Lattice Gauge Theories and Spin Models
The Wegner gauge theory- Ising spin model duality in dimensions is revisited and derived through a series of canonical transformations. The Kramers-Wannier duality is similarly obtained. The Wegner gauge-spin duality is directly generalized to SU(N) lattice gauge theory in dimensions to obtain the SU(N) spin model in terms of the SU(N) magnetic fields and their conjugate SU(N) electric scalar potentials. The exact & complete solutions of the Gauss law constraints in terms of the corresponding spin or dual potential operators are given. The gauge-spin duality naturally leads to a new gauge invariant magnetic disorder operator for SU(N) lattice gauge theory which produces a magnetic vortex on the plaquette. A variational ground state of the SU(2) spin model with nearest neighbor interactions is constructed to analyze SU(2) gauge theory.
In 1971 Franz Wegner, using duality transformations, showed that in two space dimensions lattice gauge theory can be exactly mapped into a Ising model describing spin half magnets wegner (). This is the earliest and the simplest example of the intriguing gauge-spin duality. Wegner’s work, in turn, was strongly motivated by the self-duality of planar Ising model discovered by Kramers and Wannier 30 years earlier km (). Such alternative dual descriptions have been extensively discussed in the past as they are useful to understand theories and their phases at a deeper level thmds (); hooft (); banks (); fradsuss (); kogrev (); savit (); horn (); dualsup (); baal (); sharatram (); manu (). In the context of QCD, the duality transformations have been studied to understand color confinement via dual superconductivity thmds (); hooft (); dualsup (); baal () and to extract topological degrees of freedom banks (); savit (); sharatram (); manu (). They may relate the important and relevant degrees of freedoms at high and low energies providing a better understanding of non-perturbative issues in low energy QCD. The duality ideas in the Hamiltonian framework are also relevant for the recent quest to build quantum simulators for abelian and non-abelian lattice gauge theories using cold atoms in optical lattices reznik (). In these cold atom experiments, the (dual) spin description of SU(N) lattice gauge theory should be useful as there are no exotic, quasi-local Gauss law constraints to be implemented at every lattice site coldatomgl (). The duality methods and the resulting spin models, without redundant local gauge degrees of freedom, can also provide more efficient tensor network, variational ansatzes for the low energy states of SU(N) lattice gauge theories tn ().
In this work, we start with a brief overview of Kramers-Wannier and Wegner dualities within the Hamiltonian framework. We show that these old, well-established spin-spin and gauge-spin dualities can be constructively obtained through a series of iterative canonical transformations. These canonical transformation techniques are easily generalized to SU(N) lattice gauge theory to obtain the equivalent dual SU(N) spin model without any local gauge degrees of freedom. Thus using canonical transformations we are able to treat spin, abelian and non-abelian dualities on the same footing. In dimensions the spin operators in the dual spin models are the scalar magnetic fields and their conjugate electric scalar potentials respectively. These spin operators solve the and Gauss laws.
The Kramers-Wannier and Wegner dualities naturally lead to construction of disorder operators and order-disorder algebras kogrev (); fradsuss (); horn (). In both cases the disorder operators are simply the dual spin operators creating kinks and magnetic vortices on plaquettes respectively. Note that these creation operators are highly non-local in terms of the operators of the original Ising model or gauge theory. Therefore without duality transformations they are difficult to guess. We generalize these elementary duality ideas to non-abelian gauge theories after briefly recapitulating them in the simpler contexts mentioned above. In particular, we exploit SU(N) dual spin operators to construct a new gauge invariant disorder operator for SU(N) lattice gauge theory. Further, like in lattice gauge theory, the non-abelian order-disorder algebra involving SU(N) Wilson loops and SU(N) disorder operators is worked out. The interesting role of non-localities in non-abelian duality resulting in the solutions of SU(N) Gauss laws and production of local vortices is discussed. For the sake of clarity and continuity, the SU(N) gauge theory results will always be discussed in the background of the corresponding Ising model, gauge theory results. The similar features amongst them are emphasized and the differences are also pointed out.
In the context of gauge theory in dimensions, the two essential features of Wegner duality wegner () are
it eliminates all unphysical gauge degrees of freedom mapping it into spin model with a global symmetry. There are no Gauss law constraints in the dual spin model.
it maps the interacting (non-interacting) terms in the lattice gauge theory Hamiltonian into non-interacting (interacting) terms in the spin model Hamiltonian resulting in the inversion of the coupling constant.
It is important to note that the above gauge-spin duality is through the loop description of lattice gauge theory. The original Hamiltonian is written in terms of fundamental electric fields and their conjugate magnetic vector potentials. The magnetic fields are not fundamental and obtained from magnetic vector potentials. On the other hand, in the dual Ising model the fundamental spin degrees of freedom are the magnetic fields and their conjugate electric scalar potentials. Now the electric fields are not fundamental and are obtained from the electric scalar potentials. We arrive at this dual spin description through a series of canonical transformations. They convert the initial electric fields, magnetic vector potentials into the following two mutually independent physical & unphysical classes of operators:
spin or plaquette loop operators: representing the physical magnetic fields and their conjugate electric scalar potentials over the plaquettes (see Figure 5-a),
string operators: representing the electric fields and the flux operators of the unphysical string degrees of freedom. These strings isolate all gauge degrees of freedom (see Figure 5-b).
The interactions of spins in the first set are described by Ising model. The corresponding physical Hilbert space is denoted by . The second complimentary set, containing string operators, represents all possible redundant gauge degrees of freedom. We show that the Gauss law constraints freeze all strings leading to the Wegner gauge-spin duality within . Further, the electric scalar potentials are shown to be the solutions of the Gauss law constraints. Note that no gauge fixing is required to obtain the dual description. We show that the above duality features also remain valid when these canonical/duality transformations are generalized to SU(N) lattice gauge theory. As in the case, the SU(N) Kogut-Susskind link operators get transformed into the physical spin/loop and unphysical string operators. Again the SU(N) strings are frozen and the dual SU(N) spin operators provide all solutions of SU(N) Gauss law constraints. In fact, these SU(N) canonical transformations have been discussed earlier in the context of loop formulation of SU(N) lattice gauge theories msplb (); msprd (). The motivation was to address the issue of redundancies of Wilson loops or equivalently solve the SU(N) Mandelstam constraints. We now exploit them in the context of non-abelian duality.
The plan of the paper is as follows. In section II, we discuss the canonical transformation techniques to systematically obtain Kramers-Wannier, Wegner and then SU(N) dualities. The compact U(1) lattice gauge theory duality can also be easily obtained from the SU(N) duality by ignoring the non-abelian, non-local terms. In section II.1, we start with the simplest Kramers-Wannier duality in the Ising model. The order-disorder operators, their algebras and creation, annihilation of kinks are briefly discussed for the sake of uniformity and later comparisons km (); kadanoff (); fradsuss (); kogrev (); savit (); horn (). In section II.1.1, we extend these canonical transformations to discuss Wegner duality in dimensions. We again obtain the old and well established results fradsuss (); kogrev (); savit (); wegner (); horn () with canonical transformations as the new ingredients. The gauge theory order-disorder operators, their algebras and magnetic vortices are briefly summarized. In section II.3, the canonical transformations are generalized to SU(N) lattice gauge theories leading to a SU(N) spin model. As mentioned before the SU(N) discussions are parallel to the discussions for clarity. A comparative summary of gauge-spin and SU(N) gauge-spin operators is given in Table 1. At the end of section II.3, we construct the new SU(N) disorder operator. The special case of ’t Hooft disorder operator is discussed. The Wilson-’t Hooft loop algebra is derived. The last section III is devoted to variational analyses of the truncated SU(N) spin model. A simple ‘single spin’ variational ground state of the dual SU(N) spin model is constructed. The Wilson loop in this ground state is shown to have area law behavior. In Appendix A, we discuss explicit constructions of Wegner duality through canonical transformations. In Appendix B, we discuss the highly restrictive, non-local structure of SU(N) duality transformations. We explicitly show the non-trivial cancellations of infinite number of terms required to solve the SU(N) Gauss law constraints by the dual SU(N) spin operators. In part 2 of Appendix B, another set of non-trivial cancellations are shown to hold for the SU(N) disorder operator to have a local physical action in the original Kogut-Susskind formulation. In the case of much simpler Wegner duality such cancellations are obvious. In Appendix C, we show that the ‘single spin’ variational state satisfies Wilson’s area law. In Appendix D, the expectation value of the truncated dual spin Hamiltonian is computed in the variational ground state. The expectation value of the non-local part of the spin Hamiltonian in the above disordered variational ground state is shown to vanish. This shows that non-local terms appearing with higher powers of coupling may be treated perturbatively as for the continuum.
Throughout this work, we use Hamiltonian formulation of lattice gauge theories ks () with open boundary conditions. We work in two space dimensions on a finite lattice with sites, links, plaquettes satisfying: . A lattice site is denoted by or with . The links are denoted by or with . The plaquettes are denoted by the co-ordinates of their upper right corner and sometimes by etc.. Any conjugate pair operator satisfying the corresponding conjugate canonical commutation relations will be denoted by . In Ising model, Ising gauge theory, they are the Pauli spin operators , . In SU(N) lattice gauge theory, they are the Kogut-Susskind electric fields, link operators . Similarly, the canonically conjugate SU(N) spin, string operators in the dual spin models are defined in the text.
Ii Duality and Canonical Transformations
ii.1 Kramers-Wannier duality
Kramers Wannier duality was the first and the simplest duality, apart from electromagnetism, to be constructed. As a prelude to the construction of dualities in and SU(N) lattice gauge theories, we apply canonical transformations to dimensional Ising model to get the Kramers-Wannier duality. The Ising Hamiltonian in one space dimension is in terms of the canonically conjugate operators at every lattice site satisfying,
The Hamiltonian is
The Kramers-Wannier duality is obtained by the following iterative canonical transformations along a line with and :
The above canonical transformations iteratively replace the conjugate pair or equivalently by a new conjugate pair . These new pairs are mutually independent and also satisfy the canonical relations (1). Unlike gauge theories (to be discussed in the next section), there are no spurious (string) degrees of freedom. This process is graphically illustrated in Figure 1. The relations (3) lead to,
The relations (4) can be easily inverted to give with the convention . The Ising model Hamiltonian can now be rewritten in its self-dual form in terms of the new dual conjugate pairs :
This is the famous Kramers-Wannier self duality. As expected, duality has interchanged the interacting and non interacting parts of the Hamiltonian on going from the to the dual variables. In other words, duality/canonical transformations (3) map strong coupling region to the weak coupling region and vice versa.
ii.1.1 Ising disorder operator
In Ising model the magnetization operator, is the order operator as its expectation value measures the degree of order of the variables. It is zero for and non-zero for . This implies that the phase spontaneously breaks the global symmetry: . On the other hand, the dual Hamiltonian (5) implies that it is natural to define as a disorder operator kogrev (); fradsuss (); horn (). The vacuum expectation value is the disorder parameter. We also note that the disorder operator acting on a completely ordered state (all or ), flips all spins at and creates a kink at . The resulting kink state is orthogonal to the original ordered state and the expectation value of the disorder operator in an ordered state vanishes:
ii.2 Wegner duality and Spin Model
and lattice gauge theories are the simplest theories with gauge structure and many rich features. Due to their enormous simplicity compared to or non-abelian lattice gauge theories and the presence of a confining phase, they have been used as a simple theoretical laboratory to test various confinement ideas horn (). They also provide an explicit realization of the Wilson-’t Hoofts algebra of order and disorder operators characterizing different possible phases of the SU(N) gauge theories hooft (); horn (). In 1964, Schultz, Mattis and Lieb showed that the two-dimensional Ising model is equivalent to a system of locally coupled fermions schultz (). This result was later extended to lattice gauge theory which also allows an equivalent description in terms of locally interacting fermions fradkinf (). These are old and well known results. In the recent past, lattice gauge theories have been useful to understand quantum spin models fradkinb (), quantum computations kitaev (), tensor network or matrix product states taka () and their topological properties wen (), cold atom simulations reznik () and entanglement entropy polik (). In view of these wide applications, the lattice gauge theories and associated duality transformations are important in their own right.
The lattice gauge theory involves conjugate spin operators on the link . The anti-commutation relations amongst these conjugate pairs on every link are
They further satisfy: . In order to maintain a 1-1 correspondence with SU(N) lattice gauge theory (discussed in the next section), it is convenient to identify the conjugate pairs with electric field, and vector potential, as:
Above and . A basis of the two dimensional Hilbert space on each link is chosen to be the eigenstates of with eigenvalue with acting as a spin flip operator:
The lattice gauge theory Hamiltonian is given by
In (11) represents the product of operators along the four links of a plaquette. The sum over and in (11) are the sums over all links and plaquettes respectively. The parameter is the gauge theory coupling constant. The first term and the second term in (11) represent the electric and magnetic field operators respectively. The electric field operator is fundamental while the latter is a composite of the four magnetic vector potential operators along a plaquette. After a series of canonical transformations, the above characterization of electric, magnetic field will be reversed. More explicitly, the dynamics will be described by the Hamiltonian (11) rewritten in terms of the fundamental magnetic field (the second term) and the electric field operator (the first term) will be composite of the dual electric scalar potentials (see (26a) and (II.2.3)). The same feature will be repeated in the SU(N) case discussed in the next section.
The Hamiltonian (11) remains invariant if all 4 spins attached to the 4 links emanating from a site are flipped simultaneously. This symmetry operation is implemented by the Gauss law operator :
at lattice site . In (12), represents the product over 4 links (denoted by ) which share the lattice site in two space dimensions. The gauge transformations are
Thus, under a gauge transformation at site , the 4 link flux operator on the 4 links sharing the lattice site change sign. All other remain invariant. The physical Hilbert space consists of the states satisfying the Gauss law constraints:
In other words, are unit operators within the physical Hilbert space . All operator identities valid only within are expressed by sign. We now canonically transform this simplest gauge theory with constraints (12) at every lattice site into spin model without any constraints as shown in Figure 3. To keep the discussion simple, we start with a single plaquette OABC shown in Fig. 4-a before dealing with the entire lattice. As the canonical transformations are iterative in nature, this simple example contains all the essential ingredients required to understand the finite lattice case. The four links OA, AB, BC, CO will be denoted by respectively. In this simplest case there are four gauge transformation or equivalently Gauss law operators (12) at each of the four corners O, A, B and C:
Note that these Gauss law operators satisfy a trivial operator identity:
The above identity states the obvious result that a simultaneous flippings at all 4 sites has no effect. This is because of the abelian nature of the gauge group. We now start with the four initial conjugate pairs on links and :
Using canonical transformations we define four new but equivalent conjugate pairs. The first three string conjugate pairs:
describe the collective excitations on the links and shown in Figures 4-b,a,c respectively. The remaining collective excitations over the plaquette or the loop are described by
and shown in Figure 4-c. As a consequence of the three mutually independent Gauss law constraints and , the three string electric fields are frozen to the value . Therefore there is no dynamics associated with the three strings. In other words, string degrees of freedom completely decouple from . We are thus left with the final physical spin operators which are explicitly gauge invariant. These duality transformations from gauge variant link operators to gauge invariant spin or loop operators are shown in Figure 3. To demonstrate the above results, we start with the initial link operators and as shown in Fig. (4)-a.
As was done in dimensional Ising model, we glue them using canonical transformations as follows:
The canonical transformations (18) are illustrated in Fig. 4-a. After the transformations, the two new but equivalent canonical sets , are attached to the links and respectively. They satisfy the same commutation relations as the original operators (8):
One can easily check: Further, note that the two conjugate pairs and are also mutually independent as they commute with each other. As an example, . The new conjugate pair is frozen due to the Gauss law at B: in . We now repeat (18) with replaced by respectively to define new conjugate operators and attached to the links and respectively:
As before, the new conjugate pair becomes unphysical as in . The last canonical transformations involve gluing the conjugate pairs with to define the dual and gauge invariant plaquette variables , with :
To summarize, the three canonical transformations (18), (20), (21a) and (21b) transform the initial four conjugate sets ,,, attached to the links to four new and equivalent canonical sets and attached to the links and the plaquette respectively. The advantage of the new sets is that all the three independent Gauss law constraints at and are automatically solved. They freeze the three strings leaving us only with the physical spin or plaquette loop conjugate operators . The defining canonical relations (18), (20), (21a) and (21b) can also be inverted. The inverse transformations from the new spin flux operators to link flux operators are
Similarly, the initial conjugate electric field operators on the links are
Thus the complete set of gauge-spin duality relations over a plaquette and their inverses are given in (18), (20), (21a), (21b) and (22), (II.2) respectively. Note that the Gauss law constraint at the origin does not play any role as . The total number of degrees of freedom also match. The initial gauge theory had 4 spins with 3 Gauss law constraints. In the final dual spin model the 3 gauge non-invariant strings take care of the 3 Gauss law constraints leaving us with the single gauge invariant spin described by on the plaquette . The single plaquette lattice gauge theory Hamiltonian (11) can now be rewritten in terms of the new gauge invariant spins as:
Having discussed the essential ideas, we now directly write down the general gauge-spin duality or canonical relations over the entire lattice. The details of these iterative canonical transformations (analogous to (18), (20), (21a) and (21b)) are given in Appendix A. Note that there are initial spins (one on every link) with Gauss law constraints (one at every site) satisfying the identity:
The above identity again states that simultaneous flipping of all spins around every lattice site is an identity operator because each spin is flipped twice. As mentioned earlier, it is a property of all abelian gauge theories which reduces the number of Gauss law constraints from to . In the non-abelian SU(N) case, discussed in the next section, there is no such reduction. The global SU(N) gauge transformations, corresponding to the extra Gauss law constraints at the origin , need to be fixed by hand to get the correct number of physical degrees of freedom (see section II.3.3). After canonical transformations in lattice gauge theory, there are (a) physical plaquette spins (analogous to in the single plaquette case) shown in Figure 5-a and (b) stringy spins (analogous to and in the single plaquette case) as every lattice site away from the origin can be attached to a unique string. This is shown in Figure 5-b. The degrees of freedom before and after the canonical transformations match as . All strings decouple because of the Gauss law constraints. The algebraic details of these transformations leading to freezing of all strings are worked out in detail in Appendix A.
From now onward the physical plaquette spin/loop operators are labelled by the top right corners of the corresponding plaquettes as shown in Figure 5-a). The vertical (horizontal) stringy spin operators are labelled by the top (right) end points of the corresponding links as shown in Figure 5-b. The same notation will be used to label the dual SU(N) operators in section II.3.
ii.2.1 Physical sector and dual potentials
The final duality relations between the initial conjugate sets on every lattice link and the final physical conjugate loop operators ; are (see Appendix A)
In (26a) we have defined and . The relations (26a) and (26b) are the extension of the single plaquette relations (21a) to the entire lattice. They are illustrated in Figure 6-a. The canonical commutation relations are
Further, . The canonical transformations (26a) are important as they define the magnetic field operators and its conjugate as a new dual fundamental operators. The electric field is derived from the electric scalar potentials. This should be contrasted with the original description where electric fields were fundamental and the magnetic fields were derived from the magnetic magnetic vector potentials as .
ii.2.2 Unphysical sector and string operators
The unphysical string conjugate pair operators are (see Appendix A)
The relations (28a) and (28b) are illustrated in Figure 6-b and Figure 6-c respectively. It is easy to see that in the full gauge theory Hilbert space and different string operators located at different lattice sites commute with each others. Further, one can check that all strings and plaquette operators are mutually independent and commute with each other:
ii.2.3 Inverse relations
The inverse relations for the flux operators over the entire lattice are
On the other hand, the conjugate electric field operators are
ii.2.4 Gauss laws & solutions
It is easy to see that the Gauss law constraints are automatically satisfied by the dual spin operators as shown in Figure 7-a,b. We write the electric fields around a site in terms of the electric scalar potentials:
The above duality property also generalizes to the SU(N) case. The dual SU(N) spin operators or potentials are the solutions of local SU(N) Gauss laws at all the sites except origin. However, unlike the trivial cancellations above, the non-abelian cancellations are highly nontrivial and are worked out in detail in Appendix