Classical triangular lattice antiferromagnetic Ising model as a free-fermion/superconductor system

We present a treatment of the triangular lattice antiferromagnetic Ising model (TAFIM) based on a small number of elementary ideas common to statistical and solid-state physics. The TAFIM is represented as a reduced BCS model in one space, one (imaginary) time dimension. The representation is approximate for nonzero temperature, but allows quick derivation of asymptotically exact thermodynamic functions, and the divergence of the spin–spin correlation length. The fermionic representation is exact at zero temperature. We demonstrate the existence of a two-dimensional continuum of zero-temperature equilibrium macrostates characterized by satisfied bond fractions of the three different orientations, and calculate their entropy densities.

exact thermodynamic functions, and the divergence of the spin-spin correlation length. The fermionic representation is exact at zero temperature. We demonstrate the existence of a two-dimensional continuum of zero-temperature equilibrium macrostates characterized by satisfied bond fractions of the three different orientations, and calculate their entropy densities.
Supplementary material for this article is available online Keywords: antiferromagnetic Ising model, frustration, fermions, ground state entropy (Some figures may appear in colour only in the online journal)

Introduction
Classical Ising spin models with nearest-neighbor interactions are archetypes of many fundamental phenomena of statistical mechanics. For instance, the model on a bipartite lattice with ferromagnetic (FM) couplings exemplifies order-disorder transitions, spontaneous symmetry breaking [1], and nonclassical critical phenomena [2], The antiferromagnetic (AFM) model on a triangular lattice [3,4], however, behaves completely differently. Even at zero temperature, it fails to order, because there is no spin configuration that satisfies all three bonds around even a single elementary triangle. As the simplest model with this property, the triangular lattice antiferromagnetic Ising model (TAFIM) is the archetype of geometric frustration, the presence of incompatible but equally strong elementary interactions. Frustration occurs in an enormous range of systems, from water ice [5,6] to spin systems [7][8][9][10][11][12], artificial spin ice [13][14][15], colloidal assemblies [16,17], Coulomb liquids [18], lattice gases [19], ferroelectrics [20], coupled lasers [21], and self-assembled lattices of microscopic chemical reactors [22]. For this reason, frustration is of interest to the broad spectrum of condensed matter physicists, and a treatment of the TAFIM based on basic concepts common to statistical and solid-state physics is desirable. This paper aims to provide such a treatment. The term 'frustration' was coined by P. W. Anderson according to Toulouse [7]. The present work might be called 'poor man's TAFIM', in homage [23].
The TAFIM has been studied for decades. Integral expressions for exact thermodynamic functions were determined in 1950 [3,4,24]. Wannier determined [25] the residual entropy. Spin correlation functions were studied intensively by Stephenson [26][27][28][29]. The representation of zero-temperature TAFIM configurations by strings goes back at least to the equivalence with solid-on-solid models pointed out by Blöte and Hilhorst [30] in connection with striped phases [31], which were a focus of interest in the early 1980s (see den Nijs [32] for a review).
What is offered here against that history is: a special focus on zero-temperature equilibrium macrostates, bringing out an astonishing richness which has not been explicitly discussed in the literature, and an especially simple and physically appealing approach, based on a widely familiar fundamental paradigm. The cost of this simplicity is an only approximate treatment of nonzero temperature. The approximation is nevertheless well-motivated and turns out to be surprisingly good, even up to infinite temperature. Using a mapping of bond configurations to trajectories of a system of fermions in one space, one (imaginary) time dimension, bulk thermodynamic properties (entropy and energy) are accessible through ground-state energies of the fermionic systems, and spin-spin correlation functions through fermion density fluctuations.
At T = 0, the TAFIM is equivalent, in a sense, to a simple Fermi gas. Still, it has surprising aspects. Traditional approaches are well-suited only to taking the zero-temperature limit after the thermodynamic limit. Consequently, they find only a single zero-temperature macrostate, albeit with a nonvanishing entropy density. Our approach easily takes the thermodynamic limit at T = 0, and finds a continuum of distinct macrostates. At low temperatures, and below some crossover size, this can be the more accurate description.
For T > 0, the Hamiltonian of our equivalent fermion system develops pairing terms, and is thus a superconductor. Schultz et al [33] observed the analogy with Bardeen-Cooper-Schrieffer (BCS) theory in their famous work on the square-lattice Ising model. It is much more forceful in the present case because of both the simplicity of the transformation to fermionic representation and the occurrence of the normal state at zero temperature. Perhaps some non-exactlysolvable frustrated models can be handled similarly.
We consider the essence of TAFIMs to be that every elementary triangle is frustrated, rather than that every bond is AFM, so with all possible choices of A/FM bond patterns with that property. Dealing with larger class of models turns out to be hardly any more work, once one describes configurations in terms of un/satisfied bonds rather than spins. All fully-frustrated bond patterns are locally equivalent, but there are nonlocal distinctions. We emphasize their topological nature. For example, the fermionic model corresponding to a given cylindrical TAFIM supports states only of even particle number, or odd particle number, depending upon the classification of the bond pattern.

Preview
The most specialized statistical mechanics notion that we use is that of transfer matrix. Although widely known nowadays, an adapted refresher may be helpful. Suppose the bonds of a triangular lattice can be either satisfied or unsatisfied. Orient the lattice as in figure 1(a) so that rows of horizontal bonds labelled . . . , τ, τ + 1, . . . and shaded darkly are separated by interstices of slanted bonds. If X denotes a configuration of all the bonds, the partition function is (X) . (2.1) The star indicates that the sum is constrained. Not every set of bonds is the satisfied set for some configuration. The restriction here is the subject of section 3. The idea now is to sum over configurations in a staged way: an outer sum over the configurations . . . , X τ , X τ +1 , . . . of the rows of horizontal bonds, and an inner sum over the configurations on the interstices. Since constraints on an interstitial bond depend on no horizontal bonds except those on the bordering rows, the partition function becomes Z = ...,Xτ ,Xτ+1,... · · · X τ |T|X τ +1 X τ +1 |T|X τ +2 . . . , (2.2) where T is the transfer matrix. It is certainly a matrix, indexed by configurations on two neighboring rows. Taking the Dirac notation seriously, we try to interpret the row configuration X τ as a state vector, as in quantum mechanics. This turns out to be a good idea. Another one is to identify satisfied horizontal bonds with particles, taking them to be fermions to prevent them from sitting on the same location. Then, if the system has L rows, and we use periodic boundary conditions, the partition function is Tr T L . We complete the transition to a quantum mechanical idiom by identifying T with e −H = e −iH(−i) , the evolution operator through time ∆t = −i for an appropriate Hamiltonian of the system of fermions living on a one-dimensional lattice. Then, the classical partition function Z of a two-dimensional lattice of spins becomes a quantum mechanical partition function Tr e −HL of a system of fermions on a one-dimensional lattice at temperature L −1 . At zero temperature, the fermions are semi-conserved, in the sense that they can disappear in pairs with the passage of imaginary time, but not appear. Otherwise, the fermions are noninteracting, and appropriate boundary conditions can suppress the annihilation process, in which case particle number is strictly conserved. This implies significant influence of boundary conditions on the bulk equilibrium macrostate even in the thermodynamic limit. On a cylinder of fixed circumference, taking the length to infinity, we obtain distinct bulk states with distinct entropy densities, depending on the number of fermions (see figure 10). In the planar thermodynamic limit, there is a two-dimensional continuum of 'equilibrium' macrostates (see figure 9) distinguished by the fractions of frustrated bonds with each of the three orientations, that is, not only microscopic degeneracy reflected in nonzero entropy density, but macroscopic degeneracy. This depends on taking the limit T → 0 before the thermodynamic limit. If the zero temperature limit is taken after the thermodynamic limit, only the unique one of these macrostate with maximal entropy is accessible [34], hence T → 0 limits of exact thermodynamic functions [3,24] are insensitive to all the other macrostates. We relate the entropy densities of these equilibrium macrostates to energy densities of fermion ground states and calculate them in the form of series expansions in section 6.3.
Nonzero temperature (of the spin system) is studied in section 7. In the fermionic representation, it implies having not just pair annihilation, but pair creation-the TAFIM at nonzero temperature is (equivalent to) a superconductor. We work out a simple low-temperature approximation, which turns out to be surprisingly accurate to high temperature. Figure 13 compares the results of our approximate theory to the exact thermodynamic functions. The low temperature asymptotics of the approximate theory (equation (7.29)) are exactly right. Even the superconducting order parameter c m c m+1 is meaningful, being directly proportional to the energy density of the TAFIM (7.24).
Zero temperature is a critical point of TAFIM systems in that the spin correlation length diverges as T → 0. Stephenson [26,28] discovered that the two-spin correlation function falls off only as r −1/2 at zero temperature. Section 8 studies spin correlations via density fluctuations in the equivalent fermion system. The cited T = 0 asymptotic behavior is recovered exactly. For T > 0, and using the approximate theory, we find that the correlation length diverges as e 2βJ as T → 0. This agrees with previous determinations [28,35]. However, our numerical coefficient is about 20% smaller.
But how does the fermionic representation arise at all? First, focus on which bonds are satisfied, rather than spins. This is fruitful because the constraints on the set of satisfied bonds are simple. They may be viewed as collections of strings of dual edges with no ends inside the system (see figure 2(a)). In addition, there are even/odd restrictions related to the topology of the surface on which the model is defined. This rewriting can be carried out (section 3) for arbitrary fully-frustrated triangulated surface. Each elementary triangle is frustrated, but the system does not necessarily look like a regular lattice, even locally. On a cylinder, and at zero temperature, the string representation can be modified so that the modified strings are interpreted as particle worldlines, with the length of the cylinder corresponding to 'time'. This is done in section 4. The point of zero temperature is that the particles are essentially noninteracting. It is then easy to write down directly (section 5) a Hamiltonian that generates that allowed set of worldlines.

First ideas
Graphical representations have a distinguished history in statistical mechanics of spin systems. The high-temperature expansion for a FM Ising model is conveniently described by closed loops of bonds. The standard low-temperature expansion uses the dual edges crossing unsatisfied bonds (these are domain walls). For systems with quenched disorder, such as spin glasses, it can be helpful to mark unsatisfied bonds. The resulting strings have ends in the frustrated plaquettes, but this is not a useful approach for a fully-frustrated system. This section develops such representation for arbitrary fully-frustrated triangulated surface (e.g. a portion of the plane with holes cut in it, a cylinder, a torus, a higher-genus or even nonorientable surface). 'Fully-frustrated' means that each elementary triangle is frustrated, i.e. contains one or three AFM bonds. This generality highlights the topological aspects.
Stating the main result requires some definitions. In each elementary triangle and bond, choose a center. If the surface is planar, bond b corresponds to a dual edge † b. If b borders two triangles † b is the line segment joining their centers (figure 3). Otherwise, † b joins the center of b to the center of the sole triangle it borders. Figure 2 depicts the second type extended slightly, for clarity. Each bond b intersects exactly one dual edge, namely † b. For a nonplanar triangulated surface, consider both bonds and dual edges as curved in such a way that this intersection rule holds.
A frustrated triangle must have an even number of satisfied bonds, in any configuration, see (3.3) below. Let Sat denote the set of satisfied bonds in some unspecified configuration; it can be recovered from Sat up to global spin flip. Then, the set of dual edges † Sat is a collection of end-to-end connected paths of dual edges with no terminus inside the system. Such a path is a string, and a collection thereof, a string diagram. Figure 2(a) illustrates. This is a local condition. Are there global constraints? On a cylinder, some FM/AFM bonding patterns force † Sat to intersect every circumference an even number of times. Others require odd intersection number. The statistical ensemble of spin configurations is in two-to-one correspondence with string diagrams (for † Sat) which satisfy this global constraint. Each additional hole on the surface (in a homology sense, section 3.3) brings an additional constraint, which collectively define a mod 2 cohomology class (section 3.4). Every cohomology class is realized by some FM/AFM bonding pattern.

Loops
Let C be a path of end-to-end-connected bonds. Supplemented with the spins it connects, it looks like this: where C Unsat FM denotes the set of unsatisfied FM bonds in C , and so forth. For A , B sets of bonds (or, later, dual edges), A + B is their 'sum mod 2', so a bond is in that sum if it is in one and only one of A and B. The subscript on the equals sign is a reminder that we are working mod 2. In the particular case of (3.1), this is irrelevant since the sets being added are disjoint, but it establishes the habit. Now, by adding 2C Unsat AFM or 2C Sat FM , both of which are 0 (mod 2), we obtain (3.2) If all spins and all bonds in the path are distinct, we can find a configuration on it which satisfies any desired pattern of un/satisfied bonds: Simply run along the path and cumulatively choosing successive spins to un/satisfy the previous bond as needed. Suppose we satisfy all the bonds, and detach b N from σ N and connect it to σ 0 instead. The result is a loop with at most one unsatisfied bond. In any event, if we go around a loop with a spin configuration, counting flips, there must be an even number. That is, for a loop L , and denoting the cardinality of a set by vertical bars, |L flip | = 2 0. Therefore, specializing (3.2) for a loop, we have the important identity This is equivalent to |L Unsat | = 2 |L AFM |.

Mod 2 homology
The collections of bonds which will interest from now on have no free ends, so they are loops or sums of loops. The latter are also known as cycles. For two cycles A and B, since, mod 2, both sides just count the number of FM bonds in the two cycles taken together.
In algebraic terminology, this says that the map A → |A FM | from the set of cycles to Z 2 is a homomorphism of abelian groups. Now, if L is the geometric boundary of a collection of elementary triangles, it follows from (3.4) that |L FM | = 2 0, because every elementary triangle has an even number of FM bonds. In figure 4, A is a boundary-it is the sum of the triangles inside it. B is not a boundary, however B + E is one. Whenever the difference-or sum, mod 2 they are the same thingof two cycles A and B is a boundary, they are called homologous, denoted A ∼ B. One also says that they belong to the same homology class, denoted [A ] (or [B]). Since A ∼ A ′ and B ∼ B ′ imply that A + B ∼ A ′ + B ′ , our mod 2 cycle addition transfers to homology classes: With this addition, the homology classes are the elements of the (one-dimensional) homology group of the system, H 1 . (For a general introduction to homology, see, e.g. [36][37][38].) Homology is a geometrical relation, having nothing to do with the FM or AFM nature of bonds. However, because elementary triangles have an even number of FM bonds in a fullyfrustrated system, so does a sum of such triangles, so that In exactly the same way, we obtain |C Sat | = 2 † Sat • C , for any configuration. Finally, since |C Sat | = 2 |C FM |, it follows that † Sat ∼ † FM. So, in order that a string diagram D be † Sat for some configuration, it is necessary that D be cohomologous to † FM. This is a nonlocal condition. To replace sum over configurations with sum over † Sat string diagrams, it is crucial to know if this condition is necessary. Section 3.5 shows that it is. A related question is whether every cohomology class is realized by some † FM. Section 3.6 shows that this is also true.

Every string diagram cohomologous to † FM is † Sat for some configuration
Let † D be a string diagram cohomologous to † FM. We show how to construct a spin configuration such that D = Sat. First, obtain a spanning tree T. This is a collection of bonds containing no loops (tree) and such that every spin is touched (spanning). [To construct one, start with any tree. If there is a site not touched by it, connect it by a simple path of bonds not already in the tree. Continue until all spins are touched. The procedure terminates since the system is finite.] By a simple generalization of the procedure in section 3.2, find a configuration σ such that a bond in T is satisfied precisely when it is in D. Let b be a bond not in T. Necessarily, the spins it joins are touched by T, so there is a loop L made of b and some bonds in T. Now, the first equality by hypothesis and the second by (3.3). But D and Sat agree everywhere on L except possibly b, they must agree there as well, otherwise the equality could not hold. Thus, D = Sat for the constructed configuration.

Every cohomology class is represented by a string diagram
That every cohomology class is realizable as † FM can be proved by linear algebraic reasoning and finiteness of the set of bonds, since Z 2 is a field. Since we will not need this result in the following, we omit a general proof, and consider the special case of surfaces which can be homeomorphically embedded in the plane. Figure 5 shows one such. Topologically, a cylinder is a disk with one hole, so it is covered as well. The main point is very simple. Every cycle is homologous to a sum of the boundaries of zero or more holes. We can toggle the value of each † FM • A i independently by adding the string S i to † FM.

Ground configurations
Now we leave behind general triangulated surfaces to focus on regular triangular lattices wrapped on cylinders. To the extent that we are interested in the thermodynamic limit, global topology ought not to matter, anyway. We modify the string representation of configurations developed in the previous section into one especially suitable at low temperature, and ripe for reinterpretation in terms of (almost completely) noninteracting fermions.
To help distinguish the string diagrams already introduced from the modified ones developed in this section, we refer to the former as absolute string diagrams (and the latter, naturally, as relative string diagrams). After this section, we work only with the relative ones, and will then drop the adjective. This terminology is potentially misleading insofar as it is really the interpretation which makes the diagram absolute or relative. String diagram in the abstract is still perfectly well-defined.

Local condition for ground configurations
Since every link of a configurational absolute string diagram D represents a satisfied bond, those which correspond to ground configurations are those with maximal length |D|. An unpleasant feature of this characterization is its nonlocal character; we have to count up the total length. Is there a local condition?
A sort of folklore says that in a TAFIM system ground state configuration, every elementary triangle has exactly two satisfied bonds. That is a local condition. It corresponds to absolute diagrams with string going through every triangle. Unfortunately, in general, it is not quite right [40,41]. The reason is that in a finite system, interior bonds are shared by an up-pointing triangle ( , up-triangle for short) and a down-pointing triangle ( , down-triangle), but those on the system boundary are not. This is illustrated on figure 6, which shows both spins and the corresponding absolute string diagram assuming all bonds AFM. In the little system of figure 6(a), every bond belongs to a down-triangle. Hence the lowest energy possible has a string through every down-triangle. All the bonds are then accounted for, so what happens in the up-triangle is irrelevant. This generalizes to any system for which every bond belongs to a down-triangle; we call them down-triangle-type. Similarly, there are up-triangle-type-type systems; they are just down-triangle-type systems upside-down. Toroidal systems belong to Figure 6. (a) A ground configuration containing a triangle with all three of its bonds frustrated. This is a down-triangle-type system. (b) A system which is neither uptriangle-type nor down-triangle-type. Millane and coworkers [39][40][41] have investigated such irregular systems in depth. both types, while others, such as that in figure 6(b), do not fall into either category. This last category are the hardest to analyze. In this paper, we restrict ourselves to down-triangle-type systems. For cylinders, that means we draw them with a smooth edge on top and a serrated edge on the bottom.
The point of this restriction is that we have a local criterion for ground configurations of a down-triangle-type system is easy. The motifs and are the lowest energy configurations for a down-triangle. The motif costs energy 4J relative to these. (Complete with coupling strength, bond energy is ±Jσσ ′ .) Hence, measuring energy relative to the ground state, the total energy of a down-triangle-type system is 4J times the number of 's. In particular, ground configurations have no empty down-triangles.

Relative string diagrams
Now we do some mod 2 arithmetic with string diagrams. For two string diagrams D and D ′ , D = 2 (D + D ′ ) + D ′ . Therefore, if we had a reference diagram D ref , the transformation D → D + D ref is a one-to-one mapping of the set of string diagrams onto itself. Perhaps such a transformation can give a more useful configurational representation. Conditions can be given which leave only one nontrivial candidate for D ref . For instance, we would like it to work uniformly for all cylinders. This really means that there is some sort of template from which we get D ref for every cylinder, and therefore conflate them by saying 'the' reference diagram. Also, it should respect the translation invariance of the lattice, otherwise it would be very complicated to work with (and would have difficulty satisfying the first condition). It is probably also helpful if D ref • L = 2 0 for every loop L . Then the intersection number of valid absolute and relative diagrams with a circumferential loop will be the same. These considerations identify the diagram D zigzag in figure 2(b), containing only the motifs and , and zig-zagging horizontally across the plane, or around the cylinder circumference. However, being singled out like this is not particularly important. The real question is whether the transformation is useful, and that remains to be shown.
The action of the operation +D zigzag is purely local and illustrated in figure 7. The important thing is that the motif with an energy cost changes from to . Thus, since that motif is necessary to have string loops in the diagram, relative diagrams for ground configurations do not contain loops. Supplemented with boundary conditions at the cylinder top and bottom, this guarantees that every horizontal slice through the cylinder sees the same number of strings. The number of strings entering the top is straighforwardly the number of satisfied bonds along that edge. At the bottom, a little trick is helpful. We imagine 'phantom' bonds along the lower edge, which do not count toward the energy, but allow to count strings exiting.

From strings to fermions
Henceforth, string diagram means configurational string diagrams relative to D zigzag , unless otherwise indicated.

Strings as particle worldlines
We can think of string diagrams as depictions of particle worldlines, as suggested by the labelling in figure 8. Here we make a choice to keep x constant on rows going to the lower right rather than the lower left. This will lead to an asymmetry in the hopping later. It is necessary to make one choice or the other in order that allowed positions be independent of (integer) time. Circumferential rings of bonds correspond to successive integer times τ = 0, 1, . . ., and the bond centers correspond to integer positions modulo C; x ∈ Z C = {1, 2, . . . , C}, so 0 ≡ C and C + 1 ≡ 1. In this particle interpretation, the motif corresponds to annihilation of pairs of neighboring particles. For general boundary conditions, this may occur in zero-temperature diagrams, but , representing pair creation, may not. This asymmetry is because our systems are down-type, and by convention the flat end corresponds to 'earlier'. We also recall that the reason we insist on a definite type is that it gives a local criterion for a string diagram representing a ground configuration. Without that, it is not clear how or whether we could obtain a local Hamiltonian description.
Unless otherwise noted, we assume boundary conditions are such as to force the same number of particles exiting at the bottom as entering at the top, such as simple periodic boundary conditions. In that case, annihilation events are strictly forbidden at zero temperature.

Fermions
If one of our N particles is at position x at time τ , then in the next time step, it can remain where it is, or move to x − 1. However, distinct particles cannot occupy the same position. Because our particles are moving in one dimension, this hard-core interaction can be implemented by declaring the particles to be fermions. Thus, we consider operators c † n , n = 1, . . . , C, obeying the anticommutation relations {c n , c m } = 0, {c † n , c m } = δ n,m and the Fock space generated by their action on the vacuum state |∅ (containing no particles).
A particle configuration X = (x 1 , . . . , x N ) is a set of positions, by convention in ascending order: In the original spin model, probabilistic mixtures of different configurations makes sense, and we can consider the classical state space S to consist of assignments X → p(X) of probabilities to configurations. Such a state is associated with a Fock state vector, p(X)|X , by linear extension. Ignoring normalization, the classical state space S consists effectively of linear combinations of the basis vectors |X with non-negative coefficients, whereas states in the Fock space may involve this basis with complex coefficients. This distinction is important for the interpretation of the results. The transfer matrix (equivalently, the Hamiltonian) of the fermion system has a full set of eigenstates in Fock space. If such an eigenstate |i made sense as a spin boundary condition, that is belonged to S, then that state could be forced through the entire length of the cylinder. In fact, only a few of the eigenstates belong to S. 5.3. Particle-number conserving transfer matrix 5.3.1. Real space. Now we construct the particle-number conserving ring-to-ring transfer matrix T 0 . Modifications for nonzero temperature will be taken up in section 7.
A particle at x at time τ has two options for its position at time τ + 1: x or x − 1 (see figure 2(c)). Thus, T 0 acts as The right-hand side contains every possibility, counted once. Because the fermions cannot occupy the same location, their order is maintained, up to cycling around the ring. T 0 is thus determined by the conditions There is a subtle point, however, connected with the possibility of moving from x = 1 to x = 0 ≡ C or vice versa. Since we must take Since only the parity of N matters here, there will be no problem with this rule even with pair creation/annihilation. Conditions (5.3) need to solved with that constraint imposed. However, it makes no sense to do that directly in position space.

k-space.
We pass to k-space, using where k is in the Brillouin zone The allowed values of k here are dictated by (5.5).
From this, we deduce that in terms of the occupation number operators Finally, using (1 − n(k)) + e α n(k) = e αn(k) , rewrite (5.9) as where (5.13) As the notation suggests, H 0 and P can be considered a Hamiltonian and total momentum operator, respectively for the fermion system. Ignoring the e iP/2 factor, occurrence of which will be elucidated in section 6.1.1, (5.11) says that the transfer matrix T 0 evolves the system for an imaginary time ∆t = −i. In fact, the e iP/2 factor is not particularly important in the thermodynamic limit L → ∞.

Equilibrium TAFIM ground states and their entropy densities
An equilibrium macrostate of a finite lattice system is determined by the temperature, the Hamiltonian, and boundary conditions. For systems, such as TAFIM, with many ground configurations, this concept is nontrivial also at zero temperature. The equilibrium macrostate is a uniformly weighted statistical mixture of all ground configurations compatible with the boundary conditions. The standard way to define an equilibrium macrostate for an infinite system is to consider a sequence of increasing finite systems equipped with boundary conditions such that expectations for all observables in all finite regions ('in the bulk') converge. The limiting values define the equilibrium macrostate. We can apply this to ground states as well. An alternative view, due to Dobrushin, Lanford, and Ruelle, works directly with the infinite system. A probability distribution Pr on configurations is an equilibrium state if it satisfies this consistency condition: For any finite subsystem Λ, the distribution of spins in Λ conditional on the exterior is simply a Gibbs distribution. Above the critical temperature, two-dimensional FM Ising models with zero magnetic field have a single planar limit equilibrium macrostate (henceforth possibly shortened to simply 'equilibrium state'). Below the critical temperature, including at zero temperature, there are precisely two. The TAFIM, in contrast, has a unique equilibrium macrostate for all nonzero temperatures. Asymptotically, boundary conditions do not matter. What about at zero temperature? Surprisingly, there is a two-dimensional continuum of ground states, characterized by the densities of frustrated bonds in each of the three orientations. Those fractions must average one-third (equivalently, sum to one). However, any combination satisfying that restriction can be selected by boundary conditions even in the thermodynamic limit.
The nature of the continuum of equilibrium macrostates has an interesting description in the fermionic language. As we have seen, fermion number, or charge to use a more evocative word is locally conserved. Thus, over long times, every point of a ring system must experience the same the time-average current, but the latter simply corresponds to j =n(/) −n(\), the local difference between density of bonds with the two non-circumferential orientations. This suggests that we will be dealing with eigenstates of H 0 with nonzero momentum P, but we have already cautioned that such states cannot be selected by spin boundary conditions. The resolution of this little puzzle is that we must modify the Hamiltonian by what can be interpreted as an imaginary vector potential.
The entropy densities of all these infinite-system equilibrium ground states vary, as plotted in figure 9. There is a unique such with maximal entropy density, namely that with one-third of the bonds of each orientation unsatisfied. It is the limit of the unique T > 0 equilibrium macrostates. This does not imply that the others are completely irrelevant to T > 0 physics, however. We crudely estimate that at low but nonzero temperature, something very close to one of the equilibrium ground states can be boundary-condition stabilized in a finite system with linear extent of order e 4J/T . Artificial spin ice systems [42,43] are a good place to look for this sort of control by boundary conditions. The interactions in these systems of nanoscale magnetic particles can be easily adjusted to a range where room temperature is, effectively, zero, their configurations can be studied in complete detail, and individual particles can be magnetically manipulated. of N fermions on a ring of circumference C at temperature T = L −1 is equal to the partition function Z C,L (0, N) of the zero-temperature TAFIM on a (C, L)-torus, with boundary conditions forcing N satisfied bonds on every circumference. A note about the notation: italic Z denotes a partition function of the fermion system, calligraphic Z, one for a spin system. A subtle point here, indicated by the adjective 'twisted', is the state rotation implemented by the operator e iPL/2 inside the trace. Referring to figure 2 or 8, the explanation is as follows: because of the slant of constant-x lines, (x, τ = L) is directly below (x + L/2, 0) in the natural planar or cylindrical representation of those figures. We should therefore restrict L to be even for the sake of closing the system into a torus. The twisting in the trace definingZ, indicated by the tilde, enforces equality of the state at τ = L rotated forward by L/2 with the state at τ = 0. This is twisting with respect to the x coordinates is precisely what is needed to achieve the identification of the top and bottom of the cylinder in the natural geometrically untwisted way. Now, we obtain the correspondence between thermodynamic potentials of the TAFIM system and its fermionic representation, namely On the left, we have the partition function of the fermion system at (nonzero) temperature L −1 , and particle density f C L −1 , n is its free energy density. The TAFIM is at zero temperature, so we cannot have a bare factor of T −1 in the partition function exponent. Therefore we absorb it by using the Massieu potential, generally defined as ϕ = s − βe, with e is energy density. Since we have taken our energy zero to be the ground state energy, we just have ϕ(0, n) = s, the entropy per spin. The relation (6.2) between the spin system and the fermion system continues to hold for nonzero temperature T. In that case, N is understood as referring to boundary conditions only, since for T > 0 these do not control the density in the bulk. Also, the equivalent fermion Hamiltonian becomes T dependent. That situation is investigated in section 7.
The fermion free energy is awkward to evaluate in the canonical ensemble, except in the zero-temperature limit L → ∞, where it becomes simply the energy of a filled Fermi sea: with the Fermi momentum k F = π n.   [44]). Solid curve: C → ∞ limit, given by the formula in equation (6.36). Symbols: exact numerical results for circumference 3 (□), 4 (♢) and 8 (▽). Solid (open) symbols denote periodic (antiperiodic) boundary conditions. Except at n = 1, for which entropy density is exactly (ln2)/C, the C = ∞ limit is nearly attained already for C = 8.

Series expansion of s(n).
The equation (6.4) for the C, L → ∞ limit of the TAFIM entropy density is pleasant for its simple connection to a familiar problem. Section 6.3 derives a series expansion, the first few terms of which are Here,n and the remainder satisfies 0 < R 11 (n) < 10 −4 . The maximum of s is of particular importance. The only macrostates accessible by taking the thermodynamic limit first, and then the limit T → 0, are those with maximum entropy density [34]. Assuming no spontaneous symmetry breaking, this maximum must occur for n = 2/3, because the densities of unsatisfied bonds with each of the three orientations must be equal and sum to one. From just the few terms in the formula (6.6) and the bound R 11 (1/3) < 2 × 10 −9 , the ordinary residual entropy density is lim T→0 s = s 2 3 = 0.323 0659(9), (6.8) extending the value reported by Wannier [25] by a couple of decimal places.

Remarks.
The constraint on the number of satisfied bonds arounds the circumference is controllable by boundary conditions for a cylinder; if boundary conditions enforce N = nC satisfied bonds at top and bottom, then the same is true for every circumference. However, just control of n globally does not immediately imply that it can be controlled in macroscopically large but finite regions for a sequence of cylinders with C, L → ∞. We must rule out phase separation, which would result in macroscopically large regions with values of n smaller or larger than the global value. That this does not happen is demonstrated by the strict concavity of s(n). Phase separation would give flat parts of the graph. The transfer operator (5.11) conserves other quantities besides particle number, for example momentum. This raises the question whether there is an even richer variety of thermodynamic limit states. Sadly, the answer is no. The fermion language is more expressive than the spin language, as touched upon in section 5.2. Every spin boundary condition corresponds to a Fock space state with a zero momentum component, which inevitably dominates as L → ∞.

All planar limit equilibrium macrostates
This subsection determines all the planar thermodynamic limit equilibrium states and their entropy densities. Figure 10 represents only some of these. We revert here to the earlier notation n for n(-); n(/) and n(\) can be recovered from n and j as 2n(/) = n + j, 2n(\) = n − j. (6.14) We recover the entropy as a function of the densities via Legendre transform. That is, using our established notation ϕ(µ, α) = N −1 s ln Z(0, µ, α) for the free entropy, Note that n and j are in a sans-serif font to avoid any confusion with generic summation variables later. The partition function (6.10) can be considered the Lagrangian formulation of the model. The usual Lagrangian coupling of an electromagnetic vector potential to current density iś iA · j. Our iα should be thought of as an imaginary vector potential.

Translation to fermion language.
Translation of the partition function (6.10) into fermionic language requires a minor variation on what was done in section 5.3. To give the correct weights to the movements of particles, we generalize the condition (5.3) imposed on the transfer operator T 0 to Proceed just as in section 5.3. First, gives α-dependent pseudo-energies ε α (k) defined by (cf (5.13)) ε α (k) = ε(k + 2αi) = − ln 2 cos k 2 + i α . (6.19) In contrast to the Lagrangian form in (6.10), we now have a Hamiltonian form, and we see that iα shows up in the expected form for a vector potential. The logarithm of the partition function is ln Z C (L −1 , µ, α) = CLϕ C,L (µ, α) = |q| ln 1 + e (µ−εα(q))L . (6.20) In the limit L → ∞, terms with µ < Re ε α (q) tend to zero, and therefore, in the full thermodynamic limit C, L → ∞, the density is implicitly given by Re ε α (nπ) = µ, (6.21) and the free entropy density of the spin system by We wish to obtain s(n, j) from this by Legendre transformation (6.16). Equation (6.15) give the transformation from control parameters (µ, α) to densities (n, j) abstractly. The relation (6.21) begins to make it concrete. To finish the job, note that, from (6.15), Putting things together, we obtain a simple, pretty, and useful relation: Some manipulation produces the disentangled form tan π j 2 = tanh α · tan π n 2 , (6.25a) e −2µ = 4(cos 2 π n 2 + sinh 2 α). where (6.25a) determines α in terms of (n, j), while µ has simply dropped out. The following section will detail ways to evaluate the integral here. Results are plotted in figure 9, and it is indeed strictly concave. Each slice through the graph of figure 9 from a corner to the center of the opposite edge is identical to figure 10.

Evaluation of integrals
This section gives some details on ways to evaluate the integral in (6.26), of which ( To avoid having to worry about the zeros of cosine on the real axis, assume our contours are all in either the upper or lower half-plane. Now, because we need to preserve only the real part of the difference for two integration terminal points with the same imaginary part, we are free to transform this integral in a variety of ways, such as (i) move the initial point, (ii) add an imaginary constant or imaginary multiple of w to the integrand. Therefore, since Here, we use ⊜ to mean that the real part of the difference between the two sides depends only on the imaginary part of the argument. In other words, they are equivalent for our purposes. Notice that as we assumed z in the lower half-plane, our contours are now in the upper halfplane. We consider two series representations of this integral.

'Large' |α|.
Since |e iw | ⩽ 1 for Im w ⩾ 0, we may take our reference point at +i∞ and apply the series expansion for logarithm to obtain Now, this last expression is purely imaginary whenever Re z is a multiple of π, so s(n, j) = −αj + ∞ k=1 1 k 2 e −2|α|k sin kπn, (6.31) where α = tanh −1 tan πj/2 tan π n/2 The series converges for all α, more quickly the larger |α|, yet it is reasonably practical even for α = 0.

Small |z|.
Now we consider an alternative expansion around zero. It is genuinely limited, with a radius of convergence of 2π, but converges very rapidly for small z. Integrate by parts as and recognize the second term here as the generating function of Bernoulli numbers: Then, we have, for |u| < 2π, This is especially handy for α = 0 (equivalently, j = 0), in which case Here, we have fixed the constant by recognizing that s(0) = s(1) = 0, as is clear from the physical meaning.
To bound the tail of the series, use (23.1.15) of Abramowitz and Stegun [45]: (6.37) This implies that the series in (6.36) is dominated term-by-term, by this one: The bounds reported in section 6.1.2 came from this.

Nonzero temperature
This section extends the fermionic representation of the TAFIM to nonzero temperature. Keep in mind that the temperature of the fermion system is not the temperature of the spin system; T always means the latter. The temperature of the fermion system is the reciprocal of the length of the lattice on which the spin system is defined. As we are primarily interested in the thermodynamic limit, this means that the fermion system continues to be at zero temperature here. Instead, the temperature T corresponds to a change of the fermion Hamiltonian or time-evolution operator. Our translation is not exact. In the interest of a fermionic description which is elementarily tractable, we make an approximation which captures the leading-order (roughly speaking) behavior for nonzero temperature, as (7.29) demonstrates conclusively. Nevertheless, figure 13 shows that it turns out to be a relatively good approximation even up to high temperature. Our approximation will have the form of a reduced BCS model for superconductivity. The approximation consists in neglecting more complicated interactions, rather than from a mean field approximation as in real BCS theory. Thus, there is no transition to a normal state on raising the fermion temperature, which is equivalent to shortening the cylinder on which the model lives.
The critical difference of T > 0 is that the motif has a nonzero weight, hence string diagrams can contain closed loops, as in figure 11. A priori, the motif comes with a weight κ 2 := e −4βJ . With boundary conditions enforcing the same number of particles at initial and final times, we can make things a little more symmetrical by exploiting the fact that we then have an equal number of and motifs. One can verify this by inspection of figure 11. Then, assigning weight κ to both and gives the same weight for all string diagrams. This is just for convenience, however, and we could do without it.
The small loop at C in figure 11 corresponds to creation of a pair ( ) followed by its immediate annihilation ( ). At D, there is an annihilation event followed by a creation event one time step later. Unfortunately, completely general identification of with pair-creation and with pair-annihilation is not possible, as we see from the loop labeled B, which is one event of each kind, and even more so from A, which has neither creation nor annihilation. Our basic approximation is to neglect these processes and keep only isolated and motifs, in which case the interpretation is sound-with one caveat, explained in section 7.1.
From the statistical mechanical perspective, A and B are suppressed not just energetically, but also entropically; excited triangles should not clump together. From the Hamiltonian perspective of the fermion formulation, they correspond to small corrections to the transfer matrix, being higher order in κ, and they represent complicated many-body interactions which are not qualitatively new processes. To see why these are many-body interactions, note that a wellseparated pair can annihilate, as at B, only if the intervening region is free of particles. It is Figure 11. A relative string diagram for nonzero temperature. 'Excited' triangles, containing three unsatisfied bonds, are highlighted. The approximation analyzed here disallows neighboring excited triangles, as at A and B, but pairs meeting on a horizontal bond, as in D, are permitted. Thus, the approximation does not preserve complete rotational symmetry.
plausible that our approximation captures the leading T > 0 behavior, in some sense. This is verified in a precise way in (7.29).

Approximate transfer matrix
In previous sections, T 0 implemented a time step ∆τ = 1 by moving, or not moving, each particle: ∆x = −1, 0. Now, there are two additional categories of option. The full transfer matrix T should first implement the annihilation option, then the motion option, then the creation option, in that order, because if a pair is annihilated or created, the particles composing it do not have the option of moving in that time step. That is, the transfer matrix is where T pr and T † pr implement pair destruction and creation, respectively. With abbreviation κ := e −2βJ , (7.2) they are given by As with T 0 , these are put into a more useful form in k-space. Since the operators {c i+1 c i | i ∈ Z C } are nilpotent, 1 + κc i+1 c i = exp(κc i+1 c i ); and since they commute with each other T pr = i∈ZC e κci+1ci . Applying commutativity again produces T pr = e −Hpr , where destroys a pair of fermions in the q and −q modes. Re-expanding, we finally obtain (7.6) Figure 12. A string diagram generated by the approximate transfer matrix (7.1). Shaded triangles indicate factors of κ. In addition to the intended processes, T also generates spurious events of the sort labeled A, with a weight κ 2 . Here, a and motif are juxtaposed in a way which is not legitimate in the original model. Now that we have presented the mathematical forms, it is easier to see that, not only have we neglected some processes, but have also included some undesired ones. Figure 12 illustrates a string diagram generated by the transfer matrix (7.1). Unfortunately, it generates some unwanted events, such as at A. It represents a spurious correlated pair-hopping process with a strength κ 2 . Though unwelcome, it plausibly does not have a large effect on the results.

BCS-style fermion pairing
The transfer matrix (7.1) can be written as a product of commuting factors, where T(q) is the part affecting the q and −q fermion modes. Explicitly, for q > 0, Here, we have suppressed the q labels and n = c(q) † c(q) + c(−q) † c(−q) counts the number of particles in those two modes. T(q) is the sum of a term T o (q) ('o' for 'odd') acting on the subspace with one particle in either q or −q, which is simply the constant e −ε , and a term T e (q) acting on the evenparticle-number subspace. T e (q) is conveniently represented in terms of Pauli matrices τ α . With 'spin-up' corresponding to both modes q and −q occupied, Here (restoring the argument q), Tr T e (q) = 3 2 + cos q + 2κ 2 sin 2 q, (7.10) while B(q) is the positive square root of and θ(q) ∈ [0, π] is determined by Note for future reference that A(k F ) = 1, where k F = 2π 3 is the Fermi point. It is evident that, at least for small κ, cos θ(k) goes monotonically from 1 at k = 0 to −1 at k = π, crossing zero near k F .
The larger eigenvalue of T e (q) is A + B, with corresponding eigenvector v q u q = e i θ 2 τx Thus, the ground state takes the familiar BCS form with energy density We have ignored the k = 0 mode, and will continue to do so. It occurs only for odd N anyway, and does not matter in the thermodynamic limit.

Thermodynamic functions
Armed with the ground state (7.14) and its energy (7.15), we compute the thermodynamic functions, energy density e/J, entropy density s and free entropy density ϕ, related by ϕ = s − βe. (7.16) We calculate below ϕ and e in our approximate theory, and obtain s through the above. The results are plotted in figure 13, along with exact results. For example, from Lavis and Bell [24], 8.167 (8.13.4 in the first edition) the exact energy density (with ground state value subtracted) is is equal to minus the ground state energy density of the corresponding fermion system, which is given in (7.15). At both extremes, κ = 0, 1, the energetic contribution to ϕ vanishes-at zero temperature because all states have the same energy, and at infinite temperature because β → 0 while e is bounded. This derivative can be calculated directly from the formula in (7.19). We will take a more interesting route. Since Cϕ equals the ground state energy E of the fermion system, and Ω|T|Ω = e −E , the Hellmann-Feynman theorem gives ∂ ∂κ Cϕ = ∂ ∂κ ln Ω|T|Ω = Ω|∂T/∂κ|Ω Ω|T|Ω (7.21) Now, from (7.6), and using that the various b(q) square to zero and commute with each other, and similarly for T † pr . Since, moreover, Ω is a translation-invariant eigenstate of T, (7.21) is rewritten as ∂ϕ ∂κ = −2 Ω|c n c n+1 |Ω (7.23) Therefore, e J = 2κ Ω|c n c n+1 |Ω (7.24) Evidently, the energy density is closely related with the superconducting order parameter.

7.3.3.
Low-temperature asymptotics. The asymptotic behavior of the thermodynamic functions in the vicinity of the critical point T = 0 is of especial interest. We work out the leading behavior here, starting with the energy, as it has a particularly convenient form. We cannot say that is simply O(κ) because the denominator has a zero at (κ = 0, k = k F ). The Fermi point is k F = 2π 3 , so that 1 2 + cos k F = 0, and the expansion of B(k F + q, κ) in (7.11) to O(q 2 , κ 2 ) yields B(k F + q) 2 = (sin 2 k F )(q 2 + 4κ 2 ) + . . . , (7.26) and then On the other hand, B(k) is nonzero even at κ = 0, away from k F , so there are no other singularities. Therefore, Here, ϵ is some cutoff which does not really matter since we will throw away the ln ϵ term. Using (7.20) and ϕ = s − βe, we then deduce the following set of asymptotics: The residual zero-temperature entropy density is given in (6.8). The above asymptotic formulas are in precise agreement with exact results. To see this, check that the integrand in (7.17) has a singularity at κ = 0, x = −1/2 of the same sort as the one analyzed above, and the coefficient is the same. Since e and s(0) determine everything else, the asyptotics in (7.29) are exact. A natural approach to a low-temperature expansion directly on the two-dimensional model would treat triangles with three unsatisfied bonds as elementary excitations. Since each such triangle has a Boltzmann weight κ 2 = e −4βJ , we might have anticipated that the first terms would be simply proportional to κ 2 . Since that sort of reasoning works fine for systems with ordered ground states, the appearance of logarithms in (7.29) is a surprise from this perspective. We have no intuitive justification for logarithms as such, but presumably they must be ascribed to the fact that the background into which the excitations are inserted has a nontrivial structure.

Correlation lengths and energy gaps
Suppose A(x, τ ) is TAFIM observable which can be expressed as a local operatorÂ in the fermion interpretation. An example is bond energy for circumferential bonds, which is J(1 − 2n x ), with n x the number of particles at site x. For observables of this sort, we have where Φ 0 ≡ Ω is the ground state. Thus, the connected correlation function is The rate at which this correlation function falls off is the fundamental energy gap E 1 − E 0 above the ground state, or even faster if the relevant matrix elements vanish. The energy gap is zero when (κ = 0) the fermion system is a free fermi gas. When κ > 0, the gap is the energy to create a quasiparticle, and if κ is very small, the lowest-energy quasiparticles have momentum close to k F . From (7.10) and (7.11), A(k F ) = 1 + 3 2 κ 2 , and B(k F ) = √ 3κ Therefore, by (7.15), This corresponds to a correlation length diverging as e 2βJ , showing that T = 0 is a critical point. Some connected correlations might decay at a faster rate, but none will be slower.

Spin correlations in terms of satisfied bonds
Stephenson [26,28] found that the spin-spin correlation function in the strict TAFIM does indeed behave in the way suggested by the preceding analysis: Along a bond directionê, asymptotically as ℓ → ∞, However, the spins do not correspond to local fermionic operators, so it is not immediately clear what this has to do with the analysis of section 8.1. Indeed, the very notion of a spinspin correlation function seems to make little sense from our general perspective. For instance, changing the sign of all bonds attached to σ leaves the system fully frustrated, but changes the sign of σσ ′ for every other spin σ ′ . This is why we emphasized that Stephenson's results were for the strict TAFIM, with every bond AFM. We want to make sense of the spin correlation function from the point of view of general fully frustrated systems not merely for the sake of generality, but also because the computational strategy is based on that understanding.
For the strict TAFIM (planar or not), we can rephrase the product σσ ′ of spins as Γ(C ) := (−1) |C Sat | , where C is any path of bonds from σ to σ ′ . Hence, we can rephrase the spin correlation function in terms of satisfied bonds, a characterization which fits our point of view, and which can be generalized. In general, if C and C ′ are two paths between σ and σ ′ , then C + C ′ is a loop, and we have Therefore, the rewrite of the spin product as Γ(C ) generalizes painlessly to all fully frustrated bond patterns for which FM is trivial in cohomology. On its face, Γ(C ) is very nonlocal, since it is expressed in terms of an entire path, but since it really only depends on the end points of C , that is to some extent illusory. But, how are we to think of Γ(C ) and its expectation value Γ(C ) ? If |C Sat | had the same value in every configuration, then | Γ(C ) | would be one. So, smaller values of that expectation would seem to indicate greater fluctuations in C Sat . In fact, however, a lot of information is lost in passing to Γ(C ) since it only depends on the parity of |C Sat |. We really should free ourselves from the presumption that only the parity matters. Studying fluctuations of |C Sat | for various paths C is a very natural way of studying, possibly even conceiving of, configurational fluctuations. Our approach will follow this philosophy. For C a path along a lattice direction of variable length ℓ, we obtain first suitable information on the distribution of |C Sat | itself (it is gaussian for large ℓ), and then extract Γ(C ) from that information.
In the thermodynamic limit, the correlation function along all three bond orientations is the same. The fermionic representation is especially suited to calculating Γ(C ) straight along what it regards as the spatial direction. In that case, |C | is the number of particles along the path. The spin correlation function therefore is related to fluctuations in particle density. Following that approach, we reproduce Stephenson's result (8.4) for zero temperature, while extending it to particle densities different from 2π/3: At T > 0, we obtain Stephenson's functional form, but with a slightly different coefficient in the exponent: We are unsure what to make of the discrepancy.

Approach to spin-spin correlations
Now we begin to fill in details of the calculation of (8.6) and (8.7). Takingê along the circumferential direction, and remembering that a vertical string segment represents a satisfied bond, we have where is the number of particles in the segment [1, ℓ] at some fixed time. Then, where Ω is the fermionic ground state. Splitting N ℓ into a mean and fluctuation as N ℓ = ℓn +Ñ, (8.11) we obtain e iπ N ℓ = e iπ nℓ e iπÑ . (8.12) Since N ℓ is integral, the left-hand side is necessarily real, hence so is the right hand side. However, if the exact value of e iπÑ is replaced by an asymptotic expression, that property may be lost, so we will be safe and take the real part explicitly: Ñ 2 ], and therefore The program therefore is to compute Ñ 2 , show thatÑ has a normal distribution, and thence recover (8.7) and (8.6).

Poisson summation.
The preceding is evidently based on the fact that if X is a random variable with gaussian probability density function the Fourier transform of γ. However, if N takes only integer values, the distribution of N cannot really be gaussian, and that presumably explains how e −i 3π N can equal e −iπ N . The resolution of the conundrum is to say that N takes value m with probability proportional to γ(m). Then, e −iθN ∝ m∈Z e −iθm γ(m), (8.17) and by the Poisson summation formula, the right-hand sum is equal to m∈Zγ (θ + 2π m). (8.18) This solves the problem because for large σ 2 , the sum in (8.18) is dominated by the term (or two in exceptional cases) with minimal |θ + 2π m|. In short, for e iθN , we should always reduce θ to the interval (−π, π].

Number variance.
To determine the variance ofÑ, proceed as follows. Convert the number operators to momentum representation: Rewrite the number fluctuation operator as Then, We proceed to evaluate this successively for T = 0, and then for T > 0.

Zero temperature
This subsection shows that, at zero temperature, Substituted into (8.14), this yields exactly the claimed result (8.6). However, the supposition that the fluctuations are normal (i.e. Gaussian), is cast into some doubt by the ℓ dependence of Ñ 2 . A normal distribution is usually explained as the result of many (nearly) independent small contributions. For an ordinary, say classical, particle system we expect fluctuations of particle number in mesoscopic subvolumes to be those small contributions, leading to the variance of N being proportional to system size, as in ordinary thermodynamic fluctuation theory. Evidently, the one-dimensional fermion system at zero temperature does not behave that way. Subsection 8.4.2 shows that, neverthelessÑ does have a normal distribution.

Number variance.
This subsection is devoted to a careful and detailed derivation of (8.22). The treatment has similarities to that of Villain and Bak [46].
Acting on the Fermi sea Ω, ρ(q) creates particle-hole excitations of momentum −q. If |q| is small, there are |q|C/2π distinct (that is, orthogonal) such excited states. Also, ρ(−q ′ )ρ(q) = ρ(q ′ )Ω|ρ(q)Ω is zero unless q = q ′ . Hence, as long as q < 2k F , 2(π − k F ), and therefore, from (8.21), The inequality arises because the contribution of large |q| is overestimated. However, we will see that the important contribution arises from small |q|, so in the thermodynamic limit In order to clearly isolate the singularity it is useful to introduce the family of integrals for 0 ⩽ α < β ⩽ π 2 . We will show that, for ℓ → ∞, (8.27) to obtain formula (8.22) for the number variance Ñ 2 .
In (8.27), the case α > 0 is clear, so we concentrate on α = 0. With θ : [0, π/2] → [0, 1 − (2/π) 2 ] a monotone continuous function, substitute into the integral to obtain The second integral is the important one-it evaluates to ln √ ℓ. The first is a well-defined constant (independent of ℓ), and the third is also O(1), as can be seen after integration by parts:ˆℓ This completes the demonstration of (8.27).

8.4.2.
Normality. Now, we turn to the verification thatÑ is due to many weakly dependent modes so that it really has a Gaussian distribution. To get better control of e iθÑ , we split the number fluctuation into partsÑ > andÑ < containing the density modes with wavevectors in (0, π) and (−π, 0), respectively. SinceÑ † > =Ñ < , and recalling that all the density modes commute, we have Ω e iθÑ Ω = e −iθÑ> Ω e iθÑ> Ω . (8.32) Proceed to examine the state e iθÑ> Ω. The part, H, of the fermion model Hilbert space which is relevant to us has a precise particle number (the size of the Fermi sea). This Hilbert space can be decomposed as H = ∞ n=0 H n , according to the number of particle-hole excitations, n (number of particles missing from the Fermi sea). Now,Ñ > Ω is in H 1 , of course. Applying a second fluctuation operator, we see thatÑ 2 > Ω ∈ H 2 ⊕ H 1 ; the secondÑ + may create a second particle-hole pair, or it may move the already excited particle. The crucial point to note, though, is that the norm of the component in H 1 is O(C −1 ) times that of the component in H 2 . This is because ρ(q) can create qC/2π distinct particle-hole excitations, but there is only one way it can boost the already excited particle. The argument just given can be applied over and over, showing that for any m C, (Ñ > ) m Ω is in H m , to relative order C −1 . Therefore, the component of (Ñ > ) m Ω orthogonal to H m is negligible in the thermodynamic limit.
It will now be useful to split a Fourier component ρ(q) of the density as ρ(q) = ρ(q) + + ρ(q) − + ρ(q) 0 , where ρ(q) + (the particle-hole creation part) contains the terms moving a particle from inside the Fermi sea to outside, ρ(q) − = [ρ(−q) + ] † , those moving a particle from outside to inside, and ρ(q) 0 is the remainder. Correspondingly, we writẽ w(q)ρ(q) + , (8.33) and so forth. With this new notation, we can write the conclusion reached in the previous paragraph as Here is below, ' ' means 'up to relative O(C −1 )'. Returning now to the inner product in (8.32) that we need to compute, It only remains to evaluate the norm squared, (Ñ + > ) m Ω 2 , occurring on the right, which is immediately rewritten as To evaluate this, the commutator [Ñ − < ,Ñ + > ] is needed so thatÑ − < can be moved through until it annihilates Ω. The key to this is If this holds for all θ ∈ R, it indeed verifies thatÑ, and therefore N ℓ have Gaussian distributions. Substitution of π for θ yields the claimed result (8.14).

Correlation length at nonzero temperature
The method to determine the correlation length for T > 0 is essentially the same, namely an examination of density fluctuations, and application of (8.21), using the approximate superconductor theory to determine asymptotics of Ñ 2 for small κ > 0, as ℓ → ∞. The result of the following calculation is Comparison with (8.14) implies an exponential decay σ n σ n+ℓ ∼ e −ℓ/ξ , with inverse correlation length ξ −1 ∼ π 2 4 κ. (8.43) To express the density operators in a useful way, we need another basic tool of superconductor theory, namely, the Bogoliubov-Valatin transformation expressing the quasiparticles operator in terms of c-fermions. The inverse of the preceding is where we take corresponding to θ(k) being odd in k (see (7.13)). Rewrite ρ(q) = k∈BZ c(k + q) † c(k) using the Bogoliubov-Valatin transformation and notice that ρ(q)Ω 2 only has contributions from the pieces with two quasiparticle creation operators: c(k + q) † c(k) = −u k+q v k γ(k + q) † γ(k) † + · · · . Compared to the zero-temperature (κ = 0) situation considered in section 8, an important difference of κ > 0 is that the above expression has a regular expansion at q = 0 with a nonzero constant term. To capture the leading part of Ñ 2 with respect to ℓ, we therefore simply set q = 0. With |u k v k | 2 = 1 4 sin 2 θ(k), ρ(q)Ω 2 C 4ˆs in 2 θ(k) dk π , (8.49) independent of q, as long as it is small. Assembling the pieces yields (8.52) as announced in (8.42).

Conclusion
The TAFIM is a basic model of statistical mechanics and the prototype of frustration, and for that reason is invoked by physicists in a wide variety of contexts. We have given a unified treatment of many aspects of TAFIM physics, relying on a simple mapping to a system of fermions in one space, one (imaginary) time dimension. This permits easy exact calculation and elucidation of many properties at zero temperature, which is anyway where the TAFIM is special. For example, this strategy easily uncovers the strange two-dimensional continuum of equilibrium macrostates and their entropy densities. A novel perspective on the divergence of the spin correlation length at zero temperature also emerges. From the fermionic perspective, this is where a superconducting gap closes and number fluctuations pass from extensive to logarithmic in system size.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).