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FIG. 21: Finite size scaling fit for the specific peak heights of the ordering transition. Data points represent the mean Cheak value for a given L. Error bars show one standard deviation.

Figure 21 Finite size scaling fit for the specific peak heights of the ordering transition. Data points represent the mean Cheak value for a given L. Error bars show one standard deviation.

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FIG. 2: The local proton arrangement in ice, showing oxygen atoms (large white circles) and hydrogen atoms (small black circles) arranged to obey the ice rules. The displacement of the hydrogen atoms from the mid-points of the oxygen-oxygen bonds are represented as arrows, which translate into spins on the pyrochlore lattice in Fig. 1.
FIG. 2: The local proton arrangement in ice, showing oxygen atoms (large white circles) and hydrogen atoms (small black circles) arranged to obey the ice rules. The displacement of the hydrogen atoms from the mid-points of the oxygen-oxygen bonds are represented as arrows, which translate into spins on the pyrochlore lattice in Fig. 1.
FIG. 3: (a) Specific heat and (b) entropy data for Dy2Ti207 from Ref. 29, compared with Monte Carlo simulation results for the dipolar spin ice model, with Jnn = —1.24K and Dan = 2.35K.
FIG. 3: (a) Specific heat and (b) entropy data for Dy2Ti207 from Ref. 29, compared with Monte Carlo simulation results for the dipolar spin ice model, with Jnn = —1.24K and Dan = 2.35K.
FIG. 4: The long-range ordered q = (0,0, 27/a) dipolar spin ice ground state. Projected down the z axis (a), the four tetrahedra making up the cubic unit cell appear as dark gray squares. The light gray square in the middle does not repre- sent a tetrahedron, but its diagonally opposing spins are in the same lattice plane. The component of each spin parallel to the z axis is indicated by a + and - sign. In perspective (b), the four tetrahedra of the unit cell are numbered to enable comparison with (a).
FIG. 4: The long-range ordered q = (0,0, 27/a) dipolar spin ice ground state. Projected down the z axis (a), the four tetrahedra making up the cubic unit cell appear as dark gray squares. The light gray square in the middle does not repre- sent a tetrahedron, but its diagonally opposing spins are in the same lattice plane. The component of each spin parallel to the z axis is indicated by a + and - sign. In perspective (b), the four tetrahedra of the unit cell are numbered to enable comparison with (a).
FIG. 5: The phase diagram for the dipolar spin ice model. The antiferromagnetic ground state is an all-spins-in or all- spins-out configuration for each tetrahedron. The spin ice configuration, which includes the q = (0,0,27/a) ground state, is a two spins in-two spins out configuration for each tetrahedron. The region encompassed between the quasi ver- tical dotted lines displays hysteresis in the long-range ordered state selected (q = 0 vs. q = (0,0, 27/a)) as Jan/Dnn is var- ied at fixed temperature T’.
FIG. 5: The phase diagram for the dipolar spin ice model. The antiferromagnetic ground state is an all-spins-in or all- spins-out configuration for each tetrahedron. The spin ice configuration, which includes the q = (0,0,27/a) ground state, is a two spins in-two spins out configuration for each tetrahedron. The region encompassed between the quasi ver- tical dotted lines displays hysteresis in the long-range ordered state selected (q = 0 vs. q = (0,0, 27/a)) as Jan/Dnn is var- ied at fixed temperature T’.
Simulations on the dipolar spin ice model are carried out using the standard single spin flip Metropolis algo- rithm. To mimic the experimental conditions pertinent to real materials, the simulation sample is cooled slowly. At each temperature step, the system is equilibrated carefully, then thermodynamic quantities of interest are calculated. Since Dy, can be determined once the crystal field structure of the magnetic ion is known, the nearest neighbor exchange Jpn is the only adjustable parameter in the model. The single spin flip Metropolis algorithm is able to map out three different regions of the phase diagram shown in Fig. 5. Thermodynamic data indi- cates that when the nearest neighbor exchange is AF and sufficiently large compared to the dipolar interactions (Jan < 0 and |Jan| >> Dan), the system undergoes a sec- ond order phase transition (in the three dimensional Ising universality class) to an all-in or all-out q = 0 erga state. In the spin ice regime, Jeg = Jon + Dun 2 0, each specific heat data set for different Jn, shows quali- tatively the same broad peak as observed in the nearest neighbor FM exchange model,*? which vanishes at high and low temperatures (see Fig. 6). The height, Cheax,
Simulations on the dipolar spin ice model are carried out using the standard single spin flip Metropolis algo- rithm. To mimic the experimental conditions pertinent to real materials, the simulation sample is cooled slowly. At each temperature step, the system is equilibrated carefully, then thermodynamic quantities of interest are calculated. Since Dy, can be determined once the crystal field structure of the magnetic ion is known, the nearest neighbor exchange Jpn is the only adjustable parameter in the model. The single spin flip Metropolis algorithm is able to map out three different regions of the phase diagram shown in Fig. 5. Thermodynamic data indi- cates that when the nearest neighbor exchange is AF and sufficiently large compared to the dipolar interactions (Jan < 0 and |Jan| >> Dan), the system undergoes a sec- ond order phase transition (in the three dimensional Ising universality class) to an all-in or all-out q = 0 erga state. In the spin ice regime, Jeg = Jon + Dun 2 0, each specific heat data set for different Jn, shows quali- tatively the same broad peak as observed in the nearest neighbor FM exchange model,*? which vanishes at high and low temperatures (see Fig. 6). The height, Cheax,
subtracting it off from the (total) experimental specific heat, we can uncover the underlying magnetic contribu- tion and compare to the theoretically calculated Monte Carlo specific heat data, from which Tpeak or Cheak Can be determined directly (Fig. 8). We note here, as recently observed in Ref. 78, that for Dy2Ti2O7 there should be a hyperfine nuclear contribution to the specific heat man- ifesting itself at a temperature below T XS 0.4 K in the data of Ref. 29 if one uses the typical hyperfine contact interaction expected for a Dy?+ insulating salt. The ab- sence of the high temperature 1/T? tail of the nuclear specific heat (on the descending low temperature side of the magnetic specific heat of Dy2Ti2O7) below 0.4 K in Fig. 3a is, therefore, somewhat puzzling.”
subtracting it off from the (total) experimental specific heat, we can uncover the underlying magnetic contribu- tion and compare to the theoretically calculated Monte Carlo specific heat data, from which Tpeak or Cheak Can be determined directly (Fig. 8). We note here, as recently observed in Ref. 78, that for Dy2Ti2O7 there should be a hyperfine nuclear contribution to the specific heat man- ifesting itself at a temperature below T XS 0.4 K in the data of Ref. 29 if one uses the typical hyperfine contact interaction expected for a Dy?+ insulating salt. The ab- sence of the high temperature 1/T? tail of the nuclear specific heat (on the descending low temperature side of the magnetic specific heat of Dy2Ti2O7) below 0.4 K in Fig. 3a is, therefore, somewhat puzzling.”
Our MF derivation begins with the Hamiltonian for (111) Ising spins, Eq. (2), but expressed in terms of the Ising variables, 07, and local quantization axes, 2°, Recall that indices a and b denote the sub-lattice and RY is the vector that connects spins of and or. The pyrochlore lattice is a non-Bravais lattice which is de- scribed in a rhombohedral basis with four atoms per unit cell located at positions r“ given by (0,0, 0), (1/4, 1/4, 0), (1/4, 0, 1/4), and (0,1/4, 1/4) in units of the conventional cubic unit cell of size a = RnyV8. Each of these four points define a fcc sub-lattice of cubic unit cell size a. Using H from Eq. (6), we form the free energy of our system,
Our MF derivation begins with the Hamiltonian for (111) Ising spins, Eq. (2), but expressed in terms of the Ising variables, 07, and local quantization axes, 2°, Recall that indices a and b denote the sub-lattice and RY is the vector that connects spins of and or. The pyrochlore lattice is a non-Bravais lattice which is de- scribed in a rhombohedral basis with four atoms per unit cell located at positions r“ given by (0,0, 0), (1/4, 1/4, 0), (1/4, 0, 1/4), and (0,1/4, 1/4) in units of the conventional cubic unit cell of size a = RnyV8. Each of these four points define a fcc sub-lattice of cubic unit cell size a. Using H from Eq. (6), we form the free energy of our system,
FIG. 9: A™*"(q)/D vs. q for q in the (00l) direction, in units of 27/a, for various cut-off distances, R-, and for the infinite range limit simulated by the Ewald summation tech- nique. Calculations were made at Jnn/Dnn = 0 on the spin ice side of the phase diagram, Fig. 5.
FIG. 9: A™*"(q)/D vs. q for q in the (00l) direction, in units of 27/a, for various cut-off distances, R-, and for the infinite range limit simulated by the Ewald summation tech- nique. Calculations were made at Jnn/Dnn = 0 on the spin ice side of the phase diagram, Fig. 5.
FIG. 10: The scaled maximum eigenvalues, \'""*"(q)/D, in the (hhl) plane for the same spin ice model described in Fig. 9. The dipole-dipole interactions are treated with the Ewald ap- proach.
FIG. 10: The scaled maximum eigenvalues, \'""*"(q)/D, in the (hhl) plane for the same spin ice model described in Fig. 9. The dipole-dipole interactions are treated with the Ewald ap- proach.
FIG. 11: (a) Single spin flip Monte Carlo step acceptance rate A(T) for a simulation of Dy2TizO7. The simulation be- comes frozen when the acceptance rate falls to zero. (b) The logarithm of the acceptance rate plotted versus inverse tem- perature, minus some freezing temperature Tfreeze, follows a Vogel-Fulcher type law.
FIG. 11: (a) Single spin flip Monte Carlo step acceptance rate A(T) for a simulation of Dy2TizO7. The simulation be- comes frozen when the acceptance rate falls to zero. (b) The logarithm of the acceptance rate plotted versus inverse tem- perature, minus some freezing temperature Tfreeze, follows a Vogel-Fulcher type law.
Within the Monte Carlo simulation, we use the Barkema and Newman°®°” loop algorithm originally de- signed for two dimensional square ice models, and adapt it to work in a similar manner on the three dimensional pyrochlore lattice. In the context of square ice, we tested two types of loop algorithms, the so-called long and short loop moves. In the square ice model, each vertex on a square lattice has four spins associated with it. The vertices are analogous to tetrahedron centers in the py- rochlore lattice. The ice rules correspond to “two spins pointing in, two spins pointing out” at each vertex. In the Newman and Barkema algorithm, a loop is formed by tracing a path through ice-rules vertices, alternating between spins pointing into and spins pointing out of the vertices. A “long loop” is completed when the path traced by the loop closes upon the same spin from which it started. A “short loop” is formed whenever the path traced by the loop encounters any other vertex (tetrahe- dron) already included in the loop — excluding the dan- gling tail of spins (Fig. 12). FIG. 12: Long and short loops formed by the Newmann and Barkema algorithm®®’®” on a square ice lattice. Vertices are represented by points where lattice lines cross. Each vertex has two spins pointing in and two spins pointing out, however for clarity, only spins which are included in the loops are shown. Starting vertices are indicated by large black dots. On the left is an example of a long loop, which is completed when it encounters its own starting vertex. On the right is a short loop, which is complete when it crosses itself at any point. Dark gray lines outline completed loops. The excluded tail of the short loop is shown in light gray.
Within the Monte Carlo simulation, we use the Barkema and Newman°®°” loop algorithm originally de- signed for two dimensional square ice models, and adapt it to work in a similar manner on the three dimensional pyrochlore lattice. In the context of square ice, we tested two types of loop algorithms, the so-called long and short loop moves. In the square ice model, each vertex on a square lattice has four spins associated with it. The vertices are analogous to tetrahedron centers in the py- rochlore lattice. The ice rules correspond to “two spins pointing in, two spins pointing out” at each vertex. In the Newman and Barkema algorithm, a loop is formed by tracing a path through ice-rules vertices, alternating between spins pointing into and spins pointing out of the vertices. A “long loop” is completed when the path traced by the loop closes upon the same spin from which it started. A “short loop” is formed whenever the path traced by the loop encounters any other vertex (tetrahe- dron) already included in the loop — excluding the dan- gling tail of spins (Fig. 12). FIG. 12: Long and short loops formed by the Newmann and Barkema algorithm®®’®” on a square ice lattice. Vertices are represented by points where lattice lines cross. Each vertex has two spins pointing in and two spins pointing out, however for clarity, only spins which are included in the loops are shown. Starting vertices are indicated by large black dots. On the left is an example of a long loop, which is completed when it encounters its own starting vertex. On the right is a short loop, which is complete when it crosses itself at any point. Dark gray lines outline completed loops. The excluded tail of the short loop is shown in light gray.
We use both algorithms to perform a true finite tem- perature Metropolis Monte Carlo simulation of the dipo- lar spin ice model. We code both algorithms separately, and run simulations using the regular procedure (cooled slowly from a high temperature, equilibrating carefully at every temperature step). Our preliminary Monte Carlo results for both the short and long loop are given in Fig. 13. Even though the data in this figure has low FIG. 13: Preliminary data for the low temperature magnetic specific heat (a) and energy (b) of the dipolar spin ice Monte Carlo, system size L=3, with simulation parameters set for Dy2TizO7. The data represent an average taken over ap- proximately 10° production Monte Carlo steps. Closed circles are data from a simulation of the short loop algorithm, open squares are data obtained using the long loop algorithm. Low temperature features are discussed in the next section.
We use both algorithms to perform a true finite tem- perature Metropolis Monte Carlo simulation of the dipo- lar spin ice model. We code both algorithms separately, and run simulations using the regular procedure (cooled slowly from a high temperature, equilibrating carefully at every temperature step). Our preliminary Monte Carlo results for both the short and long loop are given in Fig. 13. Even though the data in this figure has low FIG. 13: Preliminary data for the low temperature magnetic specific heat (a) and energy (b) of the dipolar spin ice Monte Carlo, system size L=3, with simulation parameters set for Dy2TizO7. The data represent an average taken over ap- proximately 10° production Monte Carlo steps. Closed circles are data from a simulation of the short loop algorithm, open squares are data obtained using the long loop algorithm. Low temperature features are discussed in the next section.
With the short loop algorithm chosen for the simula- tions, we re-investigate the Monte Carlo Metropolis al- gorithm acceptance rate. Since each loop successfully created (i.e., not aborted by encountering an ice defect) by the short loop algorithm is still subject to rejection by the Metropolis condition on the basis of its change in system energy, we expect the maximum acceptance rate to be somewhat less than the maximum efficiency of the algorithm given above. Results for the loop acceptance rate are shown in Fig. 14. Clearly, the loop algorithm be- FIG. 14: The Monte Carlo acceptance rate for the short loop algorithm (circles) as compared to the single spin flip algo- rithm (line), for a simulation using Dy2TizO7 parameters. Acceptance rates were calculated as percentages of attempted Monte Carlo steps (MCS).
With the short loop algorithm chosen for the simula- tions, we re-investigate the Monte Carlo Metropolis al- gorithm acceptance rate. Since each loop successfully created (i.e., not aborted by encountering an ice defect) by the short loop algorithm is still subject to rejection by the Metropolis condition on the basis of its change in system energy, we expect the maximum acceptance rate to be somewhat less than the maximum efficiency of the algorithm given above. Results for the loop acceptance rate are shown in Fig. 14. Clearly, the loop algorithm be- FIG. 14: The Monte Carlo acceptance rate for the short loop algorithm (circles) as compared to the single spin flip algo- rithm (line), for a simulation using Dy2TizO7 parameters. Acceptance rates were calculated as percentages of attempted Monte Carlo steps (MCS).
FIG. 15: The low temperature magnetic specific heat (a) and energy (b) of the dipolar spin ice Monte Carlo, system size L=4, with simulation parameters set for Dy2Ti2zO7. Closed circles are simulation data run with the short loop algorithm, open triangles are data obtained using the single spin flip etropolis algorithm. In the inset of (b), the energy shows an apparent discontinuity at a critical temperature JT. = 0.18 K. The broad feature in the specific heat at T = 1 K indicates the rapid development of the spin ice rule obeying states. The sharp feature at 7. is the appearance of a phase transition to a ground state being made (dynamically) accessible via the non-local loop dynamics. Note that these results are of higher statistics than those for Fig. 13, specifically, 1 x 10° equilibriation and and 3 x 10° production Monte Carlo Steps were used. In addition, the system size is increased to L = 4 as opposed to L = 3. Also, note that the location of the specific heat peak is at roughly the same temperature and is narrower than for L = 3, indicating a finite-size effect on the singular behavior of C(T).
FIG. 15: The low temperature magnetic specific heat (a) and energy (b) of the dipolar spin ice Monte Carlo, system size L=4, with simulation parameters set for Dy2Ti2zO7. Closed circles are simulation data run with the short loop algorithm, open triangles are data obtained using the single spin flip etropolis algorithm. In the inset of (b), the energy shows an apparent discontinuity at a critical temperature JT. = 0.18 K. The broad feature in the specific heat at T = 1 K indicates the rapid development of the spin ice rule obeying states. The sharp feature at 7. is the appearance of a phase transition to a ground state being made (dynamically) accessible via the non-local loop dynamics. Note that these results are of higher statistics than those for Fig. 13, specifically, 1 x 10° equilibriation and and 3 x 10° production Monte Carlo Steps were used. In addition, the system size is increased to L = 4 as opposed to L = 3. Also, note that the location of the specific heat peak is at roughly the same temperature and is narrower than for L = 3, indicating a finite-size effect on the singular behavior of C(T).
is the magnitude of the multi-component order param- eter. These three curves illustrate important finite size effects for (VW). For T < T, the different lattice sizes produce identical order parameters. By contrast, (W) for the smaller lattice size displays pronounced rounding near T, and an increased residual value for large T. The larger lattice size produces an order parameter with a clear discontinuity at T,. This discontinuity in the order paramater combined with the discontinuity of the total energy in Fig. 15b can be viewed as strong preliminary evidence for a first order transition. FIG. 16: The q = (0,0, 27/a) order parameter. Curves are shown for system size L = 2, L = 3, and L= 4.
is the magnitude of the multi-component order param- eter. These three curves illustrate important finite size effects for (VW). For T < T, the different lattice sizes produce identical order parameters. By contrast, (W) for the smaller lattice size displays pronounced rounding near T, and an increased residual value for large T. The larger lattice size produces an order parameter with a clear discontinuity at T,. This discontinuity in the order paramater combined with the discontinuity of the total energy in Fig. 15b can be viewed as strong preliminary evidence for a first order transition. FIG. 16: The q = (0,0, 27/a) order parameter. Curves are shown for system size L = 2, L = 3, and L= 4.
FIG. 18: Details of the simulation energy near the transition for different system sizes. FIG. 17: The energy probability distribution histogram for three temperatures: T = 0.178 K, T = 0.180 K (T-), and T = 0.182 K. For T > Ty (filled circles), a single peaked Gaussian histogram is present. At the transition temperature (hashed rectangles), a second peak appears which has a lower mean energy. As the temperature falls below T, (filled triangles), the peak with the higher mean energy disappears, and the system energy eventually gathers in the lowest bin.
FIG. 18: Details of the simulation energy near the transition for different system sizes. FIG. 17: The energy probability distribution histogram for three temperatures: T = 0.178 K, T = 0.180 K (T-), and T = 0.182 K. For T > Ty (filled circles), a single peaked Gaussian histogram is present. At the transition temperature (hashed rectangles), a second peak appears which has a lower mean energy. As the temperature falls below T, (filled triangles), the peak with the higher mean energy disappears, and the system energy eventually gathers in the lowest bin.
FIG. 19: Specific heat curves over the transition temperature, for L = 2 (a) and L = 3 (b) system sizes. Closed circles repre- sent data obtained with Ferrenberg and Swendsen’s histogram reweighting technique. Open triangles represent data taken using a traditional temperature cooled Monte Carlo simula- tion.
FIG. 19: Specific heat curves over the transition temperature, for L = 2 (a) and L = 3 (b) system sizes. Closed circles repre- sent data obtained with Ferrenberg and Swendsen’s histogram reweighting technique. Open triangles represent data taken using a traditional temperature cooled Monte Carlo simula- tion.
FIG. 20: Specific heat of the transition to long-range order, for system sizes L = 2, 3, and 4. It was found that peak heights for these system sizes did not vary much with the flatness of Pmu(). Critical temperatures T; were more sensitive to the flatness of the multicanonical distribution, and hence were harder to estimate than Cheak.
FIG. 20: Specific heat of the transition to long-range order, for system sizes L = 2, 3, and 4. It was found that peak heights for these system sizes did not vary much with the flatness of Pmu(). Critical temperatures T; were more sensitive to the flatness of the multicanonical distribution, and hence were harder to estimate than Cheak.
FIG. 22: The entropy calculated from integrating the simu- lated specific heat, as explained in the text. The entire value of entropy for the system (R1n2) is recovered in the high temperature limit. The inset shows the details of the entropy recovered by the transition to long-range order.
FIG. 22: The entropy calculated from integrating the simu- lated specific heat, as explained in the text. The entire value of entropy for the system (R1n2) is recovered in the high temperature limit. The inset shows the details of the entropy recovered by the transition to long-range order.
FIG. 24: A single tetrahedron projected down the z-axis. Field directions are (a) h//[100], (b) h//[110], (c) h//[111], depicted by the large arrow outline. Small arrows represent dipole moments coupled to the field. Empty circles represent decoupled spins.
FIG. 24: A single tetrahedron projected down the z-axis. Field directions are (a) h//[100], (b) h//[110], (c) h//[111], depicted by the large arrow outline. Small arrows represent dipole moments coupled to the field. Empty circles represent decoupled spins.
FIG. 25: The T=0 energies per spin of the three ice-rules ordered states of the dipolar spin ice model, as a function of applied internal field h = |h| along the (a) [100] and (b) [110] directions for Jnn = —0.52 K and Dnn = 2.35 K (i.e. HozTi2O7 parameters).
FIG. 25: The T=0 energies per spin of the three ice-rules ordered states of the dipolar spin ice model, as a function of applied internal field h = |h| along the (a) [100] and (b) [110] directions for Jnn = —0.52 K and Dnn = 2.35 K (i.e. HozTi2O7 parameters).
It should also be noted here that the q= 0 and q= X lines are parallel in Fig. 25b only for samples that are perfectly aligned with h along the [110] crystal axis. This is an important phenomenon one must consider when comparing theory and experiment, as only a small crystal misalignment will partially couple spins on the ( chains to the field. Because precise alignment of a crystal is often very difficult, the possibility of misalignment of the order of a degree must be taken into consideration when studying single crystal data with h//[110]. Repeating our ground state energy calculation for misalignment of one degree along the [100] direction, one finds a crossing of the q = X and q = 0 lines in Fig. 25b at about 1.3 T, the q = 0 configuration being of lowest energy above this field strength. FIG. 26: The T=0 energies per spin of the three ice-rules or- dered states and the three-in one-out spin state of the dipolar spin ice model as a function of applied internal field h along the [111] direction. The q = X state becomes the ground state at 0.029 T. The three-in one-out configuration becomes the ground state at around 1.4 T, breaking the ice rules for each tetrahedron. The q = X line is always 0.534 J mol~* below the q = 0 line at any given field.
It should also be noted here that the q= 0 and q= X lines are parallel in Fig. 25b only for samples that are perfectly aligned with h along the [110] crystal axis. This is an important phenomenon one must consider when comparing theory and experiment, as only a small crystal misalignment will partially couple spins on the ( chains to the field. Because precise alignment of a crystal is often very difficult, the possibility of misalignment of the order of a degree must be taken into consideration when studying single crystal data with h//[110]. Repeating our ground state energy calculation for misalignment of one degree along the [100] direction, one finds a crossing of the q = X and q = 0 lines in Fig. 25b at about 1.3 T, the q = 0 configuration being of lowest energy above this field strength. FIG. 26: The T=0 energies per spin of the three ice-rules or- dered states and the three-in one-out spin state of the dipolar spin ice model as a function of applied internal field h along the [111] direction. The q = X state becomes the ground state at 0.029 T. The three-in one-out configuration becomes the ground state at around 1.4 T, breaking the ice rules for each tetrahedron. The q = X line is always 0.534 J mol~* below the q = 0 line at any given field.
FIG. 27: The specific heat of Dy2TizO7 for h//[110], obtained using simulations with single spin flips and loop moves. Inset: the q = X order parameter calculated by the simulations. The single spin flips (SSF) are unable to find the true ground state of the system, while the loops moves (Loops) allow the system to order into the q = X structure. The large value of the high-temperature tail of the order parameter is a finte-size effect.
FIG. 27: The specific heat of Dy2TizO7 for h//[110], obtained using simulations with single spin flips and loop moves. Inset: the q = X order parameter calculated by the simulations. The single spin flips (SSF) are unable to find the true ground state of the system, while the loops moves (Loops) allow the system to order into the q = X structure. The large value of the high-temperature tail of the order parameter is a finte-size effect.
FIG. 28: The specific heat of Dy2TizO7 for h//[110], obtained using simulations with single spin flips and loop moves. Inset: the q = X order parameter calculated by the simulations. Both the single spin flips and the loop moves are able to find the true q= X ground state of the system.
FIG. 28: The specific heat of Dy2TizO7 for h//[110], obtained using simulations with single spin flips and loop moves. Inset: the q = X order parameter calculated by the simulations. Both the single spin flips and the loop moves are able to find the true q= X ground state of the system.
where the spin variables of have been dropped for nota- tional convenience. The dipole sum excludes terms with Rep = 0. Absolute convergence is forced on the sum inside the curly brackets of Eq. (Al) by use of a conver- gence factor. The form of this convergence factor differs depending on whether the dipolar sum is performed on N particles in real space (e.g., Monte Carlo and molecu- lar dynamic simulations) or in the thermodynamic limit in momentum space (mean-field theory) The dipole-dipole interaction is an infinite sum that falls off as the inverse cube of the separation distance between dipoles, 1/ [RE |%. Hence it is a conditionally convergent series. The point of the Ewald method is to convert this slowly converging lattice sum into of two absolutely (rapidly) converging series, one in real space and the other in Fourier space. The general lattice sum for (111) Ising dipoles on the pyrochlore lattice is
where the spin variables of have been dropped for nota- tional convenience. The dipole sum excludes terms with Rep = 0. Absolute convergence is forced on the sum inside the curly brackets of Eq. (Al) by use of a conver- gence factor. The form of this convergence factor differs depending on whether the dipolar sum is performed on N particles in real space (e.g., Monte Carlo and molecu- lar dynamic simulations) or in the thermodynamic limit in momentum space (mean-field theory) The dipole-dipole interaction is an infinite sum that falls off as the inverse cube of the separation distance between dipoles, 1/ [RE |%. Hence it is a conditionally convergent series. The point of the Ewald method is to convert this slowly converging lattice sum into of two absolutely (rapidly) converging series, one in real space and the other in Fourier space. The general lattice sum for (111) Ising dipoles on the pyrochlore lattice is
where }7(;,;), means the 1 = j term is excluded when a =. The remaining steps in the derivation of the q- dependent Ewald equations can be found in Ref. 52. In this work, A%°(q) was determined in at each q-point in the first zone of the (hAl) plane of the pyrochlore lattice and used in the formation of 7(q) in Eq. (11). We note that the value of A%?(q) in the limit q + 0 dependents on direction. The value of A%°(0) can be related to the demagnetization factor.9}9? In mean-field theory, one considers the Fourier trans- form of the dipole-dipole interaction; therefore, the term e#R1) is included in Eq. (Al) and plays the role of a convergence factor as discussed in Ref. 90. With a con- vergence factor that is periodic as opposed to spherical in R, the derivation of the g-dependent Ewald equations fol- lows the method of long waves introduced by Born and Huang, Ref. 73. As noted in Section III, the momen- tum dependent dipole-dipole interaction is determined between sub-lattice sites and is a component of 77°(q), where 7(q) is 4x 4 matrix. Using Eq. (A4), we can write the dipolar contribution to 7%(q) as
where }7(;,;), means the 1 = j term is excluded when a =. The remaining steps in the derivation of the q- dependent Ewald equations can be found in Ref. 52. In this work, A%°(q) was determined in at each q-point in the first zone of the (hAl) plane of the pyrochlore lattice and used in the formation of 7(q) in Eq. (11). We note that the value of A%?(q) in the limit q + 0 dependents on direction. The value of A%°(0) can be related to the demagnetization factor.9}9? In mean-field theory, one considers the Fourier trans- form of the dipole-dipole interaction; therefore, the term e#R1) is included in Eq. (Al) and plays the role of a convergence factor as discussed in Ref. 90. With a con- vergence factor that is periodic as opposed to spherical in R, the derivation of the g-dependent Ewald equations fol- lows the method of long waves introduced by Born and Huang, Ref. 73. As noted in Section III, the momen- tum dependent dipole-dipole interaction is determined between sub-lattice sites and is a component of 77°(q), where 7(q) is 4x 4 matrix. Using Eq. (A4), we can write the dipolar contribution to 7%(q) as
FIG. 29: Specific heat curves for Dy2TizO7 in an applied field h//{110], in simulations without (line) and with (circles) the boundary term (BT) in the Ewald energy summation. These simulations employed single spin flip dynamics in the Monte Carlo. The significance of this data in the context of experimental measurements on Dy2Ti2QO7 is discussed in detail in Section VI.
FIG. 29: Specific heat curves for Dy2TizO7 in an applied field h//{110], in simulations without (line) and with (circles) the boundary term (BT) in the Ewald energy summation. These simulations employed single spin flip dynamics in the Monte Carlo. The significance of this data in the context of experimental measurements on Dy2Ti2QO7 is discussed in detail in Section VI.
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