer l, which implies D ¼ 2 cos1 ðð1Þl cosðpðrþ  r  Þ=2Þ, cos H ¼ ð1Þlþ½ðrþ r Þ=2 and sin H ¼ 0. Namely, the image of C h is a point in ðH; uÞ space. Accordingly we have p Hj2h¼0 ¼ 0:

ð112Þ

This result is also valid for the case that ½r þ   ½r  is an odd integer. To summarize the argument above, we have, from Eq. (62),

MðC h Þ ¼ exp fipð1 þ ½r þ   ½r  Þg;

ð113Þ

which is equivalent with Eq. (65). Appendix C. Algebraic properties of ^sa ^a . Details are found in, for example, Ref. [17]. It is We summarize the algebraic properties of s straightforward to show that they form a Clifford algebra (75), which implies the following formulas:

s^a s^b s^a ¼ 2dab s^a  s^b :

ð114Þ

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A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

For Ba 2 R,

X

!2 ¼ kBk2 ;

^a Ba s

ð115Þ

a

where kBk is a norm of real vector, i.e., kBk ¼ fies knk ¼ 1,

exp ik

X

! ^a na s

X

¼ cos k  i

a

qffiffiffiffiffiffiffiffiffiffiffiffi P 2ffi a Ba . Furthermore, for a real unit vector n that satis-

! ^a sin k; na s

ð116Þ

a

which is 2p-periodic in k. Next formula is useful to investigate quantum maps:

^a Þs ^b exp ðiks ^a Þ ¼ ð1  dab Þs ^b þ dab fcosð2kÞs ^b  i sinð2kÞg: exp ðiks

ð117Þ

^a by complex 4  4 matrices sa in terms of complex 2  2 matriFinally, we show a representation of s ^a ¼ f0 sa f0y , where f0  ½ je1 i; jKe1 i; je2 i; jKe2 i : ces defined in Eq. (37), i.e., s

s0 ¼



I2

0

0

I2

 ;

s1 ¼



 iry ; 0

0 iry

s2 ¼



0

irx

irx

0



; s3 ¼



0

I2

I2

0



;

s4 ¼



0

irz

irz

0

 :

ð118Þ Appendix D. A derivation of Eq. (78) We derive Eq. (78) from Eq. (76). First, let Bl  lð2  qÞ=2 and Bk  kð2  pÞ=2. Hence we have iBk

^ ¼ eiðlqþkpÞ=2 eiBl s^0 =2 e U ^0 iðlqþkpÞ=2 iBl s

e

¼e

¼ eiðlqþkpÞ=2 k  i

P

^a na s

a

cos Bk  i X

! ~la s ^a ;

eiBl s^0 =2 ;

X

na

!

eiBl s^0 =2 ^

sa e

^0 =2 iBl s

sin Bk ;

a

ð119Þ

a–0

where we used Eq. (117) above, and,

k  cos Bl cos Bk  n0 sin Bl sin Bk ; ~l0  sin Bl cos Bk þ n0 cos Bl sin Bk ; ~lað–0Þ  na sin Bk :

ð120Þ

Hence we arrive Eq. (78). Appendix E. Diagonalization of +a na ^sa Let na (a ¼ 0; . . . ; 4) be real, and, normalized, i.e.,

s^ðfna gÞ 

4 X

^a ; na s

P4

2

a¼0 na

¼ 1. We will obtain the eigenvectors of

ð121Þ

a¼0

with the help of the quaternionic structure of the Hilbert space induced by the fermion time reversal 2 2 2 invariance [17]. We follow the convention of quaternions explained in Ref. [17], e.g., i ¼ j ¼ k ¼ 1, ij ¼ ji ¼ k, jk ¼ kj ¼ i, and ki ¼ ik ¼ j. In the Hilbert space H of spin-32, we employ a right quaternionic action for jwi 2 H: jwii  ijwi, and, b ðjwiÞ, which implies jwik ¼ jwiðijÞ ¼ ðjwiiÞj ¼ K b ðjwiiÞ ¼ if K b ðjwiÞg. Accordingly, for zj ; wj 2 C, jwij  K we have

1358

A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

jwi ¼ z1 je1 i þ w1 jKe1 i þ z2 je2 i þ w2 jKe2 i ¼ je1 iðz1 þ jw1 Þ þ je2 iðz2 þ jw2 Þ ¼ fq



z1 þ jw1 z2 þ jw2

 ; ð122Þ

where

fq  ½ je1 i; je2 i 

ð123Þ

is the standard frame of the quaternionic Hilbert space. Hence we obtain a natural correspondence between four-dimensional complex vector space and two-dimensional quaternionic vector space. This ^a by 2  2 quaternionic matrix sqa , i.e., induces a representation of s

s^a ¼ fq sqa f yq ;

ð124Þ

where

sq0 ¼



1

0

0 1

 ;

sq1 



0 j j



0

;

sq2 



   0 1 0 k ; sq3  ; 1 0 k 0

sq4 



0 i i

0

 :

ð125Þ

We introduce spherical variables h, g, v and u to parameterize na as n0 ¼ cos h, n1 ¼ cos v sin g sin h, n2 ¼ sin v sin g sin h n3 ¼ cos u cos g sin h, and n4 ¼ sin u cos g sin h. In terms of 2  2 quaternionic matrix, we have

"

s^ðfna gÞ ¼ fq

iðuþvÞ

ðcos g  je

cos h iðuþvÞ

eþiu ðcos g þ je

sin gÞ sin h

sin gÞeiu sin h

 cos h

# fqy ;

ð126Þ

where we introduce iðuþvÞ

h  je 2

ð127Þ

:

iðuþvÞ iðuþvÞ

Since h ¼ je

e

"

s^ðfna gÞ ¼ fq

hg

j ¼ 1, we have e hg iu

cos h

e

eþiu ehg sin h

e

¼ cos g þ h sin g. Hence we have

sin h

 cos h

# fqy :

ð128Þ

^ðfna gÞ are 1. Let jnð0Þ Now it is straightforward to see the eigenvalues of s  i be corresponding eigenvectors:

" jnð0Þ þ i

 fq "

jnð0Þ  i  fq

ehg=2 cos 2h

#

; eiu ehg=2 sin 2h

# ehg=2  sin 2h : eiu ehg=2 cos 2h

ð129aÞ ð129bÞ

Instead of the two above, we put a phase factor on them in the following: ð0Þ

jn i  jn ieiðuþvÞ=2 ;

ð130Þ

where we need to take care about the noncommutativity of multiplication in quaternions. In the complex Hilbert space, jn i are expressed as

h h jnþ i ¼ jd1 i cos þ jd2 i sin ; 2  2  h h þ jd2 i cos ; jn i ¼ jd1 i  sin 2 2

ð131aÞ ð131bÞ

where jd1 i and jd2 i are orthonormal

  g g þ jKe1 i eþiðuþvÞ=2 sin ; jd1 i  je1 i eiðuþvÞ=2 cos 2 2   g g þ jKe2 i eiðuvÞ=2 sin : jd2 i  je2 i eiðuvÞ=2 cos 2 2

ð132aÞ ð132bÞ

1359

A. Tanaka, T. Cheon / Annals of Physics 324 (2009) 1340–1359

^ðfna gÞ for the complex Hilbert space. They are obtained by the We explain the rest of eigenvectors of s time-reversal operation on jn i:

h h jKnþ i ¼ jKd1 i cos þ jKd2 i sin ; 2  2  h h þ jKd2 i cos ; jKn i ¼ jKd1 i  cos 2 2

ð133aÞ ð133bÞ

where

  g g jKd1 i ¼ je1 i þeiðuþvÞ=2 sin þ jKe1 i eþiðuþvÞ=2 cos ; 2 2   g g þiðuvÞ=2 iðuvÞ=2 þ jKe2 i e sin cos : jKd2 i ¼ je2 i e 2 2

ð134aÞ ð134bÞ

b 2 ¼ 1, fjn i; jKn ig is a complete orthogonal system for H. Because of K To compute gauge connections for fjn i; jKn ig, it is useful to summarize the basis transformation between

f  ½ jnþ i; jKnþ i; jn i; jKn i ;

ð135aÞ

f0  ½ je1 i; jKe1 i; je2 i; jKe2 i :

ð135bÞ

and

It is straightforward to obtain the following

        i i i i f ¼ f0 exp  g 4 v exp  g 3 u exp  g 2 g exp  g 1 h 2 2 2 2 where g a are 4  4 Hermite matrices

g1 



0

iI2

iI2

0

 ;

g2 



ry

0

0

ry

It is also useful to express g a in terms of

 ;

g3 



rz

0

0

rz

 ;

g4 



ð136Þ

rz

0

0

rz

 :

ð137Þ

sa : g 1 ¼ is0 s3 , g 2 ¼ is1 s3 , g 3 ¼ is3 s4 , and g 4 ¼ is1 s2 .

References [1] M.V. Berry, Proc. R. Soc. Lond. A 392 (1984) 45. [2] A. Shapere, F. Wilczek (Eds.), Geometric Phases in Physics, World Scientific, Singapore, 1989. [3] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, Z. Zwanziger, The Geometric Phase in Quantum Systems, Springer, Berlin, 2003. [4] T. Cheon, Phys. Lett. A 248 (1998) 285. [5] B. Simon, Phys. Rev. Lett. 51 (1983) 2167. [6] F. Wilczek, A. Zee, Phys. Rev. Lett. 52 (1984) 2111. [7] S. Filipp, E. Sjöqvist, Phys. Rev. A 68 (2003) 042112. [8] K. Fujikawa, Ann. Phys. (NY) 322 (2007) 1500. [9] K. Fujikawa, Phys. Rev. D 72 (2005) 025009. [10] A.J. Stone, Proc. R. Soc. Lond. A (1976) 141. [11] T. Cheon, A. Tanaka, Europhys. Lett. 85 (2009) 20001. [12] N. Manini, F. Pistolesi, Phys. Rev. Lett. 85 (2000) 3067. [13] C. Mead, D.G. Truhlar, J. Chem. Phys. 70 (1979) 2284. [14] M. Born, V. Fock, Z. Phys. 51 (1928) 165. [15] G. Herzberg, H.C. Longuet-Higgins, Disc. Farad. Soc. 35 (1963) 77. [16] C.A. Mead, Phys. Rev. Lett. 59 (1987) 161. [17] J. Avron, L. Sadun, J. Segert, B. Simon, Comm. Math. Phys. 124 (1989) 595. [18] We remark on the difference from the previous report [11] on the notations concerning to the analysis of the kicked spin-32. The spherical variables c, g, n and f in Ref. [11] should be read as h, p2  g, v and u, respectively. The set of normalized eigenvectors jvmn i (m; n ¼ 0; 1) in Eq. (36) of Ref. [11] and jn i and jKn i are related by a unitary transformation: jv00 i ¼ jnþ i cos g2 þ jKnþ i sin g2, jv01 i ¼ jnþ ið sin g2Þ þ jKnþ i cos g2, jv10 i ¼ jn i cos g2 þ jKn ið sin g2Þ, and jv11 i ¼ jn i sin g2 þ jKn i cos g2. Accordingly the choices of basis vectors have difference. This modifies the expressions of the gauge connections and holonomy matrices. [19] J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339. [20] Y. Aharonov, J. Anandan, Phys. Rev. Lett. 58 (1987) 1593. [21] A. Uhlmann, Rep. Mat. Phys. 24 (1986) 229. [22] N. Johansson, E. Sjöqvist, Phys. Rev. Lett. 92 (2004) 060406.