The quantum spin quadrumer
A fundamental motif in frustrated magnetism is the fully mutually coupled cluster of spins, with each spin coupled to every other spin. Clusters with and have been extensively studied as building blocks of square and triangular lattice antiferromagnets. In both cases, large- semiclassical descriptions have been fruitfully constructed, providing insights into the physics of macroscopic magnetic systems. Here, we develop a semiclassical theory for the cluster. This problem has rich mathematical structure with a ground state space that has non-trivial topology. We show that ground states are appropriately parametrized by a unit vector order parameter and a rotation matrix. Remarkably, in the low energy description, the physics of the cluster reduces to that of an emergent free spin- spin and a rigid rotor. This successfully explains the spectrum of the quadrumer and its associated degeneracies. However, this mapping does not hold in the vicinity of collinear ground states due to a subtle effect that arises from the non-manifold nature of the ground state space. We demonstrate this by an analysis of soft fluctuations, showing that collinear states have a larger number of soft modes. Nevertheless, as these singularities only occur on a subset of measure zero, the mapping to a spin and a rotor provides a good description of the quadrumer. We interpret thermodynamic properties of the quadrumer that are accessible in molecular magnets, in terms of the rotor and spin degrees of freedom. Our study paves the way for field theoretic descriptions of systems such as pyrochlore magnets.
The principles underlying frustrated magnetism emerge from a few prototypical models. Many of these share a common feature: they are composed of clusters of spins with each spin equally coupled to every other spin. Such a cluster is described by the Hamiltonian
Frustration emerges when , representing antiferromagnetic coupling between each pair of spins. When the clusters are coupled among themselves, this typically leads to effects such as macroscopic classical degeneracy. For example, clusters of spins occur in the square antiferromagnet and in dimerized quantum systems such as SrCu(BO)Shastry and Sutherland (1981); Kageyama et al. (1999). Clusters with occur in the Majumdar Ghosh modelMajumdar and Ghosh (1969), the triangular antiferromagnet and the kagome antiferromagnet.
For systems with and clusters, a particularly fruitful approach has been to construct large- semiclassical field theories. The field theory for systems, first derived by HaldaneHaldane (1983), is formulated in terms of a unit vector field. On the other hand, the field theory is more appropriately written in terms of an rotor field as first shown by Dombre and ReadDombre and Read (1989). A similar field theoretic approach has so far not been realized for . This is an interesting and topical problem due to its relevance to pyrochlore antiferromagnetsGardner et al. (2010), the checkerboard lattice antiferromagnetCanals (2002) and the square J-J-J modelDanu et al. (2016). In particular, it is relevant to several pyrochlore materials with Heisenberg-like couplings such as MnSbO, CdYbS, GdTiO, etc., which all have intriguing propertiesGardner et al. (2010).
Here, we derive a path integral description for the cluster which serves as a starting point for constructing semiclassical field theories. Even at the level of a single cluster, we find rich topological structure and an elegant physical description.
Ii Cluster ground states for
Classically, a spin is a vector of length . The allowed values of spin form a one-to-one and onto mapping to , the two-dimensional sphere. An arbitrary spin position can be described by two parameters, e.g., polar and azimuthal angles. As the Hamiltonian is positive semi-definite, the lowest possible classical energy is zero. In other words, a ground state is reached when the total spin is zero, i.e., . The set of all such -spin states constitutes the ground state space. Mathematically, this can be denoted as . The special features of the cluster lie in the topology of its classical ground state space. We first recapitulate the properties of the and clusters to set the stage for .
For , the ground state space is simply the set of pairs of antipodal points on the sphere. Each ground state is uniquely defined by the position of the first spin, with the ground state space being isomorphic to . This mapping brings out the topology of the ground state space, e.g., showing that it is simply connected. It also brings out its ‘manifold’ character as every point in has a two-dimensional tangent space. In physics terms, about any given ground state, we have two independent ‘soft’ fluctuations that do not cost energy. This mapping to underlies the semi-classical field theory for the antiferromagnetic Heisenberg chain. First formulated by HaldaneHaldane (1983), the field theory is written in terms of a slowly-varying field, .
For , the ground states are 120 states – the three spins lie in a plane forming the sides of an equilateral triangle. All such states can be obtained from a global rotation operation performed on a reference 120 state, say in the plane. Thus, each ground state can be uniquely mapped to an rotation matrix. The ground state space is thus isomorphic to . As before, this forms a manifold, i.e., at any point in , there exists a three dimensional tangent space. Every ground state allows for three independent soft fluctuations. Naturally, a semi-classical field theory for systems, e.g., the triangular lattice antiferromagnet and the Majumdar Ghosh model, is formulated in terms of a matrix field, Dombre and Read (1989); Rao and Sen (1994).
Iii Parametrizing the classical ground state space
The case presents a non-trivial step forward from the and cases. We first enumerate the degrees of freedom. The total space is eight dimensional () as each spin has two independent parameters. The constraint of zero total spin, , is, in fact, three independent contraints – one for each component of the total spin. With eight degrees of freedom and three constraints, the ground state space is five-dimensional. Naively, we may expect the set of ground states to form a five-dimensional manifold. However, we show below that a much more nuanced picture emerges.
Several parametrizations of the ground state space, with minor variations, are available in literatureReimers et al. (1991); Moessner and Chalker (1998a); Tchernyshyov et al. (2002); Chalker (2011); Plat et al. (2015); Danu et al. (2016); Wan and Gingras (2016). Here, we present a parametrization that leads to an elegant semi-classical description.
A generic ground state can be described using five parameters, as shown in Fig. 1. To construct this state, we initially choose all four spins to lie in the plane, with and subtending an angle while their sum points along the axis. This constrains and to also subtend the same angle with their sum pointing along . We now introduce the second degree of freedom ; we rotate and by about the axis. At the same time, we rotate and by . This operation preserves and . This prescription leads to four unit vectors,
It is easy to see that these four vectors have unit length and add to zero. This configuration describes a generic ground state, modulo a global spin rotation.
Here, we constrain and . These ranges allow for a faithful representation of allowed ground states without double counting 111 For example, a configuration with is equivalent to one with modulo a -rotation about the Y axis.. The parameters and describe relative angles between spins. However, as seen from their specified ranges, they resemble a unit-vector with polar angle and azimuthal angle , which encodes the internal state of the cluster.
Finally, the four spins in the cluster are given by , incorporating three degrees of freedom into , an rotation matrix. It can be seen that all possible ground states are described by appropriate choices of these five parameters: , , and three Euler angles describing . In this sense, the ground state space is indeed five-dimensional. Naively, we may guess that the space is simply with being the space of the vector described by and . However, as we show below, the ground state space has non-trivial topology with non-manifold character.
Some representative members of the ground state space are: (i) a tetrahedral state with spins pointing towards the corners of a regular tetrahedron, (ii) a coplanar state with spins forming the sides of a square, and (iii) a collinear state with . As discussed in Ref. Danu et al., 2016, the ground states may be classified as non-coplanar, coplanar and collinear. In particular, all coplanar states can be accessed either by setting or by setting in the above parametrization. To have a collinear state, the four spins must form two pairs of parallel spins which are anti-aligned with respect to each other. This leads to three distinct collinear states (upto a global spin rotation) – this is the number of ways of forming two pairs from four objects. The three distinct collinear states correspond to , and . We will see below that these collinear states play a key role in the topology of the ground state space.
Iv Path integral description
To describe the cluster in the large- semi-classical limit, we develop a path integral formulation. We first parametrize the spins as
Here, represents an rotation matrix while ’s represent unit vectors determined by and , as defined in Eq. 2. The vector is a new parameter that encodes net magnetization. In other words, represents the deviation from the ground state space. Note that has three independent components. Together with , , and , this accounts for the eight degrees of freedom that determine the space of all allowed configurations. As we are interested in a low energy effective theory, we take to be small. We take the spin length to be large, , while assuming so that the deviation from the ground state space is . The factor of that comes with serves as a convenient bookkeeping tool. Below, we derive the path integral partition function as an expansion in powers of , keeping terms upto in the action and neglecting all terms with lower powers of .
We have introduced a matrix, , given by . This matrix is, in fact, the projector onto the plane perpendicular to . It guarantees that the vector is normalized to . In all calculations below, we take to be large and keep terms to in the action.
The magnetization of the cluster is now given by,
where . We note here that the magnetization vector is an angular momentum variable as it is a sum of spins. Upon quantization, its components should satisfy angular momentum commutation relations.
We follow the well-established semi-classical path integral formalism for spin- spinsAuerbach (1998). For our cluster of spins, the partition function is given by
where is the action given by,
The path integral is over the three components of which are integrated over the real line at every imaginary time slice. The functions in the integrand ensure normalization.
iv.1 Berry phase term
The first term in the action is the Berry phase with defined as . Essentially, is the vector potential of a magnetic monopole at the origin, with total flux . The integral, , is a geometric quantity, equal to times the area covered by on the surface of the unit sphere.
We evaluate the Berry phase to ,
where we have defined two vector quantities, and . Repeated indices are to be summed over. The vector has an identifiable form; it is the angular velocity of a rigid body whose orientation is described by the matrix . To arrive at Eq. 7, we have used two identities: and .
The vector depends purely on ’s, and thereby on and . Remarkably, uniformly vanishes for any choice of and . This is ultimately due to the symmetric parametrization of ’s in terms of the and . Following further simplifications (see Appendix. A), the Berry phase term comes out to be
Note that the Berry phase decouples into two terms: the first only depends on the parameters and , while the second contains the matrix variable . The second term also depends on and , via the matrix . Remarkably, the first term is precisely the Berry phase of a spin- spin. We had earlier discussed that and variables resemble a unit vector order parameter. Here, from the form of the Berry phase term, we see that this vector, in fact, behaves as a spin- spin.
iv.2 Energy term
The energy term in the action is simply . The energy scales as the square of , which represents deviation from the ground state space. Notably, the Hamiltonian also depends on the ground state parameters and which determine the matrix .
iv.3 Path integral measure
The partition function, in terms of the new variables, becomes
where the index denotes imaginary time slices. The action is given by
We have introduced three angles, , and , to parametrize the rotation matrix . The parameters and determine an axis of rotation, while specifies the angle of rotation about this axis. This parametrization leads to a convenient form for the path integral measureJones (1998).
The quantity denotes the Jacobian for the transformation given by Eq. 3. To , the Jacobian for a given time slice takes the form , see Appendix. B for a detailed derivation. We only keep terms in the Jacobian. Higher order corrections, upon exponentiation, give rise to subleading terms in the action. These terms can be ignored as we only keep terms up to in the action. We find
This form has an elegant interpretation as a measure for the path integral. It contains the group-invariant measureJones (1998): . We also identify a measure for the emergent vector defined by and : . The factor can be absorbed into the infinitesimal by redefining , which is the net moment of the cluster defined in Eq. 4. Note that is a matrix that depends on and . As is an rotation matrix, its determinant is unity.
The partition function becomes
Note that we have implicitly assumed an order of integration, viz., that will be integrated out before variables. This is necessary as the definition of involves the matrix which depends on and – we will see below that this dependence brings out the non-manifold character of the ground state space.
The path integral action is given by , where the vector is defined above. Remarkably, with our choice of order of integration, the partition function apparently decouples into two parts,
Both and are well known paradigmatic forms. is the partition function of a free spin- moment. This spin is ‘emergent’ – it is not a microscopic variable, but an encoding of the internal degrees of freedom, and . represents the partition function of a spherical top (a rigid rotor with the three principal moments equal) with moment of inertia . The matrix represents angular position, while represents angular momentum. Note that is the total moment of the cluster, with its components obeying angular momentum commutation relations. It represents ‘hard modes’ that can be integrated out to obtain a zero-temperature description.
This is the main result of this article: the system of four spins coupled by mutual antiferromagnetic interactions, in the semi-classical low-energy limit, decouples into a rigid rotor and an emergent free spin- spin!
V Comparison with conventional quantum analysis
To check for consistency of the mapping to a spin and a rotor, we compare the energy spectrum given by this mapping to that obtained from a conventional quantum analysis. Conventionally, finding the spectrum of the Hamiltonian reduces to a problem of angular momentum addition. The energy eigenvalues are simply with being the total spin quantum number.
In the semi-classical approach, we have a free spin- moment and a rigid rotor. The free spin does not contribute to energy as its Hamiltonian is zero. The rigid rotor with moment of inertia does contribute, with the spectrum known to be precisely with Casimir (1931). Thus, we obtain the same low-energy spectrum from the semi-classical as well as the fully quantum approach.
To further characterize the spectrum, we obtain the degeneracy of each level using both approaches. The obtained degeneracies are compared in Table. 1. The calculation of degeneracy using the conventional ‘full quantum’ approach is discussed in Appendix. C. As for the semi-classical approach, the degeneracy of the rigid rotor problemCasimir (1931) is well known to be . This is to be multiplied by on account of the free spin. While the free spin does not contribute to energy, it modifies the degeneracy with a multiplicative factor.
As seen in Table. 1, both approaches give the same degeneracy for the ground state. However, for excited states, the two approaches agree to . As the semi-classical limit is strictly justified for large spins, we conclude that the degeneracies match. Nevertheless, the discrepancy is significant.
It shows that the quadrumer problem () is markedly different from the dimer and the trimer. For both and , the appropriate semiclassical description accurately captures the degeneracies in the spectrum.
For , conventional quantum analysis gives eigenvalues with , with level degeneracy . From the semiclassical point of view, this problem maps to a particle on a sphere with the same form of the eigenenergies and degeneracies, except that runs from .
For , the conventional quantum approach gives eigenvalues
where . The low lying states, with , have degeneracy .
Semiclassically, this problem maps to a spherical top rigid rotor. Once again, this gives the same expressions for the eigenenergies and degeneracies. However, runs from . In both cases, the low energy spectrum () is accurately captured by the semiclassical mapping. However, for , we find subleading discrepancies in the degeneracy. This could be due to two reasons:
(a) Order by disorder: For , all classical ground states are symmetry-related. As a consequence, quantum fluctuations, arising from terms with lower powers of in the action, cannot lift their degeneracy. However, for , ground states with differing values of are not related by any symmetry. This allows quantum fluctuations to have a stronger role. In principle, higher order (lower power in ) corrections can induce a preference for certain values of via the well known phenomenon of ‘order by disorder’. Such corrections could alter the form of the action, for instance, by coupling the rotor and spin degrees of freedom. This could give rise to the observed corrections in the level degeneracies.
A rigorous derivation of corrections is beyond the scope of this study. Nevertheless, we make the following observations. With regard to the ground state degeneracy, we find perfect agreement between the semiclassical and full quantum results despite the possibility of order by disorder effects. In the semiclassical picture, the ground state degeneracy arises from the emergent free spin while the rotor is in its non-degenerate ground state. This suggests that quantum fluctuations do not play a role when the rotor is in its ground state, leaving the free moment intact. When the rotor is excited, quantum fluctuations could couple it to the free spin, leading to the observed corrections in the degeneracies.
(b) Imperfect semi-classical mapping: For and , the semiclassical large- path integral precisely reduces to a particle on a sphere and a rigid rotor respectively. However, for , the mapping to a rotor and a free spin is approximate due to a subtle effect that arises from the non-trivial topology of the ground state space. This is discussed in detail in the next section. As the mapping itself is only approximate, we can have discrepancies between the semi-classical and quantum spectra. This could also lead to the observed discrepancies in degeneracies.
|State||Energy||Degeneracy: Full quantum||Degeneracy: Semi-classical|
Vi Singularities in the ground state space
There is a non-trivial subtlety in the identification of in Eq. 13 as the path integral of a rigid rotor. The rotor angular momentum is given by . The matrix here depends on the variables and . For generic values of and , has three non-zero eigenvalues. This leads to three independent components of , as required to describe the angular momentum of a rigid rotor. However, the matrix becomes singular at three isolated values of at which the spin configuration is collinear. At these points, one of the eigenvalues of vanishes, leaving us with only two degrees of freedom in ; it can no longer be identified as the angular momentum of a rigid rotor. Strictly speaking, this forbids the identification of in Eq. 13 with a rigid rotor.
This effect originates from the parametrization in Eq. 3. Suppose all four unit vectors, ’s, are collinear and aligned along . In this case, the -component of becomes redundant in Eq. 3. Note that it is the projection of onto the plane perpendicular to that enters . With all spins parallel to , we can assign any value to without changing any of the spins. This can be understood by visualizing all possible small fluctuations about a collinear state. The system can only develop a non-zero magnetization in two directions, while preserving the length of each spin. These form the two independent components of .
While our semiclassical mapping fails at collinear states, this is nevertheless a minor effect as the number of such ground states is very small. In fact, collinear ground states form a set of measure zero as they occur for three isolated values of . By neglecting this set in the integration space of Eq. 13, we can persist with our identification of the system with a spin- spin and a rigid rotor. That this is a good approximation can be seen from the excellent agreement in the spectrum as shown in Tab. 1.
Vii Soft fluctuations
To understand the topology of the ground state space, it is instructive to look at ‘soft’ fluctuations. Given a ground state configuration of the four spins in the cluster , we consider small deviations of the spins, with . As we have eight degrees of freedom in total (2 per spin), we always have eight independent fluctuation modes labeled as , with . We identify independent modes using the condition .
These small fluctuation modes can be naturally classified as ‘hard’ and ‘soft’. Hard modes take us out of the ground state space – they induce a net magnetization in the cluster, i.e., . In contrast, soft modes preserve the zero-total-spin condition and keep us within the ground state space.
In Fig. 2 and Fig. 3, we pictorially depict the soft modes around (a) a coplanar state and (b) a collinear state respectively. There are five soft modes about the coplanar state corresponding to varying , varying and three independent rotations. Indeed, this is true of all non-collinear ground states; each state allows for five soft modes which can be understood in the same way. However, collinear states allow for six soft modes as shown in Fig. 3; see App. E for explicit expressions. These correspond to two independent rotations and four independent deformations.
Mathematically, the set of soft modes describes the tangent space around a given element of the ground state space. For the ground state space to be a manifold, the tangent space at every point must have the same dimensionality. To be precise, the neighbourhood of every point must be isomorphic to , where is the dimension of the space. Here, we have an extra sixth dimension whenever the state is collinear. We assert that this demonstrates a deep mathematical property, viz., the non-manifold character of the ground state space. In Appendix D, we provide a rigorous proof that the ground state space, with collinear configurations removed, forms a five dimensional manifold. We show this using the implicit function theorem which provides a sufficient condition for manifold character.
To illustrate the singularities that occur at collinear states, we discuss a ‘spin-wave approach’ in Appendix F. Assuming that the cluster always remains in the vicinity of a collinear state, we develop a path integral description. With only two rotational degrees of freedom, the system maps to a rigid rod rather than a rigid body.
We have shown that the semi-classical description succeeds in describing the low-energy spectrum of the quadrumer. We can now reinterpret thermodynamic properties as arising from the semi-classically obtained free spin- moment and rigid rotor. From this point of view, the partition function of the quadrumer is given by
At zero temperature, the quadrumer has non-zero entropy, , arising from the degeneracy of the free spin- spin. This non-vanishing entropy can be seen in specific heat measurements on candidate materials.
To find low temperature properties, we may retain the first few values in Eq. 14. For example, retaining only , the entropy can be approximated as , where the first term is the free spin contribution and the rest is the rigid rotor contribution. Similarly, the specific heat at low temperatures comes out to be . The free spin does not contribute to specific heat as it does not contribute to energy.
Our formalism also allows us to directly calculate magnetic susceptibility. Assuming a small magnetic field of strength along , it enters the Hamiltonian as a new term, , where is the component of the magnetization. The partition function changes to , where the eigenvalues of are given by . The low temperature susceptibility comes out to be , in agreement with known results in the large- limitGarcía-Adeva and Huber (2000, 2001). The free spin does not contribute to the susceptibility as well. As the emergent spin only signifies internal coordinates (relative angles between spins), it does not couple to an external magnetic field.
Ix Summary and discussion
We have presented a path integral description for the quantum spin quadrumer. The ground state space of this system has non-trivial topology, reflected in the differing number of soft fluctuations around collinear and non-collinear ground states. This provides a simple and an experimentally realizable example of dynamics on a non-manifold space. Earlier studies of the quadrumer with Dzyaloshinskii-Moriya couplings have discussed one-dimensional ground state spaces with non-manifold characterElhajal et al. (2005); Canals et al. (2008). Our study with purely Heisenberg couplings brings out a larger five-dimensional non-manifold space.
We have shown that the quadrumer decouples into a free spin- spin and a rigid rotor. This provides a beautiful example of ‘emergence’ – the internal spin configuration manifests as a spin- order parameter in the low energy description. This spin character does not appear directly in the microscopic description; it cannot be deduced from a conventional study of the cluster and its dynamics. Indeed, although the quadrumer has been extensively studied, this property has not been brought out so far. An early indication of an emergent spin may be found in the equations for semiclassical dynamics derived in Ref.[Henley and Zhang (1998)].
We have considered a spin- cluster in which every spin is coupled to every other spin. This structure arises naturally in tetrahedral molecules/clusters which have four spins at the corners of a regular tetrahedron. Several experimental realizations are known to exist. The canonical examples are transition metal tetrahedraJian et al. (2006). Notably, NiMo realizes a near-perfect tetrahedron of momentsMüller et al. (2000); Nehrkorn et al. (2010). A tetramer is also realized in Zaharko et al. (2008). More recently, several lanthanide-based compounds have been synthesized. A Dy-based molecular magnetLin et al. (2011) realizes a tetrahedron, but with Ising-like anisotropy. These molecular magnets closely resemble our problem. Most of their experimental properties can be explained with a conventional quantum analysis of four spins (e.g., as in Ref. Schnack and Luban, 2000). Our work reinterprets these as emanating from an emergent rigid rotor and a free spin.
An exciting development is the synthesis of ‘breathing’ pyrochlore magnets with weakly coupled tetrahedra. An example from the lanthanide family is BaYbZnO in which each Yb atom forms a pseudospin- momentKimura et al. (2014); Haku et al. (2016); Rau et al. (2016); Park et al. (2016). The inter-tetrahedron coupling is so weak that the tetrahedra are essentially isolated; this is reflected in the neutron scattering intensity showing flat momentum-independent modes. Our analysis may not be directly applicable here, due to the presence of strong Dzyaloshinskii-Moriya interactions. Likewise, LiGaCrO and LiInCrO form breathing pyrochlores with Cr moments. They vary in the degree of breathing and show intriguing ordering propertiesOkamoto et al. (2013). Theoretical proposals have been put forward to explain their orderingKoga and Kawakami (2001); Tsunetsugu (2001a, 2017). Our results will help to develop a field theoretic description for these systems. The additional couplings in these materials such as anisotropies, dipolar interactions, etc., will modify our semi-classical picture and couple the rotor and the spin fields.
The real promise of our study is that it provides a starting point for semi-classical field theories. The cluster is the building block of pyrochlore magnets, the checkerboard latticeCanals (2002); Brenig and Honecker (2002); Bishop et al. (2012); Moura and Pereira (2017); Fouet et al. (2003); Hermele et al. (2004); Bernier et al. (2004), the four-leg tubePlat et al. (2015), and the square J-J-J antiferromagnetDanu et al. (2016). In particular, the Heisenberg pyrochlore magnet is of great interest as a canonical model of frustration with realizations in spinel compoundsReimers et al. (1991); Moessner and Chalker (1998b); Tsunetsugu (2002); Yamashita and Ueda (2000); Maestro and Gingras (2004); Wills et al. (2006); Henley (2005, 2006); Hizi and Henley (2007); Conlon and Chalker (2009); Lapa and Henley (2012); Berg et al. (2003). Its ground state and ordering properties continue to be debatedCanals and Lacroix (1998); Tsunetsugu (2001b); Huang et al. (2016). With new experimental realizations emergingHigo et al. (2017), we hope that a suitable field theoretic approach will throw light on this model and its intriguing properties.
Acknowledgements.We thank Pralay Chatterjee and Arghya Mondal for many helpful discussions about the mathematics of manifolds. We thank Karlo Penc, Srivatsa N. S., B. Sathiapalan and Yuan Wan for useful discussions and comments.
Appendix A Berry Phase
where is a matrix that encodes the spin. The coordinates of the spin, in the reference frame, are denoted by polar angle and azimuthal angle . This moment is rotated by the rotation matrix . The matrix is defined so as to transform to , followed by a global rotation to give . This operation is given below.
Here, , where is the unitary transformation corresponding to rotation and ’s are Pauli matrices.
It can be easily checked that . The second term in Eq. A becomes . So far, we calculated the Berry phase term for a single spin. Adding the contributions from 4 spins, we get
This follows from the ground state condition, . The first term in Eq. A can be shown to be .
this term is again for a single spin with and being the polar and azimuthal angles. We now add contributions from four spins with the corresponding polar and azimuthal angles: and :
where and are the parameters used to describe a generic ground state in Fig. 1.
Appendix B Jacobian derivation
At any given imaginary time slice, the path integral measure takes the form
Let us rewrite the spin parametrization of Eq. 3 in the following way,
where . This vector has polar angle and azimuthal angle , where and are the angles that parametrize the ground state space as shown in Fig. 1. As described in the main text, are taken to be normalized to . We have introduced three new scalar variables, and , in order to have twelve new variables. As we now have the same number of old and new variables, we can evaluate the Jacobian of the transformation. We have used and . In terms of the new variables, the integral 21 becomes
Here and parametrize the rotation matrix . define an axis of rotation, while represents the rotation angle. After finding the Jacobian , we can integrate out and by replacing in the Jacobian. The Jacobian matrix can be written in the following form,
In the above matrix, each element represents three consecutive entries along the column, corresponding to the three components of the vector. Using the transformation relations in Eqs. 22, we write the matrix in Eq. 24 as