The Berry phase, acquired by an anisotropic spin system with spin J≥1 when it is adiabatically rotated in a closed circuit, is considered to be associated with a non-quantized Dirac monopole, and has a geometrical as well as a topological component owing to the Dirac string. Here, it is argued that the Berry phase of a spin state with J≥1 can be associated with a quantized Dirac monopole when the corresponding spin is taken to be an entangled state of a composite system of 1/2 spins. Evidently, this avoids the contribution of the Dirac string, and the Berry phase is purely geometrical in nature.
It is well known that Dirac’s concept of the existence of a magnetic monopole (Dirac 1931) to explain the quantization of charge plays a very significant role in various fields of modern physics. Though the ‘real’ monopoles have not been observed, the manifestation of these ‘hypothetical’ monopoles plays a dominant role in the context of the Berry phase (Berry 1984). In Dirac’s theory, quantization of the monopole charge follows from the requirement that the wave function is single valued. Also, the quantization of the Dirac monopole is directly related to the quantization of the angular momentum, which implies the rotational invariance of the Hamiltonian (Goldhaber 1965, 1976; Hurst 1968; Zwanziger 1968, 1971; Cheng & Li 1984). Any choice of the vector potential yielding a magnetic field of the monopole must have a singularity known as the Dirac string. However, spacial isotropy requires that the Dirac string attached to the Dirac monopole must be ‘invisible’ to the electrons, which, in turn, requires that the flux carried by the Dirac string is quantized in units of ϕ0=hc/e. Otherwise, the string will become ‘observable’ through the Aharonov–Bohm effect (Aharonov & Bohm 1959), as the scattering of electrons by the string would violate spacial isotropy.
In recent times, several authors (Bruno 2004; Nesterov 2004, 2007) have pointed out that the quantization of the monopole charge is not sacrosanct, as the requirement of spacial isotropy may be lifted in some specific phenomena. Bruno (2004) has pointed out that in the case of an anisotropic spin system with spin J≥1, we may observe the effect of a ‘non-quantized’ Dirac monopole. Nesterov (2004, 2007) has argued that in the case of a non-quantization of the Dirac monopole, the Dirac string may be made invisible via the Aharonov–Bohm effect if we take into account a non-associative path-dependent wave function. It is noted that for an anisotropic spin system, if we consider that Dirac monopoles are non-quantized, the Berry phase will involve, apart from the geometrical contribution, a topological contribution depending on the topology of the circuit associated with the evolution around the Dirac string (Bruno 2004). In fact, the Berry phase is realized when the parameter space is the sphere S2, which is simply connected as π1(S2)=0, and the associated phase is purely geometrical in nature given by the solid angle subtended by the circuit. However, when the monopole is non-quantized, we will have the effect of the evolution around the Dirac string and there will be a contribution from the parameter space, which is non-simply connected, having the hometopy group π1(S1)=Z. This will contribute a term to the Berry phase given by the winding number of the circuit around the string multiplied by the non-quantized flux. According to Nesterov’s prescription, the effect of the Dirac string may be made invisible, provided we take into account the non-associative path-dependent wave function. However, this non-associativity of the path-dependent wave function requires the modification of the group nature of the U(1) gauge bundle (Nesterov 2004, 2007).
In this article, we shall argue that the quantization of the Dirac monopole is very much prevalent, even in the case of an anisotropic spin system with spin J≥1, when we consider these spins as correlated systems of composite 1/2 spins. In fact, these spin systems may be viewed as constructed out of fundamental spin 1/2 entities that are strongly correlated through entanglement. For example, the spin 1 building blocks may be constructed out of two spin 1/2 variables by symmetrication and then joining together these lattice sites to represent the whole lattice. This step of ‘joining them together’ is accomplished by establishing valence bonds between consecutive spins, which introduces a strong correlation among the two spins with J=1/2. In fact, each lattice site of J=1 building blocks may be taken to represent an ‘internal structure’ of two points corresponding to the two underlying 1/2 spins with a strong correlation among them (Affleck et al. 1988). In view of this, we may take the generalized anisotropic spin system with J≥1 as an entangled state of spin 1/2 variables in a lattice. Here, we shall argue that in this formalism, the quantization of Dirac monopoles is very much valid, and the Berry phase is given by the geometrical contribution associated with the solid angle subtended by the circuit. Evidently, the effect of the Dirac string does not arise.
In §2, we shall give a brief review of certain features of the Berry phase associated with the anisotropic spin system involving non-quantized Dirac monopoles as discussed by Bruno (2004). In §3, we shall show that when the anisotropic spin system is considered as an entangled system of spin 1/2 variables, the Berry phase is purely geometrical in nature and the underlying monopole is quantized. In §4, we shall discuss the renormalization group (RG) flow of the entanglement entropy and its implication in the Berry phase.
2. Anisotropic spin system, non-quantized Dirac monopole and the Berry phase
Here, we give a brief review of certain specific features of the Berry phase of an anisotropic spin system with spin J≥1 when we have a non-quantized Dirac monopole. This situation has been studied by Bruno (2004). As mentioned earlier when Dirac monopoles are non-quantized, we will have a contribution of the evolution around the Dirac string, which will now be ‘visible’ unless we impose certain extra conditions such as the non-associativity of the path-dependent wave function (Nesterov 2004, 2007). Let us consider the magnetic field B associated with a Dirac monopole given by2.1Any choice of the vector potential associated with the magnetic field of a Dirac monopole must have a singularity owing to a Dirac string, and we write2.2where h is the magnetic field of the string. Dirac (1931) introduced the vector potential as2.3where the unit vector n determines the direction of the string that is considered to pass from the origin of the coordinates to . The respective vector potential can be written as2.4where κ is the weight of the semi-infinite Dirac string. It is noted that κ=0 corresponds to the Schwinger choice2.5with the string being propagated from to (Schwinger 1966). It should be mentioned that the Dirac quantization condition implies 2μ∈Z, while the Schwinger condition is μ∈Z.
Let H0 be the Hamiltonian of a completely general spin system having arbitrary magnetic anisotropies and is the Hamiltonian resulting from a global rotation R. A closed circuit C in the parameter space of rotations consists of continuous sequences of rotations R(t) with t∈[0,T] and R(0)=1. For the circuit C to be closed, R(T) has to belong to the symmetry group G associated with the Hamiltonian H0. The rotations can be parametrized in the form R=(ϕ,θ,β), with β=ϕ+χ, where (θ,ϕ,χ) are the Euler angles. The polar angles (θ,ϕ) give the orientation of the unit vector z(R) of the rotated z-axis and β gives the twist angle of the x- and y-axes around z(R). The rotation operator is given by2.6with J being the total angular momentum operator.
Let (with n=1,2,…) be the normalized eigenstates of H0 of energies En. The rotated eigenstates can be written as with energies En(R)=En. It is noted that for a non-quantized magnetic monopole, the wave functions become multi-valued and hence H0=HR does not imply , but . Berry (1984) has pointed out that when a quantum system corresponding to a single-valued wave function is adiabatically transported along a closed circuit, the wave function acquires a geometric phase apart from the dynamical phase. However, for a multi-valued wave function, this simple picture will be altered and the corresponding Berry phase will have a topological part associated with the evolution around the Dirac string apart from the geometrical component. In fact, the Berry phase will now be given by (Bruno 2004)2.7It is noted that the last term here is due to the multi-valuedness of the wave function. We choose the z-axis along the expectation value of J for the state so that we write zn=Jn/Jn, and Jn=||Jn||. Since R(T) belongs to the symmetry group of H0, it must leave Jn invariant so that θn(T)=0. Therefore, from equation (2.6), we find2.8where is the total twist angle of the x- and y-axes around zn. In the parameter space M for a spin J≥1, a uniaxial anisotropy axis can have various orientations when the magnetic field is along the z-axis. Rotations of SO(3) are related to each other by proper symmetries of H0 yielding the same Hamiltonian, and are to be identified so that each element of the parameter space M is represented by a set of 2q points with in a ball of radius 2π. In general, for the symmetry group G of H0, we can write2.9and the fundamental homotopy group is given by2.10where Z2q is the group of integers of modulo 2q. It is noted that for , M=S2, and as π1(S2)=0, we arrive at the isotropic case of Berry’s model. In general, if zn is a symmetry axis of order q, we have2.11When the state is expanded in terms of the eigenstates |M,zn〉 of Jz with quantum number M for the z-axis along zn, we can write2.12If Mn is the largest value of M for which , the Berry phase is given by (Bruno 2004)2.13where is the solid angle of the curve described by zn(R). It is observed that the deviation from exact quantization Mn−Jn is a measure of the effect of anisotropy for . The first term on the right-hand side of equation (2.13) is the geometrical component of the Berry phase corresponding to a non-quantized Dirac monopole. The second term on the right-hand side of equation (2.13) corresponds to the contribution of the Dirac string and represents the topological component of the Berry phase. It is noted that the contribution of the Dirac string is proportional to the winding number pn(C) of the circuit C around the Dirac string.
3. Entangled state, quantized Dirac monopole and the Berry phase
For a spin state having spin J=n/2, where n>1 and is an integer, we consider that these are constructed out of spin 1/2 variables. For example, in the case of a spin J=1, we consider that the building blocks are constructed out of two spin 1/2 variables by symmetrization. This indicates that each lattice site has a sort of internal structure corresponding to the underlying 1/2 spins. These types of lattice sites can be taken to produce the whole lattice of spin n/2 (n>1 and is an integer) by establishing valence bonds between consecutive sites (Affleck et al. 1988). These valence bonds effectively establish an entangled state for the constituent 1/2 spins. We now study the Berry phase of the corresponding spin systems when evolved in a closed path, taking into account the quantized Dirac monopole.
It is observed that a quantized Dirac monopole can be treated as a Hopf bundle U(1) over S2 (Wu & Yang 1975, 1976; Boulware et al. 1985; Grossman ,; Wu & Zee 1985; Nakahara 1990). The quantization procedure of a fermion (Bandyopadhyay & Hajra 1987; Hajra & Bandyopadhyay 1991) in the framework of Nelson’s stochastic quantization formalism (Nelson 1966, 1967) introduces an internal variable that acts as Hopf fibration of S2. An elementary massive fermion appears here as a scalar particle attached, with a magnetic-flux line representing a soliton (Skyrmion; Bandyopadhyay & Hajra 1987; Banerjee & Bandyopadhyay 1992).
The entanglement of two spins in this picture can be visualized as to arise from the influence of the magnetic-flux line associated with one spin 1/2 particle on the magnetic-flux line associated with the other particle (Basu & Bandyopadhyay 2007, 2008). A general bipartite state can be written as3.1where α, β, γ and δ are complex coefficients satisfying the normalization condition. The states |0〉 and |1〉 correspond to the down and up spins, respectively. The measure of entanglement is given by the quantity called concurrence C and is given by the norm (Wooters 1998)3.2It has been shown that the concurrence of two nearest-neighbour spins in a lattice is related to the Berry phase acquired by a spin state in the system when it is rotated in a closed path (Basu & Bandyopadhyay 2007, 2008). In fact, the concurrence is found to be given by , where the Berry phase is . For a bipartite system of two spins, it may be considered that under the influence of the magnetic-flux line associated with one spin, the magnetic-flux line related to the other will deviate from the z-axis. We may view this as if the magnetic field is rotating with an angular velocity ω0 around the z-axis under an arbitrary angle θ. The time-dependent magnetic field is given by3.3where n(θ,t) is a unit vector that may be taken to be given by3.4The interaction can be described by the Hamiltonian3.5where σ is the vector of Pauli matrices and k=gμB, with μB being the Bohr magneton and g being the Lande factor.
The instantaneous eigenstate of a spin operator in direction n(θ,t) expanded in the σz-basis is given by3.6and3.7For the time evolution from t=0 to t=T, where T=2π/ω0, each eigenstate will acquire a geometric phase apart from the dynamical phase. We can write3.8and3.9where γ± is the geometric phase and ν± is the dynamical phase. The geometric phase is found to be given by (Sjoqvist 2000; Tong et al. 2003; Yi et al. 2004; Bertlmann et al. 2004)3.10and3.11In this entangled state, the angle θ measures the deviation of the spin axis from the z-axis.
From this result and from the relation of the Berry phase with concurrence C associated with the entanglement of two nearest neighbour spins given by C=|γ|/2π, with γ being the phase, we note that when a spin state J≥1 is represented by an entangled state of a system of n spins with spin 1/2 (n>1 and is an integer), the total concurrence accumulated in the system is . It may be mentioned that when we consider a lattice of 1/2 spins at different sites, the angle θ corresponding to different spins may be different. However, when we represent a lattice of J≥1 spins with each spin depicted as formed by joining 1/2 spins, a coherent description of the spin direction is obtained when the angle θ is the same for each constituent spin. From this, we find that the geometric phase acquired by the spin state J when it is rotated along a closed circuit is3.12 3.13and 3.14where is the solid angle subtended by the curve at the origin. Thus, we arrive at the geometrical component of the Berry phase associated with an anisotropic spin system derived by Bruno, incorporating a non-quantized Dirac monopole as given in equation (2.13). It is observed that for a quantized monopole, as discussed here, the contribution of the Dirac string does not arise. Thus, we find that when a spin with J≥1 is represented by an entangled state of spin 1/2 constituents, the Berry phase is essentially a geometrical one and involves a quantized monopole.
4. Maximally entangled state, renormalization group flow and the Berry phase
In a generalized form, the maximally entangled state (MES) of a spin 1/2 system of two spins can be written as4.1where α and β are complex coefficients and α* (β*) denotes the complex conjugate. For the MES, the concurrence C=|α|2+|β|2=1. From the relationship of the Berry phase with the concurrence , we note that the expression , for a spin 1/2 state in an entangled system, implies . It attains the maximum value 1 when θ=π, i.e. the orientation of the spin state is reversed. In an earlier paper (Banerjee & Bandyopadhyay 1992), it has been pointed out that when a scalar particle moves in a closed path, the Berry phase is given by ei2πμ, where μ corresponds to the monopole charge. It is noted that μ=1/2 corresponds to one magnetic-flux line, and when a scalar particle encircles one magnetic-flux line, the system generates a π phase representing a fermion. In an entangled state of spin 1/2 systems, we note that essentially corresponds to the effective monopole charge, which we denote as . Thus, we find that the effective monopole charge associated with a spin state in an entangled system is given by4.2which essentially represents the concurrence. In the MES, we have corresponding to θ=π. For a two spin 1/2 state representing the spin J=1, we note that when the orientation of one spin is reversed, this effectively corresponds to the Jz=0 state. Again, for θ=0 when there is no deviation of the spin axis from the z-axis, we have the state and represents the state Jz=±1. Thus, in the MES (), as well in the product state (), the Berry phase attained by a spin state by rotating around a closed path takes the trivial value . A special case arises when θ=π/2, corresponding to . In this case, the spin state acquires a π phase and the orientation of the spin becomes orthogonal to the z-axis. This corresponds to the spin–charge separation as observed in a resonating valence bond (RVB) state (Anderson 1987). Apart from these specific values of θ=(0,(π/2),π), we note that appears to be non-quantized. The Berry phase corresponds to the half of the solid angle subtended by the curve, and we can write4.3where ϕ|Σ is the magnetic flux of the monopole through some particular surface Σ spanning the closed curve. The apparent non-quantized value of effectively corresponds to the deviation of the spin axis from the z-axis and represents the anisotropic feature associated with the spin system for J≥1.
This specific feature of the effective monopole charge can be realized through the RG flow associated with the entanglement entropy. It may be noted that the entanglement content between two systems A and B in a pure state can be measured by the entanglement entropy known as the von Neumann entropy associated with the reduced density matrix ρA (equivalently ρB) given by4.4When the bipartite system is in a mixed state, the entanglement of formation given by concurrence has the property that it reduces to the von Neumann entropy in a pure state (Osborne & Nielsen 2002). It has been pointed out by Casini & Huerta (2004) that the entanglement entropy undergoes an RG flow. This implies that the concurrence associated with the nearest-neighbour spins in a mixed state will also undergo an RG flow. Evidently, the relationship between concurrence and the Berry phase suggests that, just like entanglement entropy, the monopole charge associated with the Berry phase will also undergo an RG flow. In this context, it may be added that Holzhey et al. (1994) have shown that for a spin system undergoing quantum phase transition, the entanglement entropy of a block of L spins with the rest of the system at criticality is proportional to the central charge c associated with the conformal field theory. In an earlier paper (Bandyopadhyay 2000), it has been pointed out that the monopole charge associated with the Berry phase, which is related to chiral anomaly, has its correspondence with the central charge associated with the conformal anomaly in (1+1)-dimensional conformal field theory. The central charge c is given by the relation (David 1995)4.5where m is an integer. It has been shown that the central charge c is related to the monopole charge μ through the relation4.6Evidently, μ here takes the quantized value (0,±1/2,±1,…) Zamolodchikov (1986) has pointed out that the central charge c undergoes an RG flow such that when c depends on a certain parameter λ
— c is stationary at fixed points λ* of the RG flow, i.e. ∇c(λ*)=0,
— at the fixed point c(λ*) is equal to the central charge c* of the conformal field theory, and
— c decreases along the RG flow, i.e. L(∂c/∂L)≤0, where L is a scale parameter.
In analogy to this, we can formulate a μ-theorem such that the monopole charge μ undergoes an RG flow. When μ depends on a certain parameter λ, we have
— μ is stationary at fixed pints λ* of the RG flow,
— at the fixed points μ(λ*) is equal to the monopole charge μ given by the quantized values (μ=0,±1/2,±1,…), and
— μ decreases along the RG flow, i.e. L(∂μ/∂L)≤0, where L is a scale parameter.
It is observed that when we identify , with λ depending on θ, we find that for fixed values of θ=0,(π/2) and π, we have, from equation (4.2), the specific quantized values of . We have observed that for the MES, , and for the product state, , this implies that decreases along the RG flow from the MES. We have noted that is a specific value that gives a π phase when the spin axis becomes orthogonal to the z-axis and the spin–charge separation occurs.
It may be mentioned that Milman & Mosseri (2003) have pointed out that the MES is related to the double connectedness of the SO(3) group. In fact, from equation (4.1), it is observed that the complex coefficients (α,β)∈C2, with |α|2+|β|2=1, and we have the symmetry . In view of this, the Hilbert space of all the MES can be defined as (S3/Z2)=(SU(2)/Z2)=SO(3). This indicates that there is a one-to-one correspondence between SO(3) and the MES. The difference between the well-known two classes of paths related to SO(3) implies that there is a global π phase for one class and no phase change for the other. It is noted that the π phase associated with the value θ=(π/2), as discussed here, is different from the π phase associated with the MES. In fact, this gain of the π phase in the MES is related to the number of times the state crosses the space of its orthogonal states.
From this analysis, it appears that the apparent non-quantized value of the monopole charge associated with the entangled spin system essentially depicts an RG flow from the MES to the product state. It has been argued (Carollo & Pachos 2005; Pachos & Carollo 2006) that the generation of a geometric phase in a quantum spin system is a witness of singular points associated with criticality. The presence of degeneracy at some point is accompanied by curvature in its neighbourhood, and when the system evolves along a closed path, one can detect it. The curvature gives rise to the Berry phase, which is given by the holonomy. The association of the Berry phase with criticality indicates that the entanglement entropy at the critical point can be formulated in terms of the monopole charge associated with the Berry phase (Majumdar & Bandyopadhyay 2010), and justifies its relationship with the central charge of the conformal field theory. Thus, the RG flow of the entanglement entropy, which is related to the RG flow of the central charge of the conformal field theory, implies that the quantized Dirac monopole charge also undergoes an RG flow that is manifested in an entangled spin system.
We have shown above that an anisotropic spin system for spin J≥1 can be visualized as an entangled system of 1/2 spins when J is given by a multiple of 1/2, and the Berry phase of such a system is associated with quantized Dirac monopoles in contrary to the suggestion of non-quantized Dirac monopoles for such a system. Apart from anisotropic spin systems, the multi-valued wave functions and related non-quantized Dirac monopoles in trapped Λ-type atoms has also been reported (Zhang et al. 2005). From our analysis, it appears that when such systems are treated as correlated systems, we can avoid non-quantized monopoles here also. In view of this, we can consider that the single-valuedness of a wave function and spacial isotropy, implying the quantization of Dirac monopoles, is very much sacrosanct in nature. It may be mentioned that when the requirement of rotational invariance is lifted, the spin can take any arbitrary value and in general it is not a multiple of 1/2. The situation then appears to be analogous to that of (2+1)-dimensional systems. In fact, in two spacial dimensions, angular momentum can take any arbitrary value, and we have fractional statistics, which is manifested in the fractional quantum Hall effect (FQHE). In fact, the FQHE is a phenomenon for a two-dimensional electron system under a strong magnetic field and represents a strongly correlated system. However, we can recast it in spherical geometry with quantized magnetic monopoles at the centre of the sphere when electrons are taken to reside on the surface (Haldane 1983). The Landau levels are now identified with angular momentum shells with the angular momentum l given by l=|μ|, |μ|+1,…, with μ being the monopole charge. Evidently, the lowest Landau level implies l=|μ|. In this picture, fractional statistics in the two-dimensional FQHE corresponds to strongly correlated electrons when a magnetic-flux line is shared by several electrons (Basu & Bandyopadhyay 1998, 2005).
As we have mentioned, when spins with J≥1 are taken as an entangled system of 1/2 spins, we are quite comfortable with quantized Dirac monopoles, and we can avoid the problem of the ‘visibility’ of the Dirac string. This gives rise to the Berry phase, which is purely geometrical in nature and does not depend on the topology of the circuit. In view of this, we observe that, even in the case of an anisotropic spin system, the very geometrical nature of the Berry phase is sacrosanct enough, which is ensured by the Dirac quantization rule.
- Received May 23, 2010.
- Accepted July 5, 2010.
- © 2010 The Royal Society