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Monte Carlo study of artificial two-dimensional square spin ices: may magnetic monopoles be present in these systems? Afranio R. Pereira, Lucas A. S. Mól, Winder A. Moura-Melo Departamento de Física, Universidade Federal de Viçosa, , Viçosa, MG, Brazil

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Recently, a lot of experimental effort is under way with the objective of constructing dipolar arrays of monodomain nanomagnets.

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**Spin ice materials (Dy2Ti2O7, Ho2Ti2O7) were discovered in 1997**

Spin ice materials (Dy2Ti2O7, Ho2Ti2O7) were discovered in They provided us with an instance of a new magnetic state which is distinct from paramagnets, ordered magnets, and spin glasses. theoretically, they are described by the Hamiltonian.

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1. In 2006, a group from Penn State University has built an artificial two-dimensional system that mimics the frustrated spin ice materials (Nature 439, 303 (2006)). However, the six bonds between the four islands belonging to a vertex are not all equivalent.

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Still in 2006, Möller and Moessner (PRL 96, (2006)) proposed and studied (theoretically) a modified artificial square spin ice (to my knowledge, not yet produced). The modification consists of introducing a height offset h between islands pointing along the two different lattice directions. In principle, for a special value of h=hice = 0.419a, all states obeying the ice rule would be approximately equal (degenerate). h

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In 2008, Castelnovo et al. proposed that the excitations in the 3d original spin ices are magnetic monopoles (Nature 451, 42 (2008)). Qb = - Qa < 0 Zero magnetic charge Qa =0. Qa > 0

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An interesting question would be then to ask if this kind of excitation arises in lower dimensional spin ices. Inspired by the artificial 2d array, we tried to put some light on this topic in 2008 (arXiv: (2008), JAP 106, (2009)). We found that the excitations (at zero temperature) are pairs of defects very similar to a monopole-antimonopole pair.

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Indeed, forgetting (for a while) the “background”, the force between the members of such a pair of defects obeys the 3d Coulomb law –QMR/R3 (R is the distance between the charges). The field lines are not confined to the plane. The magnetic charge is inversely proportional to the lattice parameter a (QM =constant/a).

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However, the “background creates” an additional attractive force (independent of R) between the charges. An energetic string confining the monopoles arises. In this same paper, we have also argued that, as the temperature (T) increases from zero, the string effective tension decreases, vanishing at a critical value (Tc), above which the monopoles should become free to move.

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In 2009, Möller and Moessner (PRB 80, (2009)) proposed that magnetic monopoles could arise free (independently of T) in their modified artificial square array (at hice =0.419a). Really, the linear confining force can be removed by shifting the monopoles in the third dimension so that their locations define a tetrahedra. In this same paper, the authors have also studied these excitations for an artificial kagome lattice. h hice

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In 2010, a direct observation of monopole defects in 2d artificial spin ice systems with the Kagome lattice was reported by Ladak et. al. (Nature Phys. 6, 359 (2010)).

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**Energy (1) < (2) < (3) < (4) (1) (2) (3) (4)**

Let us go back to the artificial square lattice spin ices. No direct observation of free poles was reported yet. Why? From now, we consider this system in more details. As well described by Wang et. al. , there are 4 distinct topologies for the configurations of 4 magnetic moments in a vertex (Nature 451, 42 (2008)) . Energy (1) < (2) < (3) < (4) (1) (2) (3) (4) Topologies (1) and (2) obey the ice rule. However, topology (2) is more energetic than (1).

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In our scheme, the magnetic moments of the islands are replaced by point-like dipoles (spins). Then, we describe the system by the following Hamiltonian: Shi Svi where D =0 2/4 is the coupling constant of the dipolar interaction, (from experimental data, D≈ 2 × J, ), is the lattice constant and Si represents the spins, which can assume only the values: Shi = (Sx =±1, Sy =0, Sz=0) or Svi = (Sx =0, Sy = ±1, Sz=0). To calculate the dipolar interactions between the spins, we use the Ewald summation technique.

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In a lattice of volume L2 = l2 a2 (2l2 spins), the sum is either over all l2(2l2–1) pairs of spins in the lattice for the case with open boundary conditions or over all spins and their images for the case with periodic boundary conditions. The results are similar for these 2 cases. We have studied lattices with several different sizes (L from 6a to 80a).

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**Of course, the first thing to do is to obtain the ground state**

Of course, the first thing to do is to obtain the ground state. Below, on the left, we show the configuration of the ground state. Clearly, it obeys the ice rule (all vertices with topology 1). Just for comparison, we also show other two states that obey the ice rule but with energies about 4 times larger than the energy of the actual ground state (for the case h=0).

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**The ground state looks like a checkerboard**

The ground state looks like a checkerboard. The effective magnetic charge in each vertex is naturally zero. The most elementary excitation involves inverting a single spin (violating the ice rule) to generate localized dipole magnetic charges (blue and red balls with a red arrow in between).

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The simplest excitations are then 2 neighbor vertices, one of them in the 3-in, 1-out (red) state and the other in 3-out, 1-in (blue) state. In principle, blue and red charges could be separated without further violation of the ice rule. However, there is a price to be paid. It just happens if the topology is changed along a line of vertices connecting the two charges. This change in topology cost energy!

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Indeed, differently from the 3d usual spin ice, the string connecting the charges in the 2d case is energetic, with a nonzero tension (not degenerate). The string energy must be then proportional to its length.

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Considering the system at zero temperature, we then calculate the energy of a pair of charges (defects) and its associate string as a function of their separation R and string length X. We obtain (for h=0) here L=70a: r=R/a V(R,X)= q/R +bX+ c, where Similarly to spinons in a VBS and quarks in elementary particles, the monopoles are confined in the 2d square artificial array. Then, there is confinement.

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**Thus, the total cost of a monopole pair in 2d square spin ice is the usual Coulombic term…**

…and a term resulting from the string joining the monopoles. (there is still a constant term, related to the creation energy of a pair). The magnetic charge can be calculated using the value of q obtained in expression V(R,X)= q/R +bX+ c, to compare with the Coulomb law. QM = (4

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**Some interesting numerical data for a square lattice with size L=70 a (9800 spins) and h=0.**

The lattice size is not so important. Pair creation energy: about 29D. Charge |QM|=(4π|q|/μ0)1/ =2.1 μ/a. For a=320 nm, it is about 100 times smaller than Dirac fundamental charge. On the other hand, it would be about 80 times larger than the monopole found in the 3d spin ices. A pair of defects needs an energy of about 10D to separate its constituents by one lattice constant, regardless of how far apart they are.

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**Some interesting numerical data for a square lattice with size L=70 a (9800 spins) and h=0.**

4. There is anisotropy in the Coulomb ( q/R) and linear ( bR ) interactions of two poles: the values of q and b depends on the direction in which the charges are separated. For instance, q= Da and b=9.75 D/a for a separation of charges along the same line on the lattice, while they are Da and D/a, respectively for diagonal separation. Smaller string tension

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**Now, some results for the modified system as a function of h.**

The anisotropy for the charges interaction happens to all values of h, but tends to disappear at h=h1=0.444a, where the ground state changes its configuration. Indeed, this result can be compare with the one of Möller and Moessner, hice = 0.419a.

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For h<0.444a the ground state has the same configuration as that for h=0 (GS1, see left figure). For h>0.444a, see the configuration on the right ( GS2). h GS1 GS2 These two configurations GS1 and GS2 obeying the ice rule have the same energy at h=h1=0.444a and are degenerated (as expected from the work of Möller and Moessner, all configurations obeying the ice rule should become degenerated at the critical h).

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However, we have found that it is not completely true because there is another configuration (also obeying the ice rule) with an energy a little larger than the two configurations GS1 and GS2. Ice rule but with an energy larger than the two configurations on the left. GS1 and GS2: ice rule and degenerate at h=0.444a. h

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As a consequence of this difference of energies, the string tension does not vanish even for h=0.444a. Really, it decreases as h increases but it does not vanish for any value of h. At h=h1 =0.444a, b is about 20 times smaller than that for the case h=0. Ice rule but with an energy larger than the two configurations on the left. GS1 and GS2: Ice rule and degenerate at h=0.444a.

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For h<0.444a, the red line is for diagonal separation while black line is for separation along the same line of vertices. For h>0.444a, blue line is for linear separation while green line is for diagonal separation.

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**V(h,φ)= q(h,φ)/R +b (h,φ) X+ c (h),**

We then rewrite the potential as V(h,φ)= q(h,φ)/R +b (h,φ) X+ c (h), Where φ is the angle between the line connecting the charges and the x-axis. Note that the anisotropy tends to disappear at h=h1=0.444 a (see q and b in figures below). h

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**Some comments about the system with h>0:**

It changes its ground state at h=h1=0.444a (to compare with hice =0.419a). The creation energy of a pair monopole-antimonopole decreases as h increases. The string tension b also decreases as h increases. However, it does not vanish for any value of h. At h=0.444a, where, in principle, it should vanish, b≈0.59D/a, about 20 times smaller than that for the case h=0. The anisotropy of the interaction potential between the charges does not completely disappear at h=0.444a. Indeed, the system is not completely degenerate at this value of h. Ice rule but with an energy larger than the two configurations on the left. GS1 and GS2: Ice rule and degenerate at h=0.444a. h

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**Some comments about the system with h>0:**

5. The pole charge is better defined at h=0.444a, where the anisotropy almost disappears. Thus, we get h For h≠ h1, we define with

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Thus we have numerically shown that the interaction follows a d=3 Coulomb law q/R. Besides, we have quantified several objects of the system (there is only a very small dependence on the lattice size L). Among them, the magnetic charge, string tension, pair creation energy etc. All these quantities were calculated at zero temperature (T=0) and also as function of h. Finally, we will argue that a phase transition at a critical Tc may occur. How?

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We all know that, in artificial spin ice patterns, the islands magnetization is unaffected by thermal fluctuations. All numerical results presented until now are valid for zero temperature. Then, a natural question is: what happens at finite temperatures?

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Although the moment configuration is athermal, these artificial materials can be described through an effective temperature. Such findings will be explained tomorrow afternoon by Dr. Nisoli.

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We start the thermodynamic study by making a simple analytical analysis. There are many possible ways of connecting two monopole defects with a string. Below we show some of them for a string length equal to X=24a, R=2a. Indeed, for X sufficiently large (X>>R), the number of configurations would be well approximated by the random walk result p X/a (for a 2d square lattice, p = 3).

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**For example, if h=0, kBTc ≈ 9.1 D**

Then, the string configurational entropy ( kB ln p X/a ) is proportional to X and the string free energy can be approximated by So we have an effective tension beff which vanishes for kBTc ≈ba/ln(3). For example, if h=0, kBTc ≈ 9.1 D

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We summarize the physical picture as follow: at low temperatures, just a few small pairs of defects (monopole-antimonopole pairs) are excited above the ground state. As the temperature is increased, some of them grow up, but remain relatively small due to the string tension. There is linear confinement. For high enough temperatures, above a critical value Tc, the string looses its tension by the entropic effect. Deconfinement may occur and monopoles may become free.

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**However, there is still a complication**

However, there is still a complication. I should also mention that there is another entropic effect. This is an entropic interaction between the monopoles due to the underlying spin configuration. Two monopoles should be attracted because there are more ways to arrange the surrounding dipoles in the lattice when they are close together. This interaction has the form of a Coulomb law in two dimensions (lnR) and is proportional to T. More space for disorder!!! T ln R

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To test these ideas we analyze some thermodynamics properties of the system. Once an effective dipole-dipole interaction between spins i and j within the simulation cell has been derived via the Ewald summation, we perform standard Monte Carlo techniques to calculate the averages of some quantities of the model. We calculate, as a function of temperature, the following quantities: The pair density. The average size of pairs. Specific heat.

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Density of defect pairs constituted by unit-charged monopoles for a lattice with L=40a. Right, density for doubly charged defects. These are the density of pairs or, equivalently, the density of positive (or negative) charges. Data for h=0.

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**Left: average pair size (L=20a and 40a )**

Left: average pair size (L=20a and 40a ). It starts to have an appreciable increasing around kBT=7D. Right: specific heat. The peak is also around kBT=7D. Data for h=0.

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We believe that things must be much better for systems with a height offset (h>0) for the following reasons: The creation energy of defect pairs is lower (mainly as h tends to h1 = 0.444a). The string tension is also much lower. Therefore, one would have: The number of pairs becomes appreciable at small temperatures. The string would break its tension at a temperature much smaller than that of the case with h=0. Then the other entropic effect (which is proportional to T , i.e., T ln ( R/a ) ) would be only weak, unable to avoid a possible deconfinement.

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**We would like to thank the Brazilian research agencies**

And also

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