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Ising Model: Recent Developments and Exotic Applications

Solving in his PhD thesis the one-dimensional version of a certain lattice model of ferromagnetism formulated by his supervisor Lenz [ 1 ], Ising came to the conclusion that the model fails to describe finite-temperature ferromagnetism and does not seem to be particularly important [ 2 ]. Although Ising solution was correct, his predictions on the importance of this model turned out to be grossly inadequate. Indeed, for more than a century, the Ising model, as it is now called, has provided profound insight into the behaviour of a vast variety of interacting many-body systems [ 3 ], even outside the realm of physics [ 4 ].

The present Special Issue, “Ising Model: Recent Developments and Exotic Applications”, consists of eight original research papers that contribute greatly to our understanding of the Ising model and suggest its possible new applications.

Early research on the Ising model was restricted mainly to ferromagnetic interactions and regular lattices. Gradually, however, interest shifted towards more complex models that incorporate, for example, competing interactions. Recently, there has been a growing interest in studying Ising models of heterogeneous lattices such as random graphs or scale-free networks. In the article [ 5 ] published in this Special Issue, Krasnytska et al. examine a model of heterogeneous lattices with an additional disorder related to the strength of the spins. They show that the interplay of lattice and spin strength disorders can result in a new critical behaviour. Their analysis is as an interesting extension of some earlier work based on annealed network approximations [ 6 ].

The Ising model, or its more general version—the Potts model—can be used to examine the anomalies of the IT network infrastructure, as demonstrated in the paper by Paszkiewicz [ 7 ] that also belongs to the present Special Issue. Such an approach allows one to describe the influence of various disturbances in a network such as, for example, a sudden change in people’s opinion (e.g., due to an election spot), an occurrence of malware in the IT system, or congestion in a computer network. In the era of the Internet of Things or even of Everything, this kind of modelling is likely to play an increasing role.

The next paper in our Special Issue by Valle et al. [ 8 ] demonstrates the applicability of the Ising model in econophysics. In particular, they analyse the volatility of returns and argue that this volatility can be modeled with a certain Ising model. Using the Maximum Entropy Principle and machine learning, they infer the coefficients of interactions between assets and analyse to what extent a model with pairwise interactions can explain the behaviour of financial markets. Their work even indicates that during financial turmoil, factors that are external to the financial system make predictions of financial risk much more difficult.

A number of NP-optimization problems can be translated into energy minimization problems of a certain Ising model [ 9 ]. Such an approach was used to solve certain polyomino puzzles and is described in the paper by Takabatake et al. [ 10 ] in our Special Issue. Their method uses a novel representation of the problem and is computationally less demanding than some previous approaches. They minimize the energy using Hopfield neural networks and demonstrate the effectiveness of their method for some more general polyominoes as well.

In the next paper of our Special Issue, Žukovič et al. argue that the Ising model can be used for the interpolation of spatial data [ 11 ]. Such techniques are needed when, for example, some parts of a satellite photo of a certain area are missing due to cloudiness. The authors demonstrate that their approach based on the Ising model offers a very good computational performance, even in comparison o more sophisticated techniques based on kriging or machine learning. Since the amount of data generated by satellites or drones has been rapidly growing recently, it is likely that the demand for such methods will also increase.

The idea that an image can be represented as a certain configuration of the Ising spins appears also in the paper by Choi et al. [ 12 ]. They show that a relatively small fraction of spins is sufficient for a reliable restoration of the entire image. In their paper, Choi et al. examine the security risks associated with the reconstruction of certain biometric templates, e.g., a human iris, and argue that their method should be considered as complementary to cancellable biometrics or some schemes of biometric cryptosystem.

Up and down spins in the Ising models can be also interpreted as left and right enantiomers of chiral molecules. Using such an analogy, Dutta and Gellman developed adsorption thermodynamics of chiral molecules [ 13 ]. They argue that adsorption from racemic mixtures of enantiomers is directly analogous to the Ising model’s behaviour. An important conclusion of their research, especially in the context of pharmaceutical and biochemical industries, is that enantiomer purification is to some extent similar to phase separation in the Ising model and can be achieved using achiral surfaces.

In yet another paper of the Special Issue, Kryzhanovsky et al. developed an analytical method to examine the Ising model on lattices of large dimensions d [ 14 ]. Their approach enables them to calculate the critical temperature, free energy, or heat capacity and gives some insight into the behaviour of the density of state. Although the method is less accurate for d < 4 , the agreement with numerical simulations for d > 4 is remarkably satisfactory.

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Home > Books > Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable Bodies - Recent Progress

The Ising Model: Brief Introduction and Its Application

Submitted: 14 April 2019 Reviewed: 17 December 2019 Published: 24 February 2020

DOI: 10.5772/intechopen.90875

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Though the idea to use numerical techniques, in order to solve complex three-dimensional problems, has become quite old, computational techniques have gained immense importance in past few decades because of the advent of new generation fast and efficient computers and development of algorithms as parallel computing. Many mathematical problems have no exact solutions. Depending on the complexity of the equations, one needs to use approximate methods. But there are problems, which are beyond our limits, and need support of computers. Ernst Ising published his PhD dissertation in the form of a scientific report in 1925. He used a string of magnetic moments; spin up (+1/2) and spin down (−1/2), and applied periodic boundary conditions to prove that magnetic phase transition does not exist in one dimensions. Lars Onsager, latter, exactly solved the phase transition problem in two dimensions in 1944. It is going to be a century-old problem now. A variety of potential applications of Ising model are possible now a days; classified as Ising universality class models. It has now become possible to solve phase transition problems in complex three-dimensional geometries. Though the area of spinotronics still needs more engagements of computational techniques, its limited use have provided good insights at molecular scale in recent past. This chapter gives a brief introduction to Ising model and its applications, highlighting the developments in the field of magnetism relevant to the area of solid state physics.

• surface-directed phase separation
• wetting-dewetting
• Monte Carlo simulation

Author Information

Satya pal singh *.

• Condensed Matter Physics Research Laboratory, Department of Physics and Material Science, Madan Mohan Malaviya University of Technology, Gorakhpur, Uttar Pradesh, India

*Address all correspondence to: [email protected]

1. Introduction

Ernst Ising ( Figure 1 ) was born on May 10, 1900, in Loe Koln. He started schooling in 1907 and obtained his diploma at the gymnasium there in the year 1918. After brief military training, he studied mathematics and physics at Gottingen University in the year 1919. After a short gap, he continued his studies and learnt astronomy apart of other subjects. He got focused to theoretical physics at the suggestion of Professor W. Lenz. He started investigating ferromagnetism under supervision of W. Lenz by the end of the year 1922. Ising published short paper in 1925 as a summary of his doctoral thesis [ 1 , 2 ]. He exactly calculated partition function for one-dimensional lattice system of spins. Ising had first proven that no phase transition to a ferromagnetic ordered state occurs in one dimension at any temperature.

His argument in the favor of his mathematical note was very simple. Suppose, if one of the spins get flipped at a random position because of thermal agitation, there is no force available, which can stop the neighboring spins to flip in the same direction. And this process will go on and on, and completely ordered state will not remain stable at a finite temperature. Thus no phase transition will occur at a finite temperature. Ideally speaking, any ordered state will always remain like a metastable state at finite temperature and nothing more. Molecular motion seizes at absolute zero temperature. So, one may expect that no spin fluctuations may occur at absolute zero temperature. Henceforth, the stable ordered state is a natural outcome at absolute zero temperature. But, it cannot be said to be a critical temperature in true sense. The existence of phase transition at this temperature has no physical meaning, because there is no temperature below it. After going through some approximate calculations, Ising purportedly showed that his model could not exhibit a phase transition in two and three dimensions, either. Latter, his conclusion was proven to be erroneous [ 1 , 2 ] ( Figure 2 ).

Ernst (Ernest) Ising (May 10, 1900–May 11, 1998).

Barry Simon has quoted it very well “This model was suggested to Ising by his thesis advisor, Lenz. Ising solved the one-dimensional model, and on the basis of the fact that the one-dimensional model had no phase transition; he asserted that there was no phase transition in any dimension. As we shall see, this is false. It is ironic that on the basis of an elementary calculation and erroneous conclusion, Ising’s name has become among the most commonly mentioned in the theoretical physics literature. But history has had its revenge. Ising’s name, which is correctly pronounced “E-zing”, is almost universally mispronounced “I-zing”.”

Random spin flipping in one-dimensional system.

Ising’s paper credited Wilhelm Lenz for his original idea, who had first proposed it in the year 1920. W. Lenz was Ising’s research supervisor. It has been often rendered as Lenz-Ising model in many citations. Lenz suggested that dipolar atoms in crystals are free to rotate in quantized manner. He proposed quantum treatment of dipole orientations, though in its classical version, Ising considered only two spin states, i.e., S = ±½. Ising discussed his results with Professor Lenz and Dr. Wolfgang Pauli, who was teaching at Hamburg at that time. Ising’s work was first cited by famous contemporary scientist Heisenberg. Heisenberg was first one to realize the failure of Lenz-Ising model. In order to explain ferromagnetism, he developed his own theory, using complicated interactions of spins. There are more scientists in the list, whose contribution to Lenz-Ising model or simply say Ising model must be cited here, because of their historical relevance. They have greatly enriched and contributed to this new model. This list includes scientists like Gorskly (1928), R. H. Fowler (1930), Bragg and Williams (1934), R. Peierls (1936), J. G. Krikwood (1938), Hens Bethe (1939), Kramers and Wannier (1941), and Onsager (1942). They further extended Ising model to a new class of problems.

2. Application of Ising model

Ising model has been extensively used for solving a variety of problems [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ]. Some of the problems are discussed, here, with appropriate examples.

2.1 Phase separation and wetting/dewetting

Ising model was first exploited for investigating spontaneous magnetization in ferromagnetic film (i.e. magnetization in the absence of external magnetic field). An example case of Ising model using metropolis algorithm is shown in Figure 3 . Transition temperature depends on the strength of the inter-spin exchange coupling; the dominating term governs the kinetics, when long-range interactions are introduced in the calculations. Latter, it was used to study phase separation in binary alloys and liquid-gas phase transitions (i.e., condensation of molecule in one region of space of the box). Binary alloys constitute of two different atoms. At temperature T = 0, Zn-Cu alloy; known as brass, gets completely ordered. This state is said to be β-brass. In β-brass state, each Zn atom is surrounded by eight copper atoms, placed at the corners of the unit cell of the body-centered cubic structure and vice versa. The occupation of each site can be represented by:

Variation in critical temperature vs. next nearest exchange coupling for a bcc lattice (reproduced with permission from Singh [ 3 ]).

The interaction energy between A-A, B-B, and A-B type of atoms are represented by ε AA , ε BB , and ε AB , respectively. Phase separation has been studied vastly, using Ising model [ 4 , 5 , 6 ]. A phase is simply a part of a system, separated from the other part by the formation of an interface; that essentially means that two components aggregate and form rich regions of A and B type of molecules with an interface in between them. The evolution of two distinct phases, when an initial random but homogeneous mixture is annealed below a definite temperature, is known as phase separation. Phase separation leads to discontinuity and inhomogeneity in the systems. This happens because the phase-separated regions are energetically more stable. Phase separation has been an old problem and has been extended to study diverse phenomena ranging from magnetic liquid-liquid phase separation to protein-protein phase separation in biological systems. This process has also been studied in the presence of external surfaces having affinity to one type of atom or molecule ( Figure 4a ). Both theoretical and experimental methods have been exploited and have been found in close agreement. Formation of long ridges and circular drops has been reported numerous occasions using lattice-based Ising model. For example, one may look into John W. Cahn research paper published in The Journal of Chemical Physics in the year 1965. The TEM image taken for Vycor, in which one phase had been leached out and the voids were filled with lead ( Figure 4b ).

(a) Surface-directed phase separation and dewetting in conserved binary mixture using two-dimensional lattices of size 200 × 100 nodes. The conserved components are taken in ratio 70:30 at T = 0.70. Majority component is attracted by upper and lower substrates, whereas the minority component has repulsive interaction with the two interfaces. Periodic boundary conditions are applied along X-direction. The micrograph is taken after completion of 30,000 Monte Carlo cycles using Kawasaki exchange method (the figure is reproduced with permission from Singh [ 5 ]). (b) Shows Transmission Electron Microscope (TEM) image of unsintered Vycor with one phase replaced by lead (X 200000). Reproduced with permission from W. G. Schmidt and R. J. Charles, Journal of Applied Physics 35, 2552 (1964); doi: 10.1063/1.1702905.

2.2 Lattice-based liquid-gas model

Yang and Lee first coined the term lattice gas in the year 1952. A lattice should have larger volume (V) than the number of lattice molecules (N), so that some of the nodes or lattice vertices are left empty (i.e., N < V). No lattice vertex can be occupied by more than one particle. The interaction potential between two atoms at lattice sites i and j is given by Eq. (2) :

For surface affinity of lower surface to i th liquid molecule, we chose:

The occupation number (n i ) of a lattice site i is given by:

One example case is shown in Figure 5 . Here, we chose lattice size of 128 × 128 × 48. The fluid-fluid molecule and wall-liquid molecule interactions are defined, respectively, in Eqs. (2) and (3) . In canonical ensemble, the three-dimensional lattice is swept one by one; by choosing sites regularly with one of its nearest neighbor (i.e., i = n and i + 1 = n + 1). Change in energy is calculated during exchanging of these two sites; the exchange move is accepted, if Exp [−ΔE/k B T] is found to be greater than or equal to a random number generated between [0, 1]. For all cases of studies here, ε = 1 and J 0  = 12.0, and only the lower surface is functional, while the upper surface has only hard-sphere interaction with the fluid molecules. Average number density for liquid-like molecules is taken as 0.25 [ 16 ].

Micrograph for box thickness H z  = 48 after completion of 20 K M C cycles (figure is reproduced with permission from proceedings, Singh [ 16 ]).

Figure 6 shows micrograph of self-aligned liquid columns. The system evolves from an initial homogeneous mixture of liquid- and gas-like molecules obtained by annealing the system at high temperature for few thousand MC cycles. Dynamic Monte Carlo simulation has been used with continuous but random trial movements of the molecules. The lattice-based Ising model using Eqs. (2) and (3) is also supposed to give same results, at least qualitatively.

Self-assembled channels formed in confined geometry; the system starts with a random mixture of square-well fluid (A-type) and hard-sphere (B type) particles. The chemically patterned surface has affinity to (A-type) with interaction range λ A-A  = 1.5, λ A-B  = 1.5, λ Wall-A  = 2.0; interaction strengths were taken as ε AA  = 1.0, ε AB  = 0.5, and ε Wall-A  = 3.0. Average number density of the system has been taken as ρ = 0.40. Pore width H = 4.0 and composition ratio A:B = 50:50 were taken for all cases of studies. The micrograph and density data were taken after completion of 40 × 10 5 Monte Carlo cycles (the figure is reproduced with permission from Singh et al. [ 14 ]).

2.3 Spin glasses

Crystalline solids possess short- and long-range order along its crystal axes and maintain its periodicity in three dimensions. Liquids possess only short-range order, and its molecules have no long-range correlation. Liquid molecules retain only short-range order. Gases possess neither of the two. These are the three phases, in which any matter may exist. What are the glasses then? Glasses are solids, possessing no long-range order. Molecules may only locally arrange themselves to minimize its free energy. If the molecular arrangement is completely random, then a term “random media” is assigned to that. Glasses are understood as supercooled liquids. If a liquid is frozen abruptly, so that the molecules do not get sufficient time to organize themselves, some local order can be retained inside the frozen liquid. Glasses have one peculiar property. These retain relatively higher entropy even at quite low temperatures. One example is Mn doped in metals as impurity. Mn atoms interact with other Mn so (i.e. impurity atom) via RKKY interaction J ij r ∼ cos 2 k F r k F r 3 . Because of the oscillations in it, the interactions remain random. Such spin systems are classified as spin glasses. There is great deal of frustrations in spin orientations; so on many occasions, these are also referred as “frustrated spin glasses.”

Lenz-Ising model did not remain limited to above problems only, but it was extensively used to study liquid mixtures, ternary and quaternary alloys, polymer and their mixtures, random walk problem, and many others. The important aspect of Ising model is that a variety of problems (including some problems mentioned above) can be investigated by the similar kind of modeling and approach all together. It is no longer necessary to develop a different kind of theory for each type of cooperative phenomenon. Despite of all the above, it has been ironical that the inventor of the model, Ernst Ising, gave up the idea on working it, any further presuming that his model has no physical significance. He realized after two decades that he had become famous for his model because of the results obtained by other scientist based on his model, rather by his own work. It has been a queer sensation that the results of Ising model matched with any experimental data or the model was bit artificial. As for as the exponents were concerned, they were of universal nature, and a wide variety of systems have the same Ising exponents. The experimental evidence in favor of it remained a challenge, for many decades. In the year 1974, an alloy was found, which first showed that its magnetic behavior exactly matched with the Onsager result.

3. Mathematical formulation in one dimension

Various textbooks are available nowadays, which discuss Ising model and its applications in greater details [ 19 , 20 , 21 , 22 ]. Here, brief theory of one-dimensional Ising model is presented. H, Q, and A stands for Hamiltonian, partition function, and free energy of the system, respectively:

Some thermodynamic functions are defined as follows:

3.1 Case A: free boundary with zero field

Partition function is given by:

Here, K = β J.

We now define a new variable:

Then we can assign σ to two values, i.e., ±1:

In order to consider contributions from all possible configurations {S 1 , S 2 , S 3 ………S N }, we need to provide the set of numbers {σ 1 , σ 2 , σ 3 ……..σ N-1 }; here each S i can take two values as ±1. Configuration in a lattice description means a particular set of values of all spins; if there are N numbers of vertices, there will be 2 N different configurations as a result of permutation and combination of spins. The space, thus formed with these configurations, is called configuration space. Here, summing over σ i will give only half value of Q, henceforth, we can write:

3.2 Case B: periodic boundary with zero field

Now, the partition function is given by:

Here, S N+1  = S 1

Since (S i ) 2  = 1, we can write S 1 S N =S 1 . S 2 . S 2 . S 3 . S 3 ………... S N-1 . S N-1 . S N

Here, second part in exponential has been converted into a summation series:

It can be shown that in thermodynamic limit, (i.e. N → ∞), the free energy of the system converge to a finite value. Readers are left with the exercise. So, periodic boundary condition, as shown in Figure 7 (invented by Ising), really helps one to get rid of constructing infinitely large systems. Using appropriate boundary conditions, one may obtain realistic results using large but finite number of spins.

Representation of periodic boundary conditions in a one-dimensional Ising chain.

4. Critical phenomena

A lot of research work has been dedicated to observe system behavior near critical points [ 23 , 24 , 25 , 26 , 27 ]. The relevant thermodynamic variables exhibit power-law dependences on the parameter (T − T c ) specifying the distance away from the critical point. The critical points are marked by the fact that different physical quantities pertaining to the system pose singularities at the critical point. These singularities are expressed in terms of power laws of (T − T c ) characterized by critical exponents. As, for example, magnetization <M > identified as an order parameter in magnetism, shows dependence on critical temperature (T c ), with exponent β as follows other exponents are also listed below.

Reduced temperature t ≡ ( T  −  T c )/ T c .

α: specific heat c ( t ) ∼ t −α ; B ≡ h = 0.

β: spontaneous magnetization M (t) ∼ (− t ) β , T ≤ T c , B ≡ h = 0.

γ: magnetic susceptibility χ = ∂M/∂ℎ, T ∼ | t |− γ , B ≡ h = 0.

δ: critical Isotherm M (h) ∼ |ℎ| 1/δ sgn (ℎ), t = 0.

ν: correlation length, ξ ∼ | t |− ν , B ≡ h = 0.

η: correlation function G ( r ) ∼ r (− d +2− η ) , t = 0, B ≡ h = 0.

4.1 Scaling hypothesis and renormalization group theory

Kadanoff first suggested that, when a system is near critical temperature, individual spins may be grouped into blocks of spins [ 23 ]. It is possible because of the fact that the spin-spin correlation length becomes exceedingly large near T c and details of individual spins no longer remain important. In transformed system, each block plays the role of a single spin. Now, the spin variable associated with a single block is denoted by symbol σ i . σ i can take values ±1. The new system is composed of N′ spins ( Figure 8 ).

Spin decimation process in a two-dimensional square lattice. A small cluster of 36 spins gets transformed into 9 nodal points.

Lattice constant:

In order to preserve the spatial density of the degrees of freedom of spins in the system, the spatial distances are rescaled by the factor l.

Now, the partition function can be updated as follows:

This idea was first propounded by Kadanoff, and was later developed by Wilson. This process is also referred as decimation process. A new exchange coupling constant is assigned for interaction between σ i. This new construction of lattice does not alter the free energy of the system, and it remains the same as obtained by the original method. The rescaling process helps to find relations between various exponents. More detailed discussion on this topic can be found in standard textbooks of Statistical Mechanics by Patharia, Huang, etc. Since this process involves length transformation or a change of scale, Wilson introduced the concept of renormalization group theory after removing certain deficiencies in Kadanoff’s scaling hypothesis. A greater detail of this is omitted here, because that is beyond the scope of the chapter.

5. Physical realization: simulation results based on Ising model

We now discuss some of the simulation results obtained using Ising model. Figure 9 shows spontaneous magnetization for a simple cubic crystal (i.e., scc lattice). As the strength of exchange coupling between spin-up and spin-down (J AB ) decreases, the critical temperature lowers down. Lower values of J AB weaken the spin flip-flop mechanism; henceforth the system requires further cooling, so that the spin-spin correlation overcomes the fluctuations. Spontaneous magnetization occurs in the absence of external magnetic field [ 28 ]. The confirmation of spontaneous process is further confirmed in Figure 10 . Figure 10 is plotted for spin correlation function vs. temperature of the system [ 28 ]. The critical temperature is marked by the presence of discontinuity in it. Above critical temperature, the magnetization abruptly falls to zero, which is an indication of paramagnetic state. The critical temperature in ferromagnetic thin film is known as Curie temperature. We observe similar kind of behavior with antiferromagnetic films, though below critical point (also known as Neel temperature), the net average magnetization becomes zero, because opposite spins are energetically favored in this case. The schematic diagram is shown in Figure 11 [ 28 ]. Magnetization vs. external magnetic field curves are plotted in Figure 12(a)–(d) for different sets of parameters [ 28 ].

Spontaneous magnetization in two-dimensional thin film (this figure is reproduced with permission from Singh [ 28 ]).

Correlation function vs. temperature for a two-dimensional thin film. Spontaneous magnetization is marked by discontinuity in it (this figure is reproduced with permission from Singh [ 28 ]).

Schematic representation of ferromagnetic to paramagnetic and antiferromagnetic to paramagnetic transitions.

(a) Magnetization vs. external fields at different temperature T = 0.50, 1.0, 1.5, and 2.0. (b) Magnetization vs. external fields for different exchange couplings J = 0.0, 0.25, 0.50, 0.75, and 1.0. These cases are for ferromagnetic thin films. (c) Magnetization vs. external fields at different temperature T = 0.50, 1.0, 1.5, and 2.0. (d) Magnetization vs. external fields for different exchange couplings J = 0.0, 0.25, 0.50, 0.75, and 1.0. These cases are for antiferromagnetic thin films (this figure is reproduced with permission from Singh [ 28 ]).

Simulation results obtained for a magnetically striped system as schematically shown in Figure 13 are reported in Figures 14 – 17 [ 29 ]. One or two alternate rectangular regions are created, using external field. Figure 14 shows the gradual transition at the interface, where a definite value of external field suddenly gets zero. The spin polarizations in two regions show sharp boundary. The magnetized film, in presence of magnetic field, induces the magnetic zones in proximity where its close external field is zero. Micrograph also indicates for spin-spin phase separation. The corresponding average magnetization vs. temperature and spin correlation function vs. temperature are also plotted in Figures 15 and 16 , respectively, but these studies are done using Monte Carlo simulation with semi-infinite free boundary conditions. It has been observed that these systems have relatively high critical transition temperatures. Figure 17 shows the magnetization process with two alternate magnetized zones [ 29 ].

(a) The system with one slab of size n x  × n y  × n z  = 50 × 100 × 100 exposed to an external magnetic field. (b) The system with two alternate slabs of size n x  × n y  × n z  = 50 × 100 × 100 exposed to an external magnetic field.

The micrograph of the coexisting phases in the regions of close proximity of the magnetic barrier indicating for the presence of depletion layer near the barrier.

Magnetization vs. temperature for magnetically striped system. Only one region experiences the presence of external magnetic field as illustrated in Figure 12(a) . This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [ 29 ]).

Spin correlation function vs. temperature for magnetically striped system. Only one region experiences the presence of external magnetic field as illustrated in Figure 13(a) . This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [ 29 ]).

Magnetization vs. temperature for magnetically striped system. Two alternate regions experience the presence of external magnetic field as illustrated in Figure 12(b) . This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [ 29 ]).

Low-dimensional magnetic heterostructures play vital role in spinotronics. Ferromagnets can induce magnetic ordering through a 40-nm-thick amorphous paramagnetic layer, when placed in its close proximity. One has to reconcile with long-range magnetic interaction to correctly measure the extent of induced magnetization. Readers may go through the Nature Communications article of F. Magnus et al. published in the year 2016 [ 17 ]. The magnetic properties of ferromagnetic materials with reduced dimensions get altered; when the thickness of a film is reduced below a critical value, the ferromagnetic to paramagnetic transition disappears [ 18 ]. Finite-size effects may also weaken or enhance magnetic interactions at the boundaries, as well as restrict the evolution of spin-spin correlation length. Extension of these ideas to model magnetic heterostructures, comprising of multiple magnetic and/or nonmagnetic layers, gives insight into interfacial phenomena. Many current and emerging technologies are based on this central problem. This may be very useful in understanding and exploring problems as metal-insulator transition, which is at the core of many state-of-the-art technologies. Henceforth, computational techniques, especially Ising model, can now be extended to develop and enrich science, for making new technologies. Though, its use can be said at the nascent stage, but with the advancement in computer hardware and efficient algorithms, it’s applications in areas related to spinotronics appears to be bright.

• 1. Brush SG. History of the Lenz-Ising model. Reviews of Modern Physics. 1967; 39 (4):883-893. DOI: 10.1103/RevModPhys.39.883
• 2. Niss M. History of the Lenz–Ising model 1950–1965: From irrelevance to relevance. Archive for History of Exact Sciences. 2009; 63 :243-287. DOI: 10.1007/s00407-008-0039-5
• 3. Singh SP. Curie temperature of Ising ferromagnetic film and its dependence on nn exchange coupling. AIP Conference Proceedings. 2018; 1953 :140015. DOI: 10.1063/1.5033190
• 4. Singh SP, Singh K, Roychoudhury M. Monte Carlo simulation for diffusion limited surface directed phase separation. Proceedings of the National Academy of Sciences, India—Section A. 2008; 78 :79-83
• 5. Singh SP. Spinodal theory: A common rupturing mechanism in spinodal dewetting and surface directed phase separation (some technological aspects and the significance of dipole-quadrupole interaction in spinodal dewetting). Advances in Condensed Matter Physics. 2011; 2011 :526397. DOI: 10.1155/2011/526397
• 6. Singh SP. Spatial correlation in 2D and 3 D thin films of conserved binary mixtures in presence of wetting of substrates by preferred majority component: Interpretation in real scenario. Applied Nanoscience. 2012; 2 :365-369. DOI: 10.1007/s13204-012-0094-8
• 7. Pan AC, Chandler D. Dynamics of nucleation in the Ising model. The Journal of Physical Chemistry. B. 2004; 108 :19681-19686. DOI: 10.1021/jp0471249
• 8. Sonsin AF et al. Computational analysis of 3D Ising model using Metropolis algorithms. Journal of Physics: Conference Series. 2015; 630 :012057. DOI: 10.1088/1742-6596/630/1/012057
• 9. Katsoulakis MA, Plechac P, Bellet LR. Numerical and statistical methods for the coarse-graining of many-particle stochastic systems. Journal of Scientific Computing. 2008; 37 :43-71. DOI: 10.1007/s10915-008-9216-6
• 10. Lundow PH, Markstrom K, Rosengren A. The Ising model for the bcc, fcc and diamond lattices: A comparison. Philosophical Magazine. 2009; 89 (22-24):2009-2042. DOI: 10.1080/14786430802680512
• 11. Montroll EW, Potts RB, Ward JC. Correlations and spontaneous magnetization of the two dimensional Ising model. Journal of Mathematical Physics. 1963; 4 :308-322. DOI: 10.1063/1.1703955
• 12. Huang R, Gujrati PD. Phase transitions of antiferromagnetic Ising spins on the zigzag surface of an asymmetrical Husimi lattice. Royal Society Open Science. 2019; 6 :181500. DOI: 10.1098/rsos.181500
• 13. Singh SP. Revisiting 2D lattice based spin flip-flop Ising model: Magnetic properties of a thin film and its temperature dependence. European Journal of Physics Education. 2014; 5 (3):8-19
• 14. Singh SP, Singh JK, Sharma A. Adsorption of gas like molecules on self-aligned square well fluid channels under confinement of chemically patterned substrates. Applied Nanoscience. 2013; 3 :179-187. DOI: 10.1007/s13204-012-0118-4
• 15. Singh SP. Monte Carlo simulation of microscopic viscosity and rupturing thin polymer film near melt: A molecular perspective. Physics Letters A. 2017; 381 :1321-1327. DOI: 10.1016/j.physleta.2017.02.011
• 16. SP Singh, A Kumari Singh, Formation of liquid structures and investigation of its interfacial properties using lattice based liquid-gas model, Proceedings of International Conference on Nanoscience and Nanotechnology, 29th Nov.-01st Dec. 2019, VIT, Vellore, Tamil Nadu, India
• 17. Magnus F et al. Long-range magnetic interactions and proximity effects in an amorphous exchange-spring magnet. Nature Communications. 2016; 7 :11931. DOI: 10.1038/ncomms119311
• 18. Singh SP. Specific heat capacity: Thickness critical spontaneous magnetization in striped ferromagnetic thin films. AIP Conference Proceedings. 2018; 1953 :040020. DOI: 10.1063/1.5032640
• 19. Patharia RK, Beale PD. Statistical Mechanics. 3rd ed. Academic Press; 2011
• 20. Huang K. Statistical Mechanics, 2nd Ed. Wiley; 2008
• 21. Chandler D. Introduction to Modern Statistical Mechanics. USA: Oxford University Press; 1987
• 22. Binder K, Heermann D. Monte Carlo Simulation in Statistical Physics, 1868-4513. Berlin Heidelberg: Springer-Verlag;
• 23. Kadanoff LP. Scaling Laws for Ising models near Tc*. Physics. 1966; 2 (6):263-272. DOI: 10.1103/PhysicsPhysiqueFizika.2.263
• 24. Holovatch Y. Introduction to renormalization. Condensed Matter Physics. 2006; 9 (2(46)):237-262. DOI: 10.5488/CMP.9.2.237
• 25. Kaya T. A new approach to real space renormalization group treatment of Ising model for square and simple cubic lattice. International Journal of Modern Physics B. 2018; 32 (23):1850252. DOI: 10.1142/S0217979218502521
• 26. Nauenberg M. Renormalization group solution of the one−dimensional Ising model. Journal of Mathematical Physics. 1975; 16 :703-705
• 27. Carneiro CEI, Henriques VB, Salinas SR. Renormalisation group calculations for a spin-1 Ising model with bilinear and biquadratic exchange interactions. Journal of Physics A: Mathematical and General. 1987; 20 :189-197. DOI: 10.1088/0305-4470/20/1/027
• 28. Singh SP. First observations of entropy vs free energy in lattice based modeling for spin coarsening in conserved and non-conserved binary mixtures: The phenomenological study of phase transitions in 2D thin films. Nanosystems: Physics, Chemistry, Mathematics. 2015; 6 (6):882-895. DOI: 10.17586/2220-8054-2015-6-6-882-895
• 29. Singh SP. Temperature induced spin polarization in magnetically striped ferromagnetic film: The coexisting phases and the two regime behavior. Macromolecular Symposia. 2017; 376 (1):1600193. DOI: 10.1002/masy.201600193

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Learning the Ising model with generative neural networks

Francesco d'angelo and lucas böttcher, phys. rev. research 2 , 023266 – published 2 june 2020.

• Citing Articles (18)
• INTRODUCTION
• GENERATIVE MODELS
• MONITORING LEARNING
• LEARNING THE ISING MODEL
• CONCLUSIONS AND DISCUSSION
• ACKNOWLEDGMENTS

Recent advances in deep learning and neural networks have led to an increased interest in the application of generative models in statistical and condensed matter physics. In particular, restricted Boltzmann machines (RBMs) and variational autoencoders (VAEs) as specific classes of neural networks have been successfully applied in the context of physical feature extraction and representation learning. Despite these successes, however, there is only limited understanding of their representational properties and limitations. To better understand the representational characteristics of RBMs and VAEs, we study their ability to capture physical features of the Ising model at different temperatures. This approach allows us to quantitatively assess learned representations by comparing sample features with corresponding theoretical predictions. Our results suggest that the considered RBMs and convolutional VAEs are able to capture the temperature dependence of magnetization, energy, and spin-spin correlations. The samples generated by RBMs are more evenly distributed across temperature than those generated by VAEs. We also find that convolutional layers in VAEs are important to model spin correlations whereas RBMs achieve similar or even better performances without convolutional filters.

• Received 8 December 2019
• Accepted 8 May 2020

DOI: https://doi.org/10.1103/PhysRevResearch.2.023266

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• 1 Institute of Neuroinformatics, University of Zurich and ETH Zurich, 8057, Zurich, Switzerland
• 2 Computational Medicine, UCLA, Los Angeles, California 90095-1766, USA
• 3 Institute for Theoretical Physics, ETH Zurich, 8093, Zurich, Switzerland
• 4 Center of Economic Research, ETH Zurich, 8092, Zurich, Switzerland
• * [email protected]
• † [email protected]

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Vol. 2, Iss. 2 — June - August 2020

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Restricted Boltzmann machine. An RBM is composed of one visible layer (blue) and one hidden layer (red). In this example, the respective layers consist of six visible units v i i ∈ { 1 , ⋯ , 6 } and four hidden units h i i ∈ { 1 , ⋯ , 4 } . The network structure underlying an RBM is bipartite.

Block Gibbs sampling in an RBM. We show an example of an RBM with a visible layer that consists of six visible units and a hidden layer that consists of four hidden units. Because of the bipartite network structure of RBMs, units within one layer can be grouped together and updated in parallel ( block Gibbs sampling ). Initially visible units (green) are determined by the data set. Hidden (red) and visible (blue) units are then updated in an alternating manner.

Variational autoencoder.  A VAE is composed of two neural networks: the encoder (yellow) and decoder (green). In this example, each network consists of an input and an output layer (multiple layers are also possible). The dotted black arrow between the two networks represents the reparameterization trick [see Eq. ( 17 )]. Each unit in the input layer of the decoder requires a corresponding mean μ and variance σ as input.

Evolution of physical features during training. We show the evolution of the magnetization M ( T ) (a), energy E ( T ) (b), and correlation function G ( r , T ) (c) for the training of a single RBM at T = 2.75 . Dashed lines and shaded regions indicate the mean and standard deviation of the corresponding M(RT) 2 samples and different colors in the plot of G ( r , T ) represent different radii r .

Physical features of RBM and convolutional VAE Ising samples . We use RBMs and convolutional cVAEs to generate 20 × 10 4 samples of Ising configurations with 32 × 32 spins for temperatures T ∈ { 1.5 , 2 , 2.5 , 2.75 , 3 , 4 } . We show the magnetization M ( T ) , energy E ( T ) , and correlation function G ( r , T ) for neural network and corresponding M(RT) 2 samples. In panels (a)–(c), we separately trained one RBM per temperature, whereas we used a single RBM for all temperatures in panels (d)–(f). In panels (g)–(i), we show the behavior of M ( T ) , E ( T ) , and G ( r , T ) for samples that are generated with a convolutional cVAE that was trained for all temperatures. Error bars are smaller than the markers.

Relative frequencies of samples at different temperatures. We show the relative frequencies of samples that are obtained by starting trained RBMs (a) and convoluational cVAEs (b) from random initial configurations. The data are based on 20 × 10 4 samples for each temperature.

Evolution of energy during training of a convolutional VAE. We show the evolution of the energy E ( T ) for the training of a single convolutional VAE at T = 2.75 (blue line). The dashed red line indicates the mean energy of the corresponding M(RT) 2 samples. The yellow line shows the mean energy of the convolutional cVAE trained on all temperatures for 1000 epochs.

Snapshots of Ising configurations. We show snapshots of Ising configurations for T ∈ { 1.5 , 2.5 , 4 } . The configurations in the top, middle, and bottom panels are based on M(RT) 2 , RBM, and convolutional cVAE samples, respectively.

Physical features of nonconvolutional VAE Ising samples. We use nonconvolutional cVAEs to generate 20 × 10 4 samples of Ising configurations with 32 × 32 spins for temperatures T ∈ { 1.5 , 2 , 2.5 , 2.75 , 3 , 4 } . We show the magnetization M ( T ) (a), energy E ( T ) (b), and correlation function G ( r , T ) (c) for neural-network and corresponding M(RT) 2 samples. The nonconvolutional cVAE was trained for all temperatures. Error bars are smaller than the markers.

Two-dimensional visualization of Ising samples. We show the distribution of 20 × 10 4 Ising samples for temperatures T ∈ { 1.5 , 2 , 2.5 , 2.75 , 3 , 4 } . The data in panels (a)–(c) are based on M(RT) 2 , RBM, and convolutional cVAE samples, respectively.

Distribution of magnetization at different temperatures. We show the distribution of 20 × 10 4 Ising samples for temperatures T ∈ { 1.5 , 2.0 , 2.5 , 2.75 , 3 , 4 } . The data are based on M(RT) 2 (yellow), RBM (blue), and convolutional cVAE (red) samples, respectively.

Gaussian coupling distribution . Gaussian distribution of Ising spin couplings [see Eq. ( 26 )]. The mean and standard deviation of the Gaussian distribution are both equal to one.

Physical features of RBM and convolutional VAE Ising samples for heterogeneous couplings . We use RBMs and convolutional cVAEs to generate 20 × 10 4 samples of Ising configurations with 32 × 32 spins for temperatures T ∈ { 10 − 6 , 0.5 , 1 , 1.5 , 2 , 3 } and a Gaussian coupling distribution whose mean and standard deviation are equal to one. We show the magnetization M ( T ) , energy E ( T ) , and correlation function G ( r , T ) for neural network and corresponding M(RT) 2 samples. In panels (a)–(c), we used a single RBM for all temperatures; in panels (d)–(f), we show the behavior of M ( T ) , E ( T ) , and G ( r , T ) for samples that are generated with a convolutional cVAE that was trained for all temperatures. Error bars are smaller than the markers.

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Ising Model: Recent Developments and Exotic Applications

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section " Statistical Physics ".

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Dear Colleagues,

Proposed 101 years ago and initially intended to describe magnetic ordering, the Ising model turned out to be one of the most important models of statistical mechanics. Indeed, the idea of a lattice model with nodes being discrete variables called spins, which prefer to be similarly oriented, turned out to be tremendously prolific and influential. In addition to describing various magnetic systems, the Ising model was used to analyze alloys, liquid helium mixtures, glasses, critical behaviors in various gases, or protein folding. In recent years, interest in the Ising model has by no means been waning, and it is often used to describe systems very distant from the realm of physics. To some extent, various features or attributes such as political opinions, comfort, financial decisions, ideas or culture might also be represented as discrete variables with suitably defined interactions. As a result, Ising-like models find myriads of applications in diverse research fields such as opinion formation, social network analysis and econophysics, but also computer science, computational biology and neuroscience. In the era of big data and artificial intelligence, the Ising model is bound to draw scientists’ attention for quite some time. The objective of this Special Issue is to collect papers that describe recent results related to the Ising model or introduce some original techniques for its analysis. Papers that explore some novel areas of applications of Ising models are also welcome.

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Add a method, remove a method, edit datasets, a quantum-classical hybrid algorithm with ising model for the learning with errors problem.

15 Aug 2024  ·  Muxi Zheng , Jinfeng Zeng , Wentao Yang , Pei-Jie Chang , Bao Yan , Haoran Zhang , Min Wang , Shijie Wei , Gui-Lu Long · Edit social preview

The Learning-With-Errors (LWE) problem is a crucial computational challenge with significant implications for post-quantum cryptography and computational learning theory. Here we propose a quantum-classical hybrid algorithm with Ising model (HAWI) to address the LWE problem. Our approach involves transforming the LWE problem into the Shortest Vector Problem (SVP), using variable qubits to encode lattice vectors into an Ising Hamiltonian. We then identify the low-energy levels of the Hamiltonian to extract the solution, making it suitable for implementation on current noisy intermediate-scale quantum (NISQ) devices. We prove that the number of qubits required is less than $m(3m-1)/2$, where $m$ is the number of samples in the algorithm. Our algorithm is heuristic, and its time complexity depends on the specific quantum algorithm employed to find the Hamiltonian's low-energy levels. If the Quantum Approximate Optimization Algorithm (QAOA) is used to solve the Ising Hamiltonian problem, and the number of iterations satisfies $y < O\left(m\log m\cdot 2^{0.2972k}/pk^2\right)$, our algorithm will outperform the classical Block Korkine-Zolotarev (BKZ) algorithm, where $k$ is the block size related to problem parameters, and $p$ is the number of layers in QAOA. We demonstrate the algorithm by solving a $2$-dimensional LWE problem on a real quantum device with $5$ qubits, showing its potential for solving meaningful instances of the LWE problem in the NISQ era.

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History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance

• Published: 06 December 2008
• Volume 63 , pages 243–287, ( 2009 )

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This is the second in a series of three papers that charts the history of the Lenz–Ising model (commonly called just the Ising model in the physics literature) in considerable detail, from its invention in the early 1920s to its recognition as an important tool in the study of phase transitions by the late 1960s. By focusing on the development in physicists’ perception of the model’s ability to yield physical insight—in contrast to the more technical perspective in previous historical accounts, for example, Brush (Rev Modern Phys 39: 883–893, 1967) and Hoddeson et al. (Out of the Crystal Maze. Chapters from the History of Solid-State Physics. Oxford University Press, New York, pp. 489–616, 1992)—the series aims to cover and explain in depth why this model went from relative obscurity to a prominent position in modern physics, and to examine the consequences of this change. In the present paper, which is self-contained, I deal with the development from the early 1950s to the 1960s and document that this period witnessed a major change in the perception of the model: In the 1950s it was not in the cards that the model was to become a pivotal tool of theoretical physics in the following decade. In fact, I show, based upon recollections and research papers, that many of the physicists in the 1950s interested in understanding phase transitions saw the model as irrelevant for this endeavor because it oversimplifies the nature of the microscopic constituents of the physical systems exhibiting phase transitions. However, one group, Cyril Domb’s in London, held a more positive view during this decade. To bring out the basis for their view, I analyze in detail their motivation and work. In the last part of the paper I document that the model was seen as much more physically relevant in the early 1960s and examine the development that led to this change in perception. I argue that the main factor behind the change was the realization of the surprising and striking agreement between aspects of the model, notably its critical behavior, and empirical features of the physical phenomena.

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Ashrafi, B., Hall, K., and Schweber, S. S. (n.d.): “The Solution to the Phase Transition Problem: 1965–1975” (unpublished).

Atkins K.R., Edwards M.H. (1955) “Coefficient of Expansion of Liquid Helium II,”. Physical Review 97: 1429–1434

Article   Google Scholar  

Bagatskii, M. I., Voronel’, A. V., and Gusak, V. G (1963): “Measurement of the Specific Heat C V of Argon in the Immediate Vicinity of the Critical Point”, Soviet Physics JETP-USSR 16 , 517–518. For Russian original, see (1962): Zhyrnal Eksperimental’noi i Teoreticheskoi Fiziki 43 , 728

Baker G.A. Jr. (1961) “Application of the Padé Approximant Method to the Investigation of Some Magnetic Properties of the Ising Model,”. Physical Review 124: 768–774

Baker G.A. Jr., Gammel J.L. (1961) “The Padé Approximant,”. Journal of Mathematical Analysis and Applications 2: 21–30

Article   MATH   MathSciNet   Google Scholar  

Benedek, G. B. (1966): “Equilibrium Properties of Ferromagnets and Antiferromagnets in the Vicinity of the Critical Point,” in Green and Sengers (1966), pp. 42–48.

Binney, J. J, Dowrick, N. J., Fisher, A. J., and Newman, M. E. J. (1992): The Theory of Critical Phenomena . Clarendon, Oxford

Brooks J.E., Domb C. (1951) “Order–Disorder Statistics. III. The Antiferromagnet and Order– Disorder Transitions,”. Proceedings of the Royal Society [A] 207: 343–358

Brush, S. G. (1964): “History of the Lenz–Ising Model,” UCRL-7940 , Lawrence Radiation Laboratory, Livermore, California, June.

Brush S.G. (1967) “History of the Lenz–Ising Model,”. Reviews of Modern Physics 39: 883–893

Brush S.G. (1983) Statistical Physics and the Atomic Theory of Matter. Princeton University Press, Princeton

Google Scholar  

Buckingham M.J., Fairbank W.M. (1961) “The Nature of the Lambda Transition,”. Progress in Low Temperature Physics 3: 80–113

Burley D.M. (1960) “Some Magnetic Properties of the Ising Model,”. Philosophical Magazine 5: 909–919

Chase C.E., Williamson R.C., Tisza L. (1964) “Ultrasonic Propagation near the Critical Point in Helium,”. Physical Review 13: 467–469

Cooke, A. H., Edmonds, D. T., McKim, F. R., and Wolf, W. P. (1959): “Magnetic Dipole Interactions in Dysprosium Ethyl Sulphate. 1. Susceptibility and Specific Heat between 20-Degrees-K and 1-Degrees-K,” Proceedings of the Royal Society [A] 252 , 246–259.

Cushing J.T. (1994) Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony. University of Chicago Press, Chicago

MATH   Google Scholar  

Domb C. (1949a) “Order–Disorder Statistics. I.,”. Proceedings of the Royal Society[A] 196: 36–50

Domb C. (1949b) “Order–Disorder Statistics. II. A Two-Dimensional Model,”. Proceedings of the Royal Society[A] 199: 199–221

Domb C. (1949c) “Statistical Mechanics of Some Co-operative Phenomena,”. Nature 163: 775–776

Article   MATH   Google Scholar  

Domb, C. (1952), “L’influence de la structure du réseau sur l’anomalie de la chaleur spécifique du modèle d’Ising,” in Changements de Phases , Comptes Rendus de la deuxième Réunion Annuelle tenue en commun avec la Commission de Thermodynamique de l’Union Internationale de Physique, Société de Chimie Physique, Presse Universitaires de France, Paris, pp. 8–18.

Domb C. (1955) “Statistical Physics and its Problems,”. Science Progress 43: 402–417

Domb C. (1960) “On the Theory of Cooperative Phenomena in Crystals,”. Advances in Physics 9: 149–295

Domb, C. (1966): “Critical Properties of Lattice Models,” in Green and Sengers (1966), pp. 29–41.

Domb, C. (1971): “The Curie Point,” in E. G. D. Cohen, ed., Statistical Mechanics at the Turn of the Decade , Marcel Dekker, New York, pp. 81–128.

Domb, C. (1974a): “Ising Model,” in Domb and Green (1974a), pp. 357–484.

Domb C. (1974b) “Configurational Studies of the Potts Models,”. Journal of Physics A: Mathematical, Nuclear and General 7: 1335–1348

Domb C. (1990a) “Some Reminiscences about My Early Career,”. Physica A 168: 1–21

Article   MathSciNet   Google Scholar  

Domb C. (1990b) “On Hammersley’s Method for One-Dimensional Covering Problems,”. In: Grimmett G.R., Welsh D.J.A. (eds) Disorder in Physical Systems: Essays in Honour of John M. Hammersley on the occasion of his 70th Birthday. Oxford University Press, New York, pp 33–53

Domb C. (1996) The Critical Point. Taylor and Francis, London

Domb, C. with Schweber, Silvan S. (2002): Interview, June 2002, History of Recent Science & Technology, The Dibner Institute for the History of Science and Technology, MIT, Massachusetts, available at (accessed February 18, 2008). http://authors.library.caltech.edu/5456/01/hrst.mit.edu/hrs/renormalization/Domb/

Domb C. (2003) “Some Observations on the Early History of Equilibrium Statistical Mechanics,”. Journal of Statistical Physics 110: 475–496

Domb, C. and Green, M. S., eds. (1972a): Phase Transitions and Critical Phenomena , Academic Press, London, New York, vol. 1.

Domb, C. and Green, M. S. (1972b): “Preface to Volume 1,” in Domb and Green (1972a), pp. ix-xii.

Domb, C. and Green, M. S., eds. (1974a): Phase Transitions and Critical Phenomena , Academic Press, London, New York, vol. 3.

Domb C., Sykes M.F. (1956) “On Metastable Approximations in Co-operative Assemblies,”. Proceedings of the Royal Society of London [A] 235: 247–259

Domb C., Sykes M.F. (1957a) “On the Susceptibility of a Ferromagnetic above the Curie Point,”. Proceedings of the Royal Society of London[A] 240: 214–228

Domb C., Sykes M.F. (1957b) “Specific Heat of a Ferromagnetic Substance above the Curie Point,”. Physical Review 108: 1415–1416

Domb C., Sykes M.F. (1957c) “The Calculation of Lattice Constants in Crystal Statistics,”. Philosophical Magazine 2: 733–749

DombC. Sykes M.F. (1962) “Effect of Change of Spin on Critical Properties of Ising and Heisenberg Models,”. Physical Review 128: 168–173

Dyson F.J. (1956) “General Theory of Spin-Wave Interactions,”. Physical Review 102: 1217–1230

Dyson F.J. (1995) “The Coulomb Fluid and the Fifth Painlevé Transcendent,”. In: Liu C.S., Yau S.-T (eds) Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Boston, pp 131–146

Elcock E.W. (1956) Order–Disorder Phenomena. Methuen, London

Essam J.W., Fisher M.E. (1963) “Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point,”. Journal of Chemical Physics 38: 802–812

Fairbank, W. M., Buckingham, M. J., and Kellers, F. (1957): “Specific Heat of Liquid He 4 Near the Lambda Point,” in J. R. Dillinger, ed., Proceedings of the Fifth International Conference on Low-Temperature Physics and Chemistry , University of Wisconsin Press, Madison, pp. 50–52.

Fisher M.E. (1959) “The Susceptibility of the Plane Ising Model,”. Physica 25: 521–524

Fisher M.E. (1964a) “Specific Heat of a Gas near the Critical Point,”. Physical Review 136: A1599–A1604

Fisher M.E. (1964b) “Correlation Functions and the Critical Region of Simple Fluids,”. Journal of Mathematical Physics 5: 944–962

Fisher, M. E. (1965): “The Nature of Critical Points,” in W. E. Brittin, ed., Lectures in Theoretical Physics VII C , University of Colorado Press, Boulder, pp. 1–159.

Fisher M.E. (1967) “The Theory of Equilibrium Critical Phenomena,”. Reports in Progress in Physics 30: 615–730

Fisher, M. E. (1996): “Foreword: About the Author and the Subject,” Domb (1996), pp. xiii-xviii.

Fisher M.E., Sykes M.F. (1962) “Antiferromagnetic Susceptibilities of the Simple Cubic and Body-Centered Cubic Ising Lattices,”. Physics 28: 939–956

Gaunt D.S., Fisher M.E., Sykes M.F., Essam J.W. (1964) “Critical Isotherm of a Ferromagnet and of a Fluid,”. Physical Review Letters 13: 713–715

Gelfert A. (2005) “Mathematical Rigor in Physics: Putting Exact Results in Their Place,”. Philosophy of Science 72: 723–738

Goldenfeld N. (1992): Lectures on Phase Transitions and the Renormalization Group , Perseus Books, Reading, Massachusetts.

Green, M. S. (1966): “Introduction,” in Green and Sengers (1966), pp. xi–xi.

Green, M. S. and Sengers, eds., J. V. (1966): Critical Phenomena. Proceedings of a Conference Held in Washington, D. C., April 1965 , National Bureau of Standards

Habgood H.W., Schneider W.G. (1954) “PVT measurements in the Critical Region of Xenon,”. Canadian Journal of Chemistry 32: 98–112

Hill T.S. (1956) Statistical Mechanics. McGraw-Hill, New York

Hoddeson L., Schubert H., Heims S.J., Baym G. (1992) “Collective Phenomena,”. In: Hoddeson L., Braun E., Teichmann J., Weart S. (eds) Out of the Crystal Maze. Chapters from the History of Solid-State Physics. Oxford University Press, New York, pp 489–616

Hunt K.L. (1953) “Collective Electron Ferromagnetism: A Generalization of the Treatment and an Analysis of Experimental Results,”. Proceedings of the Royal Society of London [A] 32: 103–117

Ising, E. (1924): “Beitrag zur Theorie des Ferro- und Paramagnetismus,” Ph.D. Thesis, University of Hamburg.

Ising E. (1925) “Beitrag zur Theorie des Ferromagnetismus,”. Zeitschrift für Physik 31: 253–258

Jaeger G. (1998) “The Ehrenfest Classification of Phase Transitions: Introduction and Evolution,”. Archive for History of Exact Sciences 53: 51–81

Kaufman B. (1949) “Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis,”. Physical Review 76: 1232–1243

Kaufman B., Onsager L. (1949) “Crystal Statistics. III. Short-Range Order in a Binary Ising Lattice,”. Physical Review 76: 1244–1252

Kirkwood J.G. (1950) “Critique of the Free Volume Theory of the Liquid State,”. Journal of Chemical Physics 18: 380–382

Kouvel J.S., Fisher M.E. (1964) “Detailed Magnetic Behavior of Nickel near its Curie Point,”. Physical Review 136: A1626–A1632

Krieger M.H. (1996) Constitutions of Matter. University of Chicago Press, Chicago

Landau L.D. (1937) “Zur Theorie der Phasenumwandlungen. I.,”. Physikalische Zeitschrift der Sowjetunion 11: 26–47

Landau L.D., Lifshitz E.M. (1958) Statistical Physics. Pergamon, London

Lasheen M.A., Vanden Broek J., Gorter C.J. (1958) “The magnetic Susceptibility and Relaxation of a MnCl 2 · H 2 O Single Crystal in the Paramagnetic and Antiferromagnetic States,”. Physica 24: 1061–1075

Lebowitz J.L (1995) “Lars Onsager November 27, 1903 - October 5, 1976: In Memoriam,”. Journal of Statistical Physics 78: 1–3

Lee T.D., Yang C.N. (1952) “Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising model,”. Physical Review 87: 410–419

Lenz W. (1920) “Beitrag zum Verständnis der magnetischen Erscheinungen in festen Köpern,”. Physikalische Zeitschrift 21: 613–615

Linder B. (1954) “Order–Disorder Phenomena,”. Journal of Chemical Physics 22: 970–973

Lipa J.A., Swanson D.R., Nissen J.A., Chui T.C.P., Israelsson U.E. (1996) “Heat Capacity and Thermal Relaxation of Bulk Helium very near the Lambda Point,”. Physical Review Letters 76: 944–947

LonguetHiggins, H. C. and Fisher, M. E. (1996): “Lars Onsager: 27 November, 1903–5 October, 1976, “in Hemmer, P. C., Holden, H. and Kjelstrup Ratkje, S., eds. (1996): The Collected Works of Lars Onsager .World Scientific, Singapore, pp. 9–34.

Moldover M.R., Little W.A. (1965) “Specific Heat of He 3 and He 4 in the Neighborhood of their Critical Points,”. Physical Review Letters 15: 54–56

Montroll E.W., Potts R.B., Ward J.C. (1963) “Correlations and Spontaneous Magnetization of the Two-Dimensional Ising Model,”. Journal of Mathematical Physics 4: 308–323

Moser H. (1936) “Messung der wahren spezifischen Wärme von Silber, Nickel, β-Messing, Quartzkristall und Quartzglas zwischen + 50 und 700° C nach verfeinerten Methode,”. Physikalische Zeitschrift 37: 737–753

Muto T., Takagi Y. (1955) “The Theory of Order–Disorder Transitions in Alloys,”. Solid State Physics 1: 194–284

Nielsen, A. I. and Timmermann, S. (2002): En historisk undersøgelse af udviklingen af L. D. Landaus teori for kontinuerte overgange .Unpublished report, Department of Mathematics and Physics, Roskilde University, Roskilde, Denmark.

Niss M. (2005) “History of the Lenz–Ising Model 1920–1950: From Ferromagnetic to Cooperative Phenomena,”. Archive for History of Exact Science 59: 267–318

Onsager L. (1944) “Crystal Statistics. I. A Two-Dimensional Model with an Order–Disorder Transition,”. Physical Review 65: 117–149

Onsager L. (1949) discussion remark. Nuovo Cimento, Suppl. 6: 261

Pelissetto A., Vicari E. (2002) “Critical Phenomena and Renormalization-group Theory,”. Physics Reports 368: 549–727

Pais, A. (1958): “The Scientific Work of T. D. Lee and C. N. Yang,” Nuclear Physics 5 , 297–300.

Potter H.H. (1934) “The magneto-caloric effect and other magnetic phenomena in iron,”. Proceedings of the Royal Society of London [A] 146: 262–387

Robinson, W. K. and Friedberg, S. A. (1960): “Specific Heats of NiCl 2 · 6 H 2 O and CoCl 2 ·6 H 2 O between 1.4° and 20° K ,” Physical Review Letters 117 , 402–408.

Rowlinson, J. S. (1966): “Critical States of Simple Fluids and Fluid Mixtures: a Review of the Experimental Position,” in Green and Sengers (1966), pp. 9–12.

Ruelle D. (1969) Statistical Mechanics: Rigorous Results. Benjamin, New York

Rushbrooke G.S., Wood P.J. (1958) “On the Curie Points and High Temperature Susceptibilities of Heisenberg Model Ferromagnetics,”. Molecular Physics 1: 257–283

Sauer T. (2001) “The Feynman Path Goes Monte Carlo,”. In: Janke W., Pelster A., Scmidt H-J., Bachmann M. (eds) Fluctuating Paths and Fields. Festschrift Dedicated to Hagen Kleinert on the Occasion of His 60th Birthday. World Scientific, Singapore, New Jersey, London, Hong Kong, pp 29–42

Schweber S.S. (1994) QED and the Men Who Made It. Princeton University, Princeton

Skalyo J., Friedberg S.A. (1964) “Heat Capacity of the Antiferromagnet CoCl 2 · H 2 O near its Néel Point,”. Physical Review Letters 13: 113–135

Stanley H.E. (1971) Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, New York, Oxford

Sucksmith W., Clark C.A., Oliver D.J., Thompson J.E. (1953) “Spontaneous Magnetization— Techniques and Measurements,”. Reviews of Modern Physics 25: 34–41

Suchsmith W., Pearce R.R. (1938) “The Paramagnetism of the Ferromagnetic Elements,”. Proceedings of the Royal Society of London[A] 167: 189–204

Sykes M.F., Fisher M.E. (1962) “Antiferromagnetic Susceptibility of Plane Square and Honeycomb Ising Lattices,”. Physica 28: 919–938

Sykes C., Wilkinson H. (1937) “The Transformation in the Beta Brasses,”. Journal of the Institute of Metals 61: 223–240

Sykes C., Wilkinson H. (1938) “The Specific Heat of Nickel from 100 Degrees C. to 600 Degrees C,”. Proceedings of the Physical Society of London 50: 834–851

Temperley H.N.V (1956) Changes of State. Cleaver-Hume, London

Tisza L. (1951) “On the General Theory of Phase Transitions”. In: Smoluchowski R., Mayer J.E., Weyl W.A. (eds) Phase Transformations in Solids [Symposium held at Cornell University, 1948]. Wiley, New York, pp 1–37

Van Vleck J.H. (1945) “A Survey of the Theory of Ferromagnetism,”. Reviews of Modern Physics 17: 27–47

Van Vleck J.H. (1953) “Models of Exchange Coupling in Ferromagnetic Media,”. Reviews of Modern Physics 25: 220–228

Wannier G.H. (1945) “The Statistical Problem in Cooperative Phenomena,”. Reviews of Modern Physics 17: 50–60

Widom B. (1964) “Degree of the Critical Isotherm,”. Journal of Chemical Physics 41: 1633–1634

Widom B., Rice O.K. (1955) “Critical Isotherm and the Equation of Liquid-Vapor Systems,”. Journal of Chemical Physics 23: 1250–1255

Wolf W.P. (2000) “The Ising Model and Real Magnetic Materials,”. Brazilian Journal of Physics 30: 794–810

Yang C.N. (1952) “The Spontaneous Magnetization of a Two-Dimensional Ising Model,”. Physical Review 85: 808–816

Yang, C. N. (1972): “Introductory Note on Phase Transitions and Critical Phenomena” in Domb and Green (1972a), pp. 1–5.

Yang, C. N. (1983): “Commentary,” in C. N. Yang, Selected Papers 1945–1980 , Freeman, San Francisco, pp. 1–82.

Yang C.N. (1995) “Remarks about Some Developments in Statistical Mechanics,”. AAPPS Bulletin 5: 2–3

Yang C.N., Yang C.P. (1964) “Critical Point in Liquid-Gas Transitions,”. Physical Review Letters 13: 303–305

Ziman J.M. (1965) “Mathematical Models and Physical Toys,”. Nature 206: 1187–1192

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Niss, M. History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance. Arch. Hist. Exact Sci. 63 , 243–287 (2009). https://doi.org/10.1007/s00407-008-0039-5

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Title: symmetry topological field theory and non-abelian kramers-wannier dualities of generalised ising models.

Abstract: For a class of two-dimensional Euclidean lattice field theories admitting topological lines encoded into a spherical fusion category, we explore aspects of their realisations as boundary theories of a three-dimensional topological quantum field theory. After providing a general framework for explicitly constructing such realisations, we specialise to non-abelian generalisations of the Ising model and consider two operations: gauging an arbitrary subsymmetry and performing Fourier transforms of the local weights encoding the dynamics of the theory. These are carried out both in a traditional way and in terms of the three-dimensional topological quantum field theory. Whenever the whole symmetry is gauged, combining both operations recovers the non-abelian Kramers-Wannier duals à la Freed and Teleman of the generalised Ising models. Moreover, we discuss the interplay between renormalisation group fixed points of gapped symmetric phases and these generalised Kramers-Wannier dualities.

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Storm-scale convection-allowing models (CAMs) are an important tool for predicting the evolution of thunderstorms and mesoscale convective systems that result in damaging extreme weather. By explicitly resolving convective dynamics within the atmosphere they afford meteorologists the nuance needed to provide outlook on hazard. Deep learning models have thus far not proven skilful at km-scale atmospheric simulation, despite being competitive at coarser resolution with state-of-the-art global, medium-range weather forecasting. We present a generative diffusion model called StormCast, which emulates the high-resolution rapid refresh (HRRR) model—NOAA’s state-of-the-art 3km operational CAM. StormCast autoregressively predicts 99 state variables at km scale using a 1-hour time step, with dense vertical resolution in the atmospheric boundary layer, conditioned on 26 synoptic variables. We present evidence of successfully learnt km-scale dynamics including competitive 1-6 hour forecast skill for composite radar reflectivity alongside physically realistic convective cluster evolution, moist updrafts, and cold pool morphology. StormCast predictions maintain realistic power spectra for multiple predicted variables across multi-hour forecasts. Together, these results establish the potential for autoregressive ML to emulate CAMs – opening up new km-scale frontiers for regional ML weather prediction and future climate hazard dynamical downscaling.

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• Published: 13 August 2024

A scalable universal Ising machine based on interaction-centric storage and compute-in-memory

• Wenshuo Yue   ORCID: orcid.org/0000-0002-4339-0489 1 , 2   na1 ,
• Teng Zhang   ORCID: orcid.org/0000-0003-2011-5940 1   na1 ,
• Zhaokun Jing 1 ,
• Yuxiang Yang 1 ,
• Zhen Yang 1 ,
• Yongqin Wu 4 ,
• Weihai Bu 4 ,
• Kai Zheng 4 ,
• Jin Kang 4 ,
• Yibo Lin 1 ,
• Yaoyu Tao   ORCID: orcid.org/0000-0001-7500-5250 2 ,
• Bonan Yan   ORCID: orcid.org/0000-0002-3052-9330 1 , 2 ,
• Ru Huang   ORCID: orcid.org/0000-0002-8146-4821 1 , 2 &
• Yuchao Yang   ORCID: orcid.org/0000-0003-4674-4059 1 , 2 , 5 , 6  

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Ising machines are annealing processors that can solve combinatorial optimization problems via the physical evolution of the corresponding Ising graphs. Such machines are, however, typically restricted to solving problems with certain kinds of graph topology because the spin location and connections are fixed. Here, we report a universal Ising machine that supports arbitrary Ising graph topology with reasonable hardware resources using a coarse-grained compressed sparse row method to compress and store sparse Ising graph adjacency matrices. The approach, which we term interaction-centric storage, is suitable for any kind of Ising graph and reduces the memory scaling cost. We experimentally implement the Ising machine using compute-in-memory hardware based on a 40 nm resistive random-access memory arrays. We use the machine to solve max-cut and graph colouring problems, with the latter showing a 442–1,450 factor improvement in speed and 4.1 × 10 5 –6.0 × 10 5 factor reduction in energy consumption compared to a general-purpose graphics processing unit. We also use our Ising machine to solve a realistic electronic design automation problem—multiple patterning lithography layout decomposition—with 390–65,550 times speedup compared to the integer linear programming algorithm on a typical central processing unit.

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Data availability.

Source data are available via Zenodo at https://doi.org/10.5281/zenodo.10686168 (ref. 41 ). Other data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The core source code that implements the full-FPGA version of the UIM architecture is available via Zenodo at https://doi.org/10.5281/zenodo.10686168 (ref. 41 ). Other codes are available from the corresponding authors upon reasonable request.

Karp, R. M. Reducibility Among Combinatorial Problems (Springer, 2010).

Aadit, N. A. et al. Massively parallel probabilistic computing with sparse Ising machines. Nat. Electron. 5 , 460–468 (2022).

Article   Google Scholar  

Schuetz, M. J., Brubaker, J. K. & Katzgraber, H. G. Combinatorial optimization with physics-inspired graph neural networks. Nat. Mach. Intell. 4 , 367–377 (2022).

Korte, B. H., Vygen, J., Korte, B. & Vygen, J. Combinatorial Optimization Vol. 1 (Springer, 2011).

Lin, S. & Kernighan, B. W. An effective heuristic algorithm for the traveling-salesman problem. Oper. Res. 21 , 498–516 (1973).

Article   MathSciNet   Google Scholar  

Kokash, N. An Introduction to Heuristic Algorithms (Univ. Trento, 2005).

Tatsumura, K., Yamasaki, M. & Goto, H. Scaling out Ising machines using a multi-chip architecture for simulated bifurcation. Nat. Electron. 4 , 208–217 (2021).

Mohseni, N., McMahon, P. L. & Byrnes, T. Ising machines as hardware solvers of combinatorial optimization problems. Nat. Rev. Phys. 4 , 363–379 (2022).

Hu, F., Wang, B.-N., Wang, N. & Wang, C. Quantum machine learning with D-wave quantum computer. Quantum Eng. 1 , e12 (2019).

Johnson, M. W. et al. Quantum annealing with manufactured spins. Nature 473 , 194–198 (2011).

Inagaki, T. et al. A coherent Ising machine for 2000-node optimization problems. Science 354 , 603–606 (2016).

Dutta, S. et al. An Ising Hamiltonian solver based on coupled stochastic phase-transition nano-oscillators. Nat. Electron. 4 , 502–512 (2021).

Moy, W. et al. A 1,968-node coupled ring oscillator circuit for combinatorial optimization problem solving. Nat. Electron. 5 , 310–317 (2022).

Takemoto, T., Hayashi, M., Yoshimura, C. & Yamaoka, M. 2.6 A 2 × 30k-spin multichip scalable annealing processor based on a processing-in-memory approach for solving large-scale combinatorial optimization problems. In IEEE International Solid-State Circuits Conference (ISSCC) 52–54 (IEEE, 2019).

Takemoto, T. et al. 4.6 a 144 kb annealing system composed of 9 × 16 kb annealing processor chips with scalable chip-to-chip connections for large-scale combinatorial optimization problems. In IEEE International Solid-State Circuits Conference (ISSCC) Vol. 64, 64–66 (IEEE, 2021).

Yamamoto, K. et al. 7.3 statica: a 512-spin 0.25 m-weight full-digital annealing processor with a near-memory all-spin-updates-at-once architecture for combinatorial optimization with complete spin-spin interactions. In IEEE International Solid-State Circuits Conference (ISSCC) 138–140 (IEEE, 2020).

Chou, J., Bramhavar, S., Ghosh, S. & Herzog, W. Analog coupled oscillator based weighted Ising machine. Sci. Rep. 9 , 14786 (2019).

Su, Y., Kim, H. & Kim, B. CIM-spin: a scalable CMOS annealing processor with digital in-memory spin operators and register spins for combinatorial optimization problems. IEEE J. Solid-State Circuits 57 , 2263–2273 (2022).

Su, Y., Mu, J., Kim, H. & Kim, B. A scalable CMOS Ising computer featuring sparse and reconfigurable spin interconnects for solving combinatorial optimization problems. IEEE J. Solid-State Circuits 57 , 858–868 (2022).

Wang, T. & Roychowdhury, J. OIM: oscillator-based Ising machines for solving combinatorial optimisation problems. In Unconventional Computation and Natural Computation: 18th International Conference, UCNC 2019 Vol. 18, 232–256 (Springer International Publishing, 2019).

Wang, T., Wu, L. & Roychowdhury, J. New computational results and hardware prototypes for oscillator-based Ising machines. In Proc. 56th Design Automation Conference (DAC) 1–2 (ACM, 2019).

Tatsumura, K., Dixon, A. R. & Goto, H. FPGA-based simulated bifurcation machine. In Proc. International Conference on Field Programmable Logic and Applications (FPL) 59–66 (IEEE, 2019).

Yamamoto, K. et al. A time-division multiplexing Ising machine on FPGAs. In Proc. International Symposium on Highly Efficient Accelerators and Reconfigurable Technologies 1–6 (ACM, 2017).

Cook, C., Jin, W. & Tan, S. X.-D. GPU-based Ising computing for solving balanced min-cut graph partitioning problem. Preprint at https://arxiv.org/abs/1908.00210 (2019).

Cook, C., Zhao, H., Sato, T., Hiromoto, M. & Tan, S. X.-D. GPU-based Ising computing for solving max-cut combinatorial optimization problems. Integration 69 , 335–344 (2019).

Verma, N. et al. In-memory computing: advances and prospects. IEEE Solid-State Circuits Mag. 11 , 43–55 (2019).

Roy, K., Chakraborty, I., Ali, M., Ankit, A. & Agrawal, A. In-memory computing in emerging memory technologies for machine learning: an overview. In Proc. 57th Design Automation Conference (DAC) 1–6 (IEEE, 2020).

Yan, B. et al. Resistive memory-based in-memory computing: from device and large-scale integration system perspectives. Adv. Intell. Syst. 1 , 1900068 (2019).

Sebastian, A., Le Gallo, M., Khaddam-Aljameh, R. & Eleftheriou, E. Memory devices and applications for in-memory computing. Nat. Nanotechnol. 15 , 529–544 (2020).

Jiang, M., Shan, K., He, C. & Li, C. Efficient combinatorial optimization by quantum-inspired parallel annealing in analogue memristor crossbar. Nat. Commun. 14 , 5927 (2023).

Barahona, F., Grötschel, M., Jünger, M. & Reinelt, G. An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36 , 493–513 (1988).

Goemans, M. Improved approximation algorithms for maximum cut and satisability problems using semidenite programming. J. Assoc. Comput. Mach. 42 , 330–343 (1995).

Coudert, O. Exact coloring of real-life graphs is easy. In Proc. 34th Design Automation Conference (DAC) 121–126 (ACM, 1997).

Kahng, A. B., Park, C.-H., Xu, X. & Yao, H. Layout decomposition approaches for double patterning lithography. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 29 , 939–952 (2010).

Ciesielski, M. J., Yang, S. & Perkowski, M. A. Multiple-valued Boolean minimization based on graph coloring. In Proc. International Conference on Computer Design: VLSI in Computers and Processors 262–263 (IEEE, 1989).

Smith, M. D., Ramsey, N. & Holloway, G. A generalized algorithm for graph-coloring register allocation. In Proc. ACM SIGPLAN Conference on Programming Language Design and Implementation 277–288 (ACM, 2004).

Johnson, D. S. & Trick, M. A. Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge Vol. 26 (American Mathematical Society, 1996).

Zhang, K., Qiu, M., Li, L. & Liu, X. Accelerating genetic algorithm for solving graph coloring problem based on CUDA architecture. In Bio-Inspired Computing — Theories and Applications 578–584 (Springer, 2014).

Ma, Y., Zeng, X. & Yu, B. Methodologies for layout decomposition and mask optimization: a systematic review. In Proc. International Conference on Very Large Scale Integration (VLSI-SoC) 1–6 (IEEE, 2017).

Li, W. et al. OpenMPL: an open-source layout decomposer. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 40 , 2331–2344 (2020).

Yue, W. Scalable universal Ising machine enabled by interaction-centric storage and compute-in-memory technology. Zenodo https://doi.org/10.5281/zenodo.10686168 (2024).

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Acknowledgements

This work was supported by the National Key R&D Programme of China (grant no. 2023YFB4502200), National Natural Science Foundation of China (grant nos 61925401, 92064004, 61927901, 8206100486, 92164302 and 92364102), Beijing Natural Science Foundation (grant no. L234026) and the 111 Project (grant no. B18001). Fund grant nos 2023YFB4502200, 61925401, 92064004, 92164302 and L234026 were awarded to Yuchao Yang. Fund grant no. 61927901 was awarded to R.H. Fund grant no. 8206100486 was awarded to Y.T. Fund grant no. 92364102 was awarded to B.Y.

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These authors contributed equally: Wenshuo Yue, Teng Zhang.

Authors and Affiliations

Beijing Advanced Innovation Center for Integrated Circuits, School of Integrated Circuits, Peking University, Beijing, China

Wenshuo Yue, Teng Zhang, Zhaokun Jing, Yuxiang Yang, Zhen Yang, Yibo Lin, Bonan Yan, Ru Huang & Yuchao Yang

Institute for Artificial Intelligence, Peking University, Beijing, China

Wenshuo Yue, Yaoyu Tao, Bonan Yan, Ru Huang & Yuchao Yang

College of Electron and Information Engineering, Hebei University, Baoding, China

Semiconductor Technology Innovation Center (Beijing) Corporation, Beijing, China

Yongqin Wu, Weihai Bu, Kai Zheng & Jin Kang

Guangdong Provincial Key Laboratory of In-Memory Computing Chips, School of Electronic and Computer Engineering, Peking University, Shenzhen, China

Yuchao Yang

Center for Brain Inspired Intelligence, Chinese Institute for Brain Research (CIBR), Beijing, China

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Contributions

B.Y. and Yuchao Yang directed the research. W.Y. and B.Y. conceived the idea and planned the study. W.Y., Z.J. and B.Y. designed the RRAM chip. W.Y. and B.Y. designed the UIM architecture. W.Y. implemented the closed-loop CIM-based UIM system. Y.L. extended the UIM to the EDA application. T.Z., Yuxiang Yang, Z.Y., Y.W., W.B., K.Z., J.K., R.H. and Yuchao Yang developed the RRAM integration processes. T.Z., W.Y., Yuxiang Yang, Z.Y., Y.W. and Y.T. optimized the RRAM device. W.Y. and K.W. developed the video demonstration. K.W. optimized the testing platform. All authors reviewed and edited the paper.

Corresponding authors

Correspondence to Bonan Yan or Yuchao Yang .

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Nature Electronics thanks Bin Gao, Luke Theogarajan and Masanao Yamaoka for their contribution to the peer review of this work.

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Supplementary information

Supplementary information.

Supplementary Figs. 1–22, Notes 1–25 and Tables 1–12.

Supplementary Video 1

Computing process in the closed-loop system of the UIMe.

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Yue, W., Zhang, T., Jing, Z. et al. A scalable universal Ising machine based on interaction-centric storage and compute-in-memory. Nat Electron (2024). https://doi.org/10.1038/s41928-024-01228-7

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How to cite ChatGPT

Use discount code STYLEBLOG15 for 15% off APA Style print products with free shipping in the United States.

We, the APA Style team, are not robots. We can all pass a CAPTCHA test , and we know our roles in a Turing test . And, like so many nonrobot human beings this year, we’ve spent a fair amount of time reading, learning, and thinking about issues related to large language models, artificial intelligence (AI), AI-generated text, and specifically ChatGPT . We’ve also been gathering opinions and feedback about the use and citation of ChatGPT. Thank you to everyone who has contributed and shared ideas, opinions, research, and feedback.

In this post, I discuss situations where students and researchers use ChatGPT to create text and to facilitate their research, not to write the full text of their paper or manuscript. We know instructors have differing opinions about how or even whether students should use ChatGPT, and we’ll be continuing to collect feedback about instructor and student questions. As always, defer to instructor guidelines when writing student papers. For more about guidelines and policies about student and author use of ChatGPT, see the last section of this post.

Quoting or reproducing the text created by ChatGPT in your paper

If you’ve used ChatGPT or other AI tools in your research, describe how you used the tool in your Method section or in a comparable section of your paper. For literature reviews or other types of essays or response or reaction papers, you might describe how you used the tool in your introduction. In your text, provide the prompt you used and then any portion of the relevant text that was generated in response.

Unfortunately, the results of a ChatGPT “chat” are not retrievable by other readers, and although nonretrievable data or quotations in APA Style papers are usually cited as personal communications , with ChatGPT-generated text there is no person communicating. Quoting ChatGPT’s text from a chat session is therefore more like sharing an algorithm’s output; thus, credit the author of the algorithm with a reference list entry and the corresponding in-text citation.

When prompted with “Is the left brain right brain divide real or a metaphor?” the ChatGPT-generated text indicated that although the two brain hemispheres are somewhat specialized, “the notation that people can be characterized as ‘left-brained’ or ‘right-brained’ is considered to be an oversimplification and a popular myth” (OpenAI, 2023).

OpenAI. (2023). ChatGPT (Mar 14 version) [Large language model]. https://chat.openai.com/chat

You may also put the full text of long responses from ChatGPT in an appendix of your paper or in online supplemental materials, so readers have access to the exact text that was generated. It is particularly important to document the exact text created because ChatGPT will generate a unique response in each chat session, even if given the same prompt. If you create appendices or supplemental materials, remember that each should be called out at least once in the body of your APA Style paper.

When given a follow-up prompt of “What is a more accurate representation?” the ChatGPT-generated text indicated that “different brain regions work together to support various cognitive processes” and “the functional specialization of different regions can change in response to experience and environmental factors” (OpenAI, 2023; see Appendix A for the full transcript).

Creating a reference to ChatGPT or other AI models and software

The in-text citations and references above are adapted from the reference template for software in Section 10.10 of the Publication Manual (American Psychological Association, 2020, Chapter 10). Although here we focus on ChatGPT, because these guidelines are based on the software template, they can be adapted to note the use of other large language models (e.g., Bard), algorithms, and similar software.

The reference and in-text citations for ChatGPT are formatted as follows:

• Parenthetical citation: (OpenAI, 2023)
• Narrative citation: OpenAI (2023)

Let’s break that reference down and look at the four elements (author, date, title, and source):

Author: The author of the model is OpenAI.

Date: The date is the year of the version you used. Following the template in Section 10.10, you need to include only the year, not the exact date. The version number provides the specific date information a reader might need.

Title: The name of the model is “ChatGPT,” so that serves as the title and is italicized in your reference, as shown in the template. Although OpenAI labels unique iterations (i.e., ChatGPT-3, ChatGPT-4), they are using “ChatGPT” as the general name of the model, with updates identified with version numbers.

The version number is included after the title in parentheses. The format for the version number in ChatGPT references includes the date because that is how OpenAI is labeling the versions. Different large language models or software might use different version numbering; use the version number in the format the author or publisher provides, which may be a numbering system (e.g., Version 2.0) or other methods.

Bracketed text is used in references for additional descriptions when they are needed to help a reader understand what’s being cited. References for a number of common sources, such as journal articles and books, do not include bracketed descriptions, but things outside of the typical peer-reviewed system often do. In the case of a reference for ChatGPT, provide the descriptor “Large language model” in square brackets. OpenAI describes ChatGPT-4 as a “large multimodal model,” so that description may be provided instead if you are using ChatGPT-4. Later versions and software or models from other companies may need different descriptions, based on how the publishers describe the model. The goal of the bracketed text is to briefly describe the kind of model to your reader.

Source: When the publisher name and the author name are the same, do not repeat the publisher name in the source element of the reference, and move directly to the URL. This is the case for ChatGPT. The URL for ChatGPT is https://chat.openai.com/chat . For other models or products for which you may create a reference, use the URL that links as directly as possible to the source (i.e., the page where you can access the model, not the publisher’s homepage).

Other questions about citing ChatGPT

You may have noticed the confidence with which ChatGPT described the ideas of brain lateralization and how the brain operates, without citing any sources. I asked for a list of sources to support those claims and ChatGPT provided five references—four of which I was able to find online. The fifth does not seem to be a real article; the digital object identifier given for that reference belongs to a different article, and I was not able to find any article with the authors, date, title, and source details that ChatGPT provided. Authors using ChatGPT or similar AI tools for research should consider making this scrutiny of the primary sources a standard process. If the sources are real, accurate, and relevant, it may be better to read those original sources to learn from that research and paraphrase or quote from those articles, as applicable, than to use the model’s interpretation of them.

We’ve also received a number of other questions about ChatGPT. Should students be allowed to use it? What guidelines should instructors create for students using AI? Does using AI-generated text constitute plagiarism? Should authors who use ChatGPT credit ChatGPT or OpenAI in their byline? What are the copyright implications ?

On these questions, researchers, editors, instructors, and others are actively debating and creating parameters and guidelines. Many of you have sent us feedback, and we encourage you to continue to do so in the comments below. We will also study the policies and procedures being established by instructors, publishers, and academic institutions, with a goal of creating guidelines that reflect the many real-world applications of AI-generated text.

For questions about manuscript byline credit, plagiarism, and related ChatGPT and AI topics, the APA Style team is seeking the recommendations of APA Journals editors. APA Style guidelines based on those recommendations will be posted on this blog and on the APA Style site later this year.

Update: APA Journals has published policies on the use of generative AI in scholarly materials .

We, the APA Style team humans, appreciate your patience as we navigate these unique challenges and new ways of thinking about how authors, researchers, and students learn, write, and work with new technologies.

American Psychological Association. (2020). Publication manual of the American Psychological Association (7th ed.). https://doi.org/10.1037/0000165-000

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