Gauge-free duality in pure square spin ice: Topological currents and monopoles

Degenerate artificial square ice is perhaps the simplest two-dimensional system in which the disorder constrained by the ice rule¹ and its violations as monopole excitations are studied.^2,3 While spin ice pyrochlores^4–7 had opened new vistas in the study of geometric frustration and constrained disorder, more recently, artificial spin ice—systems of interacting, magnetic nanoislands^8–13—have provided controllable platforms that can be characterized at the constituent level.

Although the field has developed to include new forms of frustration and geometries, allowing for the realization of magnets often not found in nature^14–21 and revealing the novel phenomena absent in its crystal analog,^12,22 recent fabrication and characterization advances^23–25 have brought back the austere simplicity of celebrated early models to the fore, such as kagome and square ice.

The square geometry of spin ice was among⁸ the first to be realized artificially,⁹ when it was shown that non-ice rule²⁶ vertices are suppressed after AC demagnetization.^9,27,28 However, because moments impinging perpendicularly in the vertex interact more strongly than moments impinging collinearly, the degeneracy of the ice manifold is lifted, and antiferromagnetic (AFM) vertices are favored, leading to a phase transition in the Ising class^29,30 toward an ordered antiferromagnetic ground state.^31–34 

Soon after, Möller and Moessner³⁵ proposed to offset the height of half of the nanoislands to regain the ice rule degeneracy.^36,37 The idea was recently realized.^38,39 Meanwhile, ice rule degeneracy in square ice has also been demonstrated in rectangular lattices,^40,41 via cleverly placed interaction modifiers placed in the vertices,⁴² with nano-holes in a connected spin ice of nanowires,⁴³ or by rotation of the moments.⁴⁴ Recently, pure square ice, i.e., square ice with only nearest neighbor interaction, was realized in a quantum annealer⁴⁵ and used to demonstrate purely entropic monopole interactions. Then, various different regimes can be achieved by tweaking specs and couplings, leading to antiferromagnetic states, line states, or ice manifolds of different spectral characterizations.⁴⁶ 

The scope of this work is to provide a unifying and expandable framework that can cover many different square ice systems around the ice rule degeneracy point^35,45 by considering topological charges and currents as the relevant degrees of freedom and subsuming the spin structure into effective, entropic interactions among them. This approach flows naturally from the gauge-free duality of the system, which is absent in three dimension (3D).

We limit ourselves to pure square ice, that is, a 16 vertex model, where interactions are limited to spins within the same vertex.^29,47,48 We consider no long-range interactions and thus no 3D-Coulomb (i.e., 1/r) interaction among monopoles or currents (we have considered such cases elsewhere⁴⁹). However, because of the emergent nature of these objects, we show that they interact via 2D-Coulomb (i.e., ∼ln r) entropic interactions.

Within this model, we compute free energies, entropic interactions, structure factors, susceptibilities, correlations, screening, and relaxation dynamics. We discuss strengths and limits of this approach. We show new results but also re-derive results that were previously achieved in similar systems through a variety of methods. These had included phenomenological approaches via coarse grained field, height models, or analogies with chemical physics approaches.^2,50–60 We also particularized some results that we had already found on generic graphs.⁶¹ 

In 2D, there is gauge-free duality absent in 3D pyrochlore ice. It is related to the rather gravid mathematical fact that in 2D, the Helmholtz decomposition has no gauge freedom. That in turn follows from the fact that orthogonal directions are uniquely defined in 2D. It is amply used in 2D continuum theories, from fluid dynamics to the XY model,⁶² and is behind the entire edifice of complex analysis.

In 2D, we can always write a continuum vector field $S⃗$ in terms of longitudinal and transverse potentials h_‖ and h_⊥,

where $ê3=ê1∧ê2$⁠, $ê1,ê2$ are orthonormal bases of the plane, and we call $S⃗‖andS⃗⊥$ the longitudinal and perpendicular components of the field, respectively. Unlike in the 3D case, where the perpendicular part of $S⃗$ is the curl of a vector potential, there is no gauge freedom, as shown in Eq. (1).

If we define the charge and current distributions of the field as

then

If for a vector $w⃗$ we call

the perpendicular of $w⃗$⁠, we then have

which express the duality between charges and currents, or longitudinal and perpendicular components of the field, under a π/2 rotation.

This duality has a discretized analog in square spin ice.

Square spin ice (Fig. 1) is a set of classical, binary spins $S⃗e$ aligned on the N_e edges e of a square lattice of N_v = N_e/2 vertices labeled v. Spins form four vertex topologies often classified⁹ as t-I, …, t-IV, where t-I and t-II obey the ice rule^26,35,37,48 (i.e., have two spins pointing in and two pointing out).

In degenerate square ice, ice rule vertices are degenerate and energetically favored, and they lead to a ground state¹ of constrained disorder, called the ice manifold. The latter is a Coulomb phase,^50–53 i.e., a topological state labeled, in lieu of an order parameter, by a height field.^56,63

Consider Q_v[S], the topological charge of the vertex v, defined as the number of spins pointing to the vertex minus those pointing out. Then, an ice rule vertex v has Q_v = 0.

We can similarly define the topological current I_p[S] of a minimal square plaquette p as the number of spins pointing clockwise around the edge of the plaquette minus that pointing counterclockwise.⁶⁴ 

For a spin configuration $S⃗$⁠, consider its perpendicular configuration $S⃗⊥$ (Fig. 1), for which old plaquettes are now vertices and old vertices are now plaquettes. We then have

as in Eq. (5).

For each configuration of spins that obeys the ice rule, a unique (up to a constant) height function h_⊥ can be defined on the plaquettes such that

is true (⁠$pp′̂$ is the unit vector pointing from plaquette p to p′ separated by the edge e as in Fig. 1). This follows from the fact that $S⊥$ is “irrotational:” the line-sum of spins $S⃗⊥$ (gray in Fig. 1) along a closed loop is zero.

Similarly, if a spin configuration has zero topological current on each plaquette (I_p[S] = 0 ∀p), a height function h_‖v can be defined on the vertices by

where e is the edge connecting the vertices vv′.

Thus, in the ice manifold, there are no charges, currents are disordered, and $S⊥$ is the discrete gradient of h_⊥, which “labels” the disorder of currents in the absence of charges. Conversely, in the current-free manifold, S is the discrete of h_‖, which labels the disorder of charges in the absence of currents.

Equations (7) and (8) are thus the discrete analogue to Eq. (1) in the continuum. A significant difference is that they are well defined only in the charge-free or the current-free state. We will show in Sec. V A 2 how to generalize them to any spin ensemble, inclusive of monopole and current excitations, and thus at non-zero temperature.

The height formalism can usefully, if heuristically, describe the pure ice manifold. Height models are said to be in a “rough” (degenerate) or “flat” (ordered) phase, which is jargon derived from the theory of roughening transition historically associated with these models by various exact mappings.^65,66

The ice manifold of the square ice, i.e., the six-vertex model, is known to be equivalent to a dimer cover model^48,67 and thus in a rough phase.^50,56,57 A widespread,^56,57,65,66 although by no means, rigorously justified ansatz ascribes to a configuration in the ice manifold, an entropy⁶⁸ that is quadratic in the height function. In our language, we can write

where h_⊥ is homogenized into a continuum field and χ₀ is the positive uniform susceptibility (seen later). Clearly, the same is true for the current-free manifold by replacing h_⊥ with h_‖.

Equation (9) can be understood in terms of the zero temperature partition function

where $H⃗$ is an external field.

Note that Eqs. (9) and (10) are not obviously unproblematic in a 2D (and thus gauge-free) theory. In 3D, we would be safe as gauge invariance of the transverse part of the field forbids the proliferation of relevant operators at the fixed point (which, incidentally, is why a Higgs boson is needed in the standard model). Equation (9) merely happens to work in reproducing correlations that can also in part be computed exactly^48,69 (see also Ref. 57 and references therein for a discussion).

A series of interesting deductions come from Eqs. (9) and (10). For height function correlations, in reciprocal space, one immediately finds

From that and the continuum limit of Eq. (7) (or $S⃗=−ê3∧∇⃗h⊥$⁠), one obtains the spin correlator as⁷⁰ 

which is purely transversal.

Note that from Eq. (11), in real space, correlations of the height function are logarithmic, or

Spin correlations in real space can be obtained from Eq. (12) or more easily as partial transversal derivatives (or $∇⃗⊥=ê3∧∇⃗$⁠) of Eq. (11), obtaining

where $P(x)$ is the kernel of the dipole–dipole interaction in 2D, or

Analogously, in the case of pyrochlore spin ice, the spin correlations are the kernel of the 3D dipolar interaction.⁵⁰ 

The spin correlations are therefore algebraic, making the ice manifold a critical phase of infinite correlation length. On the other hand, the correlation length for currents is zero: from Eqs. (11) and (3), we have

which implies the infinitely localized screening of any pinned current.

In terms of currents, from Eq. (9), the entropy for the ice manifold can be rewritten as

i.e., as a pairwise 2D-Coulomb interaction among the currents.

The phenomenological picture is thus the following: in the ice manifold, charges are absent, disorder can be labeled by currents, and their 2D-Coulomb mutual interaction determines the entropy.

From Eqs. (9) and (1), the entropy of a configuration in the ice manifold can be written in terms of $S⃗$ as

with the constraint $∇⃗⋅S⃗=0$⁠.

Equation (18), unlike the previous formulas, is also valid in 3D. There, it has been labeled as Jaccard entropy in water ice^60,71–74 and later in pyrochlore spin ice^50,51,53,55 necessary to produce purely transverse correlations in the ice manifold. We see, therefore, that in square ice, the Jaccard entropy describes, in fact, a 2D-Coulomb interaction among disordered currents. The uniform susceptibility χ₀ is thus related to the so-called Φ constant.⁷⁴ 

We note that in the previous deductions, we have taken some cavalier liberties with the boundary terms. With collaborators, we have already shown how fixing the boundaries can induce a net charge in the bulk via a geometrical expression of Gauss’s Law.⁴⁵ 

In the following, we will show how to deduce a field theory for charges and currents in which the heuristic equations (1) and (3) make sense, the intuitive height function formalism [Eqs. (9)–(18)] finds a solid ground, and it is generalized for T > 0.

Section II C pertains to degenerate states, charge-free or current-free, but does not account for excitations and thermal fluctuations.¹ For that, we need an energetic description.

The following Hamiltonian,

reflects the current-charge duality by placing a cost or advantage on topological currents and monopoles (ϵ and κ are energies). In terms of an Ising model, it is equivalent to a J₁, J₂, J₃ model where (Fig. 1) J₁ = ϵ − κ, J₂ = −κ, and J₃ = ϵ. By the duality, the symmetry by orthogonalization corresponds to ϵ ↔ κ. The Hamiltonian describes various cases, often close to the experimental reality.

If κ = 0 and ϵ > 0, the ground state is the ice manifold shown in Fig. 1 (black arrows). Equivalently, by gauge-free duality, if ϵ = 0 and κ > 0, the ground state is an extensively degenerate ice manifold for $S⃗⊥$⁠, shown through gray arrows in Fig. 1.

If κ = 0 and ϵ < 0, the ground state is the charge-full state, i.e., the ordered, antiferromagnetic tessellation of t-IV vertices. If ϵ = 0 and κ < 0, the ground state is the current-full state, i.e., the ordered, antiferromagnetic tessellation of t-I vertices, which is the orthogonal of the charge-full state.

For κ > 0 and ϵ > 0, both charges and currents are suppressed. Because J₁ < J₃, ferromagnetic t-II vertices are promoted over t-I. Because J₂ < 0, t-II vertices want to align, and the ground state is the fourfold ferromagnetic state made of t-II vertices ferromagnetically aligned (more loosely; as the ground state is current-free and charge-free, we have Δh_‖ = Δh_⊥ = 0, which implies a uniform $S⃗$⁠.). This case has not been investigated experimentally, although it can certainly be realized in a quantum annealer.⁴⁵ It might approximate, however, experimental situations where t-II vertices can be favored,^38,42,75 leading to a line state that is disordered but of sub-extensive entropy.

For κ < 0 and ϵ > 0, the ice rule is enforced at low T, currents are promoted, J₁ > J₃, and the ground state is an ordered antiferromagnetic (AFM) tessellation of t-I vertices. Large |κ| describes early square ice realizations.⁹ Small |κ| might approximate spin ices that are designed so that ice rule vertices are degenerate^35,38 but where the dipolar interaction still favors closed magnetic fluxes, thus promoting topological currents and an ordered ground state.

When κ = ϵ, we have J₁ = 0, and the set of vertical and horizontal arrows becomes two decoupled systems. When κ = ϵ > 0, each subsystem is ferromagnetic, and the ground state is a fourfold fully polarized state. When κ = ϵ < 0, each subsystem is antiferromagnetic, and the ground state is a fourfold antiferromagnetic state, which includes the two orientations of the charge-full state and the two orientations of the current-full state (one is the orthogonal of the other, as expected on the symmetry line κ = ϵ).

Here, we will not study the full monopole-current model of Eq. (19), which leads to a rich phase diagram in the βϵ × βκ plane (where β = 1/T and T is the temperature measured in units of energy) to be compared with other models.^29,76 We will consider only ϵ > 0 and |κ|/ϵ to be small and investigate how ice manifold features are retained by small perturbations around the spin ice point κ = 0.

Because charges and currents represent an emergent description of pure square ice, we deduce a field theory for which they are the relevant degrees of freedom.

We generalize our previous approach for general graphs^49,61 to include currents. The partition function from Eq. (19) reads

and it is the generator of correlations,

The fields $H⃗,Vq,Vi$ are measured in units of energy.

To obtain a continuum field theory, in the sum of (20), we insert the tautology

and then sum over the spins, obtaining

where $dqdi=(2π)−Nv∏vdqv∏pdip$⁠. $Ω̃[q,i]$ is a generalized density of states for q_vandi_p, given by

and it is therefore the functional Fourier transform of

(the product runs on all the edges $e=vv′̂$ once, and ∇_vv′ϕ : = ϕ_v′ − ϕ_v, ∇_pp′ψ : = ψ_p′ − ψ_p, and $Hvv′:=H⃗e⋅vv′̂$⁠, while $pp′̂=−êz∧vv′̂$⁠).

Note that by construction, $⟨Qv1…Qvn⟩=⟨qv1…qvn⟩$⁠, and $⟨Ip1…Ipn⟩=⟨ip1…ipn⟩$⁠. Note also that the use of $H⃗$⁠, V_q, and V_i is superabundant: $Vq→Vq+Vq′,Vi→Vi+Vi′$ is equivalent to $H⃗→H⃗+∇⃗Vq′−ê3∧∇Vi′$⁠, but we keep it because it is useful.

We have gone from binary variables to a theory of continuous emergent topological charges and currents constrained by an entropy

which conveys the effect of the underlying spin ensemble. Equivalently, in the language of field theory, charges q_v and currents i_p interact entropically via the fields

of the generalized free energy

Note that $F[ϕ,ψ]$ can have imaginary values. As a consequence, although the variables ϕandψ are themselves real, their expectation values ⟨ϕ⟩ and ⟨ψ⟩ are imaginary, and thus, the entropic fields $⟨Vqe⟩,⟨Vie⟩$ are real. Indeed, by integrating over q in Eq. (23) and applying the second and third equation in (21), one finds

when κ > 0. Similar Gaussian gymnastics also prove that

We thus see that while $i$ϕ and $i$ψ correlate charges and currents, the vector fields −$i$T∇_vv′ϕ and −$i$T∇_pp′ψ correlate spins that would otherwise be trivially paramagnetic.

In the spirit of field theory, we could therefore say the following: $Vqe$ is an entropic potential acting on charges, and $Vie$⁠, on currents; correspondingly, there is an entropic field $B⃗e$ acting on the spins, which in the long wavelength approximation is given by

When ϵ > 0 and κ > 0 (see Sec. VI C for κ ≤ 0), integrating over charges and currents in Eq. (23) returns Gaussians in the fields of variance ⟨ϕ²⟩ = ϵ/T and ⟨ψ²⟩ = κ/T: as temperature increases, the entropic fields mediating spin correlations become smaller, as one would expect. In fact, in the infinite temperature limit (but with βH constant), the Gaussian distributions in the field become Dirac deltas, and Eq. (23) becomes the standard “paramagnetic” partition function,

Note that for H = 0, Eq. (32) returns the correct entropy per spin at infinite temperature, s = ln 2.

In the high T limit, we can linearize Eq. (30) and write, in the long wavelength limit,

Then, from $q=−∇⃗⋅S⃗$⁠, $j=ê3⋅∇⃗∧S⃗$⁠, and Eq. (29), we find screened Poisson equations for q and I,

(here, $νqext=∇⃗⋅H⃗$⁠, where q_ext is the external charge, $νiext=∇⃗∧H⃗$⁠, where i_ext is the external current, and ν is an energy), and therefore “correlation lengths”⁷⁷ as

This expression for ξ_‖ was already achieved in other works, for different geometries, via different means,^52,53 and we had generalized it on a graph.⁶¹ Note that for κ < 0, ξ_⊥ would be imaginary, pointing to a periodicity typical of the AFM ensemble, as we will discuss in section VIC. Instead, when κ = 0, the integral over i returns a functional delta function on ψ, and ψ disappears from the equations, which for all purpose are equivalent to taking ξ_⊥ = 0.

We now explore these heuristic deductions more precisely. Because we are interested in the monopole liquid below the ice manifold threshold T ≃ 2ϵ but above other possible low T transitions, a high T approximation is a good starting point. Since it corresponds to small entropic fields, we expand ln Ω[ϕ, ψ] at quadratic order and Fourier transform via $gx=∫BZg̃(k⃗)e−ik⃗⋅xd2k/(2π)2$⁠, where BZ is the Brillouin zone, g is a generic field, and x = v, p, l represents vertices, edges, or plaquettes, respectively.

We obtain the approximated partition function at zero loop,

where

is an effective Hamiltonian for the new variables and the vector $γ⃗$ has components

for α = x, y, which contains information on the square symmetry of the lattice [in the long wavelength limit: $γ⃗≃k⃗+O(k3)$].

The first line of Eq. (37) contains the free energies for the uncoupled charge, currents, and entropic fields.

The second line contains the coupling between charges, currents, and their entropic fields; importantly, no ϕψ cross term survives at quadratic order in the high T approximation, and charge and currents are independent at second order, leading to the so-called magnetic fragmentation.^78–80 

The third line contains the coupling with the external field $H⃗$ (here, we neglect V). Because $−iγ⃗⋅w̃⃗$ and $−iγ⃗∧w̃⃗$ are the generalized divergence and curl, respectively, on the lattice in moment space for a generic field $w⃗$⁠, the entropic fields for currents (resp. charges) couple to the curl (resp. divergence) of the external field.

The quantity χ₀, already encountered in Eq. (9) on the opposite limit (T = 0), is the uniform susceptibility with respect to β H. In Eq. (37), it is χ₀ = 1, but we include it, nonetheless, in the equation because it can be χ₀ ≠ 1 at low temperatures (see below).

Integrating Z₂ over $ϕ̃,ψ̃$ when $H̃=0$ returns the effective free energies for q and i in the absence of external field, which are decoupled at second order,

where the two terms can be written as

In field theory language, the first terms in Eq. (40) are the “masses” of the monopole or of the current. The second terms imply that the underlying spin ensemble mediates a pairwise entropic interaction among charges (or currents), which is the 2D-Coulomb potential. In real space at large distances (where γ² ≃ k²), the entropic interactions for charges and currents are

The origin of entropic interactions is in the emergent nature of charges and currents, which exist in a spin vacuum. An assignation of charges and/or currents to the system changes the number of ways in which the underlying spin ensemble can be compatibly arranged. Thus, the entropy associated with a fixed distribution of charges and currents depends on their mutual position and distance. While this is obvious, it is not obvious that changes in entropy can be written via pairwise terms, as shown above, leading to an effective pairwise interaction. We see that subsuming the effect of the underlying spins leads to a “2D electrodynamics” formalism for charges and currents.

From Eq. (39) and equipartition, we obtain the correlations for charges and currents,

where $χ̃‖(k),χ̃⊥(k)$ are defined as

Correctly, from Eq. (42), we have$⟨|q̃(0⃗)|2⟩=0$ since $⟨|q̃(0⃗)|2⟩=⟨∑vqv2⟩=0$ (and the same for currents). Figure 2 shows a heat map for $⟨|q̃(k⃗)|2⟩$⁠.

Furthermore, if ⟨q²⟩ is the average charge per vertex, or $⟨q2⟩=1Nv∑v⟨qv2⟩$⁠, then

From it, and Eq. (42), we obtain ⟨q²⟩ → 4 for T ↑ ∞, which is correct. Indeed, ⟨q²⟩ = 4 is the value deducible from a multiplicity argument (2²/2 + 4²/8 = 4). Because of the duality, the same is true for currents.

Note that $χ̃‖(k),χ̃⊥(k)$ are the longitudinal and perpendicular static susceptibilities (multiplied by T, thus accounting already for the Curie–Weiss law) for $S̃‖=S̃⃗⋅γ̂$ (the longitudinal, charge-full, current-fee part of the magnetization) and $S̃⊥=S̃⃗⋅⊥γ̂$ (the perpendicular, charge-free, and current-full magnetization), respectively.⁸¹ Indeed, by integrating the Gaussian integral in Eq. (36) and remembering that $γ̂αγ̂α′+γ̂α⊥γ̂α′⊥=δαα′$⁠, one obtains the total free energy as a function of the external field at lowest order as

From it, the spin correlations could be obtained as

In the limit T → ∞, Eq. (46) correctly returns to $⟨S̃α(k⃗)S̃α′(k⃗)⟩→δαα′$⁠, or uncorrelated spins.

Equation (46) also implies that the magnetic susceptibilities in reciprocal space can be obtained from experimentally measured spin–spin correlations as

and similarly, from (42),

Finally, from the spin correlations, we can obtain the magnetic structure factor,

which at small k corresponds to $χ̃⊥$⁠.

Taking the long wavelength approximation, Eq. (46) implies the following effective free energy for the coarse grained spins:

which in the long wavelength limit becomes

The first two terms in the previous equation are energetic. The third is the Jaccard entropy^2,60 mentioned above. In the ice manifold, it is the only surviving term since ξ_⊥ = 0 and $∇⋅S⃗=0$⁠, thus returning to Eq. (18), previously found heuristically.

In Eq. (51), ξ_⊥ = 0 reduces to the functional that was found elegantly via methods of chemical physics for pyrochlore ice by Bramwell.^58,59 Indeed, when expressed in terms of spins $S⃗$ rather than currents and charges iandq, the two formalisms are expected to coincide.

Expressing the coarse grained $S⃗$ via height functions as in Eq. (1), we obtain the free energy for the height functions in the long wavelength limit, at quadratic order,

When ξ_⊥ = 0, Eq. (52) represents the generalization for T > 0 of the heuristic equation (9). When T = 0h_‖ is constant, Eq. (52) reduces to Eq. (9). Moreover, in fact, Eq. (52)tends to Eq. (9) when T → 0. For T > 0, monopoles appear, and thus, the longitudinal height function is also present.

When T ↓ 0, fluctuations of the entropic fields should diverge. One could then perform a proper study of the higher order expansion of Ω(ϕ, ψ) in terms of Feynman diagrams. Because ∈ Ω has the form of a locking potential, and at low temperature $∇⃗ϕ,∇⃗ψ$ are no longer limited, vertex operators might intervene.⁹⁹ Because of the complex shape of Ω(ϕ, ψ), such a program is challenging. It might also be uninteresting in real systems.

That is because at very small T, our model becomes insufficient for real systems. First, in magnetic systems, long-range interactions become relevant at low temperature, and they can induce ordering at equilibrium.⁸² Second, regardless of the range of interaction, creation of monopoles is suppressed when T ≪ ϵ. Therefore, spin flips that do not require large (compared to T) activation energy either pertain to spins impinging on a monopole and that move the monopole or to collaborative flips of entire loops, which are exponentially unlikely because of the size of the loop. Thus, one expects a freeze-in as T ↓ 0, in analogy with what is seen in pyrochlores,⁸³ but our approach is at equilibrium.

We will therefore intend T as small but not too small and proceed by assuming that the functional form of an effective theory is quadratic and has the same functional form as the theory mentioned above, but those interactions among fluctuations lead to a “dressing” in the constants.

Because correlation lengths diverge at low T, from Eq. (44), for charges, we have

which implies a redefinition of constants in the theory at low T: ϵandκ are dressed as

Note that this dressing corresponds to a quadratic theory that has the correct mean square charge and current already from equipartition. Note also that Eq. (53) has the form of a Debye screening length for a 2D-Coulomb potential whose coupling constant is proportional to T, which is the case for our entropic potential. We have shown elsewhere⁴⁹ that Eq. (53) can follow from a Debye–Hückel approach to the entropic potentials (see also the  Appendix).

Note finally that χ₀ must be finite at low T (see also below) and that ⟨q²⟩ is decently approximated by the naive $q2̄$ computed by assuming uncorrelated vertices, each with proper multiplicity and Boltzmann weight. At low T, $q2̄∼(16/3)exp(−2ϵ/T)$⁠. We thus have $ξ‖∼expϵT$ for T ↓ 0. A similar exponential behavior for the correlation length was indeed suggested experimentally by analyzing the pinch points in the structure factor of pyrochlore ice.⁸⁴ This exponential singularity points to the topological nature of the T = 0 ice manifold.

In conclusion, at low T we can use all the equations of Sec. V A, if expressed in terms of ξ_‖ and ξ_⊥, where

and similarly,

For χ₀ and χ₀ ∼ 1, T ↑ ∞ while it remains finite at T = 0 (we show below that, e.g., for T = 0andκ = 0, we have χ₀ = 2).

In the  Appendix, we further motivate how this choice is mathematically reasonable. Note also that no Kosterlitz–Thouless transition of monopole or current unbinding is present⁶² because the interaction is entropic. Transitions are driven by the interplay between temperature and energy, but here, interaction is itself entropic and thus thermal.

In this case, the ground state, aka ice manifold, is the degenerate six-vertex model,¹ described in Sec. II. In the purest form—i.e., without the interference of long-range interaction—it was recently realized in a quantum annealer,⁴⁵ where the entropic effects were cleanly studied. Nanomagnetic realizations^35,38,39 imply long-range interaction, whose ulterior effects have been described elsewhere.⁴⁹ 

When κ = 0, the functional integration over ∏_pdi_p in Eq. (23) produces the delta functions ∏_pδ(ψ_p). Further integration of ∏_pdψ_p returns it to the following partition function:

where $Ω̃[q]$ is the density of states for the charges, and it is given by

which is therefore the functional Fourier transform of

In other words, the currents disappear from the picture, and the entropic potential −$i$Tψ is replaced in the equations by the external potential V_i acting on currents. Then, proceeding based on the quadratic approximation derived before and using the third of Eq. (21), for the current correlations, one obtains

which for small wave vectors reduces to Eq. (16).

More generally, the reader will find that all the equations of the theory developed above apply to the pure ice case by taking ξ_⊥ = 0. For example, from Eq. (43), $χ̃⊥(k⃗)=χ0$⁠, and the perpendicular susceptibility in real space is a delta function, consistent with zero correlation length.

Considering charge correlations and screening, the first of Eqs. (42) can be rewritten as

where the first term Fourier-transforms to a Kronecker delta, whereas the second term implies the charge correlation at large distance,

Note that the modified Bessel function K₀ is the screened 2D-Coulomb potential. It is exponentially screened, or $K0(x)∼π/2xexp(−x)$⁠, making ξ_‖ the correlation/screening length, as presaged above. In 3D, we would have a screened 3D-Coulomb form,¹⁰⁰ exp(−|v₁ − v₂|/ξ_‖)/|v₁ − v₂|. This result is general: for spin ice on a generic graph, for which Laplacian operators can also be defined, entropic interactions lead to screened Coulomb correlations.⁶¹ Note that from Eq. (43), the kernel of the longitudinal susceptibility functional is also screened 2D-Coulomb, given by the modified Bessel function K₀ with correlation length ξ_‖.

In artificial realizations, it is possible to pin a charge Q_pin in v₀. Then, it is easy to show that the pinned charge generates a charge distribution

Remarkably, the screening comes entirely from the entropic interaction. This has been verified in a quantum annealer, where charges can be pinned.⁴⁵ Figure 3 reports experimental results of screening, which verify Eq. (62). At high ϵ/T, the curve becomes flat, consistent with Eqs. (61) and (62) as the correlation length exceeds the finite size of the sample.⁸⁵ 

We can gain some knowledge of screening at small distances by approximating around the K = (±π, ±π) points of the BZ. There, γ(k)² is maximum, and $γ(k⃗+K)2=8−k2$⁠. This leads to a screening function