We investigate the Ising model on a square lattice with antiferromagnetic exchange between nearest and next-nearest neighbors and show that at low temperatures stripe-like and droplet-like superstructures appear. We show that the competing interactions introduce a strong frustration that could plausibly describe the systems with dipole{dipole type interactions and pay particular attention to arrays of interacting Josephson &frac14;-rings

Kugel, Kliment I

Kusmartsev, Feodor V

O'Hare, Anthony

English

Stirling Online Research Repository (RIOXX)

Vol. 115 (2009) ACTA PHYSICA POLONICA A No. 1Proceedings of the European Conference “Physics of Magnetism” (PM’08), Poznań 2008Two-Dimensional Ising Model with Competing Interactionsas a Model for Interacting π-RingsA. O’Harea, F.V. Kusmartseva and K.I. Kugela,baDepartment of Physics, Loughborough UniversityLeicestershire, LE11 3TU, UKbInstitute for Theoretical and Applied ElectrodynamicsRussian Academy of SciencesIzhorskaya Str. 13, Moscow, 125412 RussiaWe investigate the Ising model on a square lattice with antiferromagnetic exchange between nearestand next-nearest neighbors and show that at low temperatures stripe-like and droplet-like superstructuresappear. We show that the competing interactions introduce a strong frustration that could plausibly describe thesystems with dipole–dipole type interactions and pay particular attention to arrays of interacting Josephson π-rings.PACS numbers: 05.50.+q, 75.10.Hk, 74.81.Fa, 03.67.Lx1. IntroductionIt is well known that the interplay between the com-peting interactions in many magnetic and superconduct-ing materials as well as in complex systems leads to arich phase diagram with a large number of phases andnon-trivial types of ordering. As a vivid example of suchsystems, we investigate here the two-dimensional (2D)Ising model with competing antiferromagnetic nearest--neighbor (J) and diagonal (J ′) interactions on a squarelattice. It is now well accepted that such a model exhibitstwo phases with magnetic long-range order: a semiclas-sical Néel-like magnetic order at small J ′ and a antifer-romagnetically coupled ferromagnetic chains (vertical orhorizontal stripes) at large J ′. These two ordered phasesare separated by an intermediate quantum paramagneticphase without the long-range order, the nature of whichis still under discussion.An example of a system that may be described by sucha model is a chain or a planar array of electrically isolatedJosephson π-rings which could be treated as a set of mag-netic moments oriented perpendicular to the plane (i.e.as the Ising spins) and interacting via magnetic dipoleforces (taking care to include next-nearest neighbor in-teractions) [1].Here, we show that this model exhibits an additionallow temperature disordered phase due to the proximityof the lowest energy excited states to the ground state(see also [2]).2. Model and a 5-site plaquetteThe Hamiltonian for the Ising model with nearestneighbor and next-nearest (diagonal) neighbor interac-tions is given byH = J∑〈nn〉sisj + J ′∑〈dn〉sisj + h∑isi, (1)where J, J ′ > 0, s is a two-value Ising variable s = ±1,〈nn〉 and 〈dn〉 denote the summation over sites i and jbeing respectively nearest neighbors and diagonal neigh-bors.For a square lattice with fixed lattice spacing, the ratiornn/rdn =√2, since dipole–dipole interactions decay as≈ 1/r3 we have J/J ′ = 2√2 for the square lattice. Forthe sake of generality, we will consider various values ofJ/J ′ which may be applicable to distorted square lattices.The simplest system, which may already have non--trivial features but contains all the physics of the modelinvestigated here, is a cluster consisting of five sites. Sucha cluster has a similar geometry to the square lattice, i.e.it is a site with four neighbors (a plaquette). Further-more, we choose this plaquette for the further analysisbecause it also retains the symmetry and the charac-teristic geometry of the lattice as a whole (the numberof nearest-neighbor and diagonal bonds are equal, seeFig. 1).Fig. 1. Square plaquette of the minimum size (withequal numbers of nearest- and next-nearest neighborbonds).(150)Two-Dimensional Ising Model with Competing Interactions . . . 151The possible configurations of this 5-site plaquette areeasily listed (we have 25 = 32 configurations with dif-ferent values of energy). Knowing the energy levels andtheir degeneracy it is possible to write the partition func-tion and calculate the free energy and heat capacity ofthis plaquette. Plotting the positions of heat capacitymaxima in the T/J ′−J/J ′ plane yields a kind of phasediagram for a small plaquette (Fig. 2). Indeed, the po-sitions of C(T/J ′, J/J ′) peaks could be related to theactual phase transitions occurring in infinite systems.Let us note that at J/J ′ < 2, the lowest energy corre-sponds to a state in which ordering occurs via chains ofspins with the same direction, whereas at J/J ′ > 2, theusual two-sublattice antiferromagnetism is the most fa-vorable. However, at non-zero temperatures, there is noboundary between these two phases. Instead, we have awhole quadrant in the T/J ′−J/J ′ plane (with the ver-tex at point (0, 2)), where one could expect numerousphases in the infinite lattice, intermediate between thetwo-sublattice and chain-like structures. The situationis very much like that of the ANNNI model [3], wherethe “devil’s staircase” of different phases arises at finitetemperatures between two main phases existing at zerotemperature. Region 3 within this quadrant is boundedby a curve corresponding to the positions of an additional(high-temperature) peak in the temperature dependenceof the heat capacity. Therefore, this is another indica-tion that in the infinite system, one should expect in thisrange a phase with a certain kind of ordering, maybequite unusual, rather than simply a disordered paramag-netic phase.Fig. 2. Phase diagram (positions of the heat capac-ity peaks) for a 5-site plaquette. Region 1 correspondsto antiferromagnetic ordering, region 2 corresponds tostripe ordering, while in region 3, one could expect forthe corresponding infinite square lattice multiple phases(similar to those in the ANNNI model) or a phase witha complicated spin ordering.For the plaquette cluster, the antiferromagnetic groundstate, having the zero entropy (non-degenerate), is verydifficult to reach. So the system is practically alwaysdisordered due to a proliferation of defects. However,the broad peak in specific heat indicates that a true phasetransition may arise for large systems.3. Larger plaquettes and a glassy stateQualitatively, the situation remains the same as weincrease lattice size. We calculated the specific heat foran 8 × 8 lattice via the transfer matrix method and a16× 16 lattice via the Monte Carlo method.The phase diagram obtained for the 8 × 8 lattice isplotted in Fig. 3. The comparison of the phase diagramsfor 8 × 8 and 16 × 16 lattices is illustrated in Fig. 4.We find that at low temperatures there are many energylevels lying close to the ground state effectively trappingthe system in one of these locally (meta-)stable statesassociated with local energy minima separated by largeenergy barriers.Fig. 3. Phase diagrams for an 8 × 8 lattice calculatedby the transfer matrix method.Fig. 4. Peaks in the specific heat of the 2D Ising modelwith competing interactions on a square lattice using a16 × 16 Monte Carlo simulation and an 8 × 8 transfermatrix. The low temperature “phase transition” is nota peak in the specific heat but rather a crossover tran-sition.At low temperature, one would expect the system toexist in an ordered state with long-range order. How-ever, the appearance of defects (locally stable droplets)152 A. O’Hare, F.V. Kusmartsev, K.I. Kugelreduce correlations to a few spins, but temporal corre-lations due to freezing of the spins can be very strong.Local squared magnetization is given by the average ofthe autocorrelation functions. This is the order param-eter proposed by Edwards and Anderson [4]: qEA =limt→∞ limV→∞ [〈si(t0)si(t0 + t)〉].Spin-glass ordering, if it can be described within theframework of equilibrium statistical mechanics, corre-sponds to a non-zero value of the Edwards–Andersonorder parameter. We can see from Fig. 5 that MonteCarlo simulations clearly show the Edwards–Andersonorder parameter exploding as T/J ′ → 0. This is indica-tive of spin-glass ordering.Fig. 5. Edwards–Anderson order parameter for the 2DIsing model with next-nearest neighbor interactions onan 16 × 16 square lattice. The explosion of this orderparameter at low temperature is an indication of theformation of a spin-glass state.The low-temperature superstructures that appear inthis model are characterized by the formation of locallystable droplets in the background order. This low tem-perature phase is akin to spin-glass ordering as shownby the Edwards–Anderson order parameter though otherorder parameters may exist.4. ConclusionThe zero value of entropy at low temperatures showsthat the ground state at low temperatures is ordered ei-ther as stripes, J/J ′ < 2 or as the Néel antiferromagnet,J/J ′ > 2. However, the recent findings in the frame-work of this model [2] clearly demonstrate that the en-ergy of defects and dislocations in the lattice is close tothe ground state of the system. Therefore, many locallystable (or metastable) states associated with local energyminima separated by energy barriers may appear forminga glass-like state.Careful preparation of the lattice could result in thesystem being trapped in this glass-like state which maybe used as an initial state for adiabatic quantum com-puting [2]. Recently, a scalable design that may be usedin such a way [5] has been proposed and realized thoughthe range of problems that may be solved may be lim-ited. One such application is a Travelling Salesman Prob-lem [6], which can be represented in the form of morecomplicated Ising model, with a set of a coupling con-stants [7].AcknowledgmentsThe work was supported by the ESF network-programAQDJJ, European project CoMePhS, ISTC (grantG1335), and RFBR (project 08-02-00212).References[1] J.R. Kirtley, C.C. Tsuei, Ariando, H.J.H. Smilde,H. Hilgenkamp, Phys. Rev. B 72, 214521 (2005).[2] A. O’Hare, F.V. Kusmartsev, K.I. Kugel, M.S. Laad,Phys. Rev. B 76, 064528 (2007).[3] W. Selke, Phys. Rep. 170, 212 (1988).[4] S.F. Edwards, P.W. Anderson, J. Phys. F 5, 965(1975).[5] V. Zakosarenko, N. Bondarenko, S.H.W. van derPloeg, A. Izmalkov, S. Linzen, J. Kunert, M. Graj-car, E. Il’ichev, H.-G. Meyer, Appl. Phys. Lett. 90,022501 (2007).[6] Tien D. Kieu, quant-ph/0601151 (2006).[7] R. Martoňàk, G.E. Santoro, E. Tosatti, Phys. Rev. E70, 057701 (2004).

Two-Dimensional Ising Model with Competing Interactions as a Model for Interacting pi-Rings

Stirling Online Research Repository

Vol. 115 (2009) ACTA PHYSICA POLONICA A No. 1Proceedings of the European Conference “Physics of Magnetism” (PM’08), Poznan´ 2008Two-Dimensional Ising Model with Competing Interactionsas a Model for Interacting pi-RingsA. O’Harea, F.V. Kusmartseva and K.I. Kugela,baDepartment of Physics, Loughborough UniversityLeicestershire, LE11 3TU, UKbInstitute for Theoretical and Applied ElectrodynamicsRussian Academy of SciencesIzhorskaya Str. 13, Moscow, 125412 RussiaWe investigate the Ising model on a square lattice with antiferromagnetic exchange between nearestand next-nearest neighbors and show that at low temperatures stripe-like and droplet-like superstructuresappear. We show that the competing interactions introduce a strong frustration that could plausibly describe thesystems with dipole–dipole type interactions and pay particular attention to arrays of interacting Josephson pi-rings.PACS numbers: 05.50.+q, 75.10.Hk, 74.81.Fa, 03.67.Lx1. IntroductionIt is well known that the interplay between the com-peting interactions in many magnetic and superconduct-ing materials as well as in complex systems leads to arich phase diagram with a large number of phases andnon-trivial types of ordering. As a vivid example of suchsystems, we investigate here the two-dimensional (2D)Ising model with competing antiferromagnetic nearest--neighbor (J) and diagonal (J ′) interactions on a squarelattice. It is now well accepted that such a model exhibitstwo phases with magnetic long-range order: a semiclas-sical Ne´el-like magnetic order at small J ′ and a antifer-romagnetically coupled ferromagnetic chains (vertical orhorizontal stripes) at large J ′. These two ordered phasesare separated by an intermediate quantum paramagneticphase without the long-range order, the nature of whichis still under discussion.An example of a system that may be described by sucha model is a chain or a planar array of electrically isolatedJosephson pi-rings which could be treated as a set of mag-netic moments oriented perpendicular to the plane (i.e.as the Ising spins) and interacting via magnetic dipoleforces (taking care to include next-nearest neighbor in-teractions) [1].Here, we show that this model exhibits an additionallow temperature disordered phase due to the proximityof the lowest energy excited states to the ground state(see also [2]).2. Model and a 5-site plaquetteThe Hamiltonian for the Ising model with nearestneighbor and next-nearest (diagonal) neighbor interac-tions is given byH = J∑〈nn〉sisj + J ′∑〈dn〉sisj + h∑isi, (1)where J, J ′ > 0, s is a two-value Ising variable s = ±1,〈nn〉 and 〈dn〉 denote the summation over sites i and jbeing respectively nearest neighbors and diagonal neigh-bors.For a square lattice with fixed lattice spacing, the ratiornn/rdn =√2, since dipole–dipole interactions decay as≈ 1/r3 we have J/J ′ = 2√2 for the square lattice. Forthe sake of generality, we will consider various values ofJ/J ′ which may be applicable to distorted square lattices.The simplest system, which may already have non--trivial features but contains all the physics of the modelinvestigated here, is a cluster consisting of five sites. Sucha cluster has a similar geometry to the square lattice, i.e.it is a site with four neighbors (a plaquette). Further-more, we choose this plaquette for the further analysisbecause it also retains the symmetry and the charac-teristic geometry of the lattice as a whole (the numberof nearest-neighbor and diagonal bonds are equal, seeFig. 1).Fig. 1. Square plaquette of the minimum size (withequal numbers of nearest- and next-nearest neighborbonds).(150)Two-Dimensional Ising Model with Competing Interactions . . . 151The possible configurations of this 5-site plaquette areeasily listed (we have 25 = 32 configurations with dif-ferent values of energy). Knowing the energy levels andtheir degeneracy it is possible to write the partition func-tion and calculate the free energy and heat capacity ofthis plaquette. Plotting the positions of heat capacitymaxima in the T/J ′−J/J ′ plane yields a kind of phasediagram for a small plaquette (Fig. 2). Indeed, the po-sitions of C(T/J ′, J/J ′) peaks could be related to theactual phase transitions occurring in infinite systems.Let us note that at J/J ′ < 2, the lowest energy corre-sponds to a state in which ordering occurs via chains ofspins with the same direction, whereas at J/J ′ > 2, theusual two-sublattice antiferromagnetism is the most fa-vorable. However, at non-zero temperatures, there is noboundary between these two phases. Instead, we have awhole quadrant in the T/J ′−J/J ′ plane (with the ver-tex at point (0, 2)), where one could expect numerousphases in the infinite lattice, intermediate between thetwo-sublattice and chain-like structures. The situationis very much like that of the ANNNI model [3], wherethe “devil’s staircase” of different phases arises at finitetemperatures between two main phases existing at zerotemperature. Region 3 within this quadrant is boundedby a curve corresponding to the positions of an additional(high-temperature) peak in the temperature dependenceof the heat capacity. Therefore, this is another indica-tion that in the infinite system, one should expect in thisrange a phase with a certain kind of ordering, maybequite unusual, rather than simply a disordered paramag-netic phase.Fig. 2. Phase diagram (positions of the heat capac-ity peaks) for a 5-site plaquette. Region 1 correspondsto antiferromagnetic ordering, region 2 corresponds tostripe ordering, while in region 3, one could expect forthe corresponding infinite square lattice multiple phases(similar to those in the ANNNI model) or a phase witha complicated spin ordering.For the plaquette cluster, the antiferromagnetic groundstate, having the zero entropy (non-degenerate), is verydifficult to reach. So the system is practically alwaysdisordered due to a proliferation of defects. However,the broad peak in specific heat indicates that a true phasetransition may arise for large systems.3. Larger plaquettes and a glassy stateQualitatively, the situation remains the same as weincrease lattice size. We calculated the specific heat foran 8 × 8 lattice via the transfer matrix method and a16× 16 lattice via the Monte Carlo method.The phase diagram obtained for the 8 × 8 lattice isplotted in Fig. 3. The comparison of the phase diagramsfor 8 × 8 and 16 × 16 lattices is illustrated in Fig. 4.We find that at low temperatures there are many energylevels lying close to the ground state effectively trappingthe system in one of these locally (meta-)stable statesassociated with local energy minima separated by largeenergy barriers.Fig. 3. Phase diagrams for an 8 × 8 lattice calculatedby the transfer matrix method.Fig. 4. Peaks in the specific heat of the 2D Ising modelwith competing interactions on a square lattice using a16 × 16 Monte Carlo simulation and an 8 × 8 transfermatrix. The low temperature “phase transition” is nota peak in the specific heat but rather a crossover tran-sition.At low temperature, one would expect the system toexist in an ordered state with long-range order. How-ever, the appearance of defects (locally stable droplets)152 A. O’Hare, F.V. Kusmartsev, K.I. Kugelreduce correlations to a few spins, but temporal corre-lations due to freezing of the spins can be very strong.Local squared magnetization is given by the average ofthe autocorrelation functions. This is the order param-eter proposed by Edwards and Anderson [4]: qEA =limt→∞ limV→∞ [〈si(t0)si(t0 + t)〉].Spin-glass ordering, if it can be described within theframework of equilibrium statistical mechanics, corre-sponds to a non-zero value of the Edwards–Andersonorder parameter. We can see from Fig. 5 that MonteCarlo simulations clearly show the Edwards–Andersonorder parameter exploding as T/J ′ → 0. This is indica-tive of spin-glass ordering.Fig. 5. Edwards–Anderson order parameter for the 2DIsing model with next-nearest neighbor interactions onan 16 × 16 square lattice. The explosion of this orderparameter at low temperature is an indication of theformation of a spin-glass state.The low-temperature superstructures that appear inthis model are characterized by the formation of locallystable droplets in the background order. This low tem-perature phase is akin to spin-glass ordering as shownby the Edwards–Anderson order parameter though otherorder parameters may exist.4. ConclusionThe zero value of entropy at low temperatures showsthat the ground state at low temperatures is ordered ei-ther as stripes, J/J ′ < 2 or as the Ne´el antiferromagnet,J/J ′ > 2. However, the recent findings in the frame-work of this model [2] clearly demonstrate that the en-ergy of defects and dislocations in the lattice is close tothe ground state of the system. Therefore, many locallystable (or metastable) states associated with local energyminima separated by energy barriers may appear forminga glass-like state.Careful preparation of the lattice could result in thesystem being trapped in this glass-like state which maybe used as an initial state for adiabatic quantum com-puting [2]. Recently, a scalable design that may be usedin such a way [5] has been proposed and realized thoughthe range of problems that may be solved may be lim-ited. One such application is a Travelling Salesman Prob-lem [6], which can be represented in the form of morecomplicated Ising model, with a set of a coupling con-stants [7].AcknowledgmentsThe work was supported by the ESF network-programAQDJJ, European project CoMePhS, ISTC (grantG1335), and RFBR (project 08-02-00212).References[1] J.R. Kirtley, C.C. Tsuei, Ariando, H.J.H. Smilde,H. Hilgenkamp, Phys. Rev. B 72, 214521 (2005).[2] A. O’Hare, F.V. Kusmartsev, K.I. Kugel, M.S. Laad,Phys. Rev. B 76, 064528 (2007).[3] W. Selke, Phys. Rep. 170, 212 (1988).[4] S.F. Edwards, P.W. Anderson, J. Phys. F 5, 965(1975).[5] V. Zakosarenko, N. Bondarenko, S.H.W. van derPloeg, A. Izmalkov, S. Linzen, J. Kunert, M. Graj-car, E. Il’ichev, H.-G. Meyer, Appl. Phys. Lett. 90,022501 (2007).[6] Tien D. Kieu, quant-ph/0601151 (2006).[7] R. Martonˇa`k, G.E. Santoro, E. Tosatti, Phys. Rev. E70, 057701 (2004).

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Two-Dimensional Ising Model with Competing Interactions as a Model for Interacting pi-Rings

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