{"history":[],"article":{"id":"7fb86f79-8c38-4473-b035-652cca1af326","title":"Lattice model (physics)","slug":"lattice-model-physics","content":":::summary\n:::\n\n# \nIn [mathematical physics](https://bliptext.com/articles/mathematical-physics), a **[lattice](https://bliptext.com/articles/lattice-group) model** is a [mathematical model](https://bliptext.com/articles/mathematical-model) of a physical system that is defined on a [lattice](https://bliptext.com/articles/lattice-group), as opposed to a [continuum](https://bliptext.com/articles/continuum-theory), such as the [continuum](https://bliptext.com/articles/continuum-theory) of [space](https://bliptext.com/articles/space) or [space](https://bliptext.com/articles/space)time.\nLattice models originally occurred in the context of [condensed matter physics](https://bliptext.com/articles/condensed-matter-physics), where the [atoms](https://bliptext.com/articles/atom) of a [crystal](https://bliptext.com/articles/crystal) automatically form a lattice.\nCurrently, lattice models are quite popular in [theoretical physics](https://bliptext.com/articles/theoretical-physics), for many reasons.\nSome models are [exactly solvable](https://bliptext.com/articles/exactly-solvable), and thus offer insight into physics beyond what can be learned from [perturbation theory](https://bliptext.com/articles/perturbation-theory).\nLattice models are also ideal for study by the methods of [computational physics](https://bliptext.com/articles/computational-physics), as the discretization of any continuum model automatically turns it into a lattice model.\nThe exact solution to many of these models (when they are solvable) includes the presence of [solitons](https://bliptext.com/articles/soliton).\nTechniques for solving these include the [inverse scattering transform](https://bliptext.com/articles/inverse-scattering-transform) and the method of [Lax pairs](https://bliptext.com/articles/lax-pair), the [Yang–Baxter equation](https://bliptext.com/articles/yang-baxter-equation) and [quantum groups](https://bliptext.com/articles/quantum-group).\nThe solution of these models has given insights into the nature of [phase transitions](https://bliptext.com/articles/phase-transitions), [magnetization](https://bliptext.com/articles/magnetization) and [scaling behaviour](https://bliptext.com/articles/scaling-behaviour), as well as insights into the nature of [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\nPhysical lattice models frequently occur as an approximation to a continuum theory, either to give an [ultraviolet cutoff](https://bliptext.com/articles/ultraviolet-cutoff) to the theory to prevent divergences or to perform [numerical computations](https://bliptext.com/articles/numerical-analysis).\nAn example of a continuum theory that is widely studied by lattice models is the [QCD lattice model](https://bliptext.com/articles/qcd-lattice-model), a discretization of [quantum chromodynamics](https://bliptext.com/articles/quantum-chromodynamics).\nHowever, [digital physics](https://bliptext.com/articles/digital-physics) considers nature fundamentally discrete at the Planck scale, which imposes [upper limit to the density of information](https://bliptext.com/articles/talk-bekenstein-bound), aka [Holographic principle](https://bliptext.com/articles/holographic-principle).\nMore generally, [lattice gauge theory](https://bliptext.com/articles/lattice-gauge-theory) and [lattice field theory](https://bliptext.com/articles/lattice-field-theory) are areas of study.\nLattice models are also used to simulate the structure and dynamics of polymers.\n# [A three-dimensional lattice filled with two molecules A and B, here shown as black and white spheres. Lattices such as this are used - for example - in the Flory–Huggins solution theory | upload.wikimedia.org/wikipedia/commons/8/86/LatticemodelAB.png]\n\n# Mathematical description\nA number of lattice models can be described by the following data:\n

A [lattice](https://bliptext.com/articles/lattice-group) \\Lambda, often taken to be a [lattice](https://bliptext.com/articles/lattice-group) in d-dimensional Euclidean space or the d-dimensional torus if the [lattice](https://bliptext.com/articles/lattice-group) is periodic.\nConcretely, \\Lambda is often the [cubic lattice](https://bliptext.com/articles/integer-lattice).\nIf two points on the lattice are considered 'nearest neighbours', then they can be connected by an edge, turning the lattice into a [lattice graph](https://bliptext.com/articles/lattice-graph).\nThe vertices of \\Lambda are sometimes referred to as sites.\n
A spin-variable space S. The configuration space \\mathcal{C} of possible system states is then the space of functions.\nFor some models, we might instead consider instead the space of functions where E is the edge set of the graph defined above.\n
An energy functional, which might depend on a set of additional parameters or 'coupling constants' \\{g_i\\}.\n

\n\n## Examples\nThe [Ising model](https://bliptext.com/articles/ising-model) is given by the usual cubic lattice graph where \\Lambda is an infinite cubic lattice in or a period n cubic lattice in T^d, and E is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context).\nThe spin-variable space is.\nThe energy functional is\nThe spin-variable space can often be described as a [coset](https://bliptext.com/articles/coset).\nFor example, for the Potts model we have.\nIn the limit, we obtain the XY model which has S = SO(2).\nGeneralising the XY model to higher dimensions gives the n-vector model which has.\n\n## Solvable models\nWe specialise to a lattice with a finite number of points, and a finite spin-variable space.\nThis can be achieved by making the lattice periodic, with period n in d dimensions.\nThen the configuration space \\mathcal{C} is also finite.\nWe can define the [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics)\nand there are no issues of convergence (like those which emerge in field theory) since the sum is finite.\nIn theory, this sum can be computed to obtain an expression which is dependent only on the parameters \\{g_i\\} and \\beta.\nIn practice, this is often difficult due to non-linear interactions between sites.\nModels with a closed-form expression for the partition function are known as [exactly solvable](https://bliptext.com/articles/exactly-solvable).\nExamples of exactly solvable models are the periodic 1D Ising model, and the periodic 2D Ising model with vanishing external magnetic field, H = 0, but for dimension d>2, the Ising model remains unsolved.\n\n## Mean field theory\nDue to the difficulty of deriving exact solutions, in order to obtain analytic results we often must resort to [mean field theory](https://bliptext.com/articles/mean-field-theory).\nThis mean field may be spatially varying, or global.\n\n### Global mean field\nThe configuration space \\mathcal{C} of functions \\sigma is replaced by the [convex hull](https://bliptext.com/articles/convex-hull) of the spin space S, when S has a realisation in terms of a subset of.\nWe'll denote this by.\nThis arises as in going to the mean value of the field, we have.\nAs the number of lattice sites, the possible values of fill out the convex hull of S. By making a suitable approximation, the energy functional becomes a function of the mean field, that is, The partition function then becomes\nAs, that is, in the [thermodynamic limit](https://bliptext.com/articles/thermodynamic-limit), the [saddle point approximation](https://bliptext.com/articles/saddle-point-approximation) tells us the integral is asymptotically dominated by the value at which is minimised:\nwhere is the argument minimising f.\nA simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field.\nWriting configurations as, truncating terms of then summing over configurations allows computation of the partition function.\nSuch an approach to the periodic Ising model in d dimensions provides insight into [phase transitions](https://bliptext.com/articles/phase-transitions).\n\n### Spatially varying mean field\nSuppose the [continuum limit](https://bliptext.com/articles/continuum-limit) of the lattice \\Lambda is.\nInstead of averaging over all of \\Lambda, we average over neighbourhoods of.\nThis gives a spatially varying mean field.\nWe relabel with \\phi to bring the notation closer to field theory.\nThis allows the partition function to be written as a [path integral](https://bliptext.com/articles/path-integral-formulation)\nwhere the free energy F[\\phi] is a [Wick rotated](https://bliptext.com/articles/wick-rotated) version of the action in [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\n\n# Examples\n\n## Condensed matter physics\n\n## Polymer physics\n

[Bond fluctuation model](https://bliptext.com/articles/bond-fluctuation-model)
[2nd model](https://bliptext.com/articles/2nd-model)

\n\n## High energy physics\n

[QCD lattice model](https://bliptext.com/articles/qcd-lattice-model)

","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:03:31.807Z","updated_at":"2025-01-25T19:03:31.807Z","search_vector":"'lattic':1 'model':2 'physic':3","search_content":" \n \n\n# \nIn [mathematical physics](https://bliptext.com/articles/mathematical-physics), a **[lattice](https://bliptext.com/articles/lattice-group) model** is a [mathematical model](https://bliptext.com/articles/mathematical-model) of a physical system that is defined on a [lattice](https://bliptext.com/articles/lattice-group), as opposed to a [continuum](https://bliptext.com/articles/continuum-theory), such as the [continuum](https://bliptext.com/articles/continuum-theory) of [space](https://bliptext.com/articles/space) or [space](https://bliptext.com/articles/space)time.\nLattice models originally occurred in the context of [condensed matter physics](https://bliptext.com/articles/condensed-matter-physics), where the [atoms](https://bliptext.com/articles/atom) of a [crystal](https://bliptext.com/articles/crystal) automatically form a lattice.\nCurrently, lattice models are quite popular in [theoretical physics](https://bliptext.com/articles/theoretical-physics), for many reasons.\nSome models are [exactly solvable](https://bliptext.com/articles/exactly-solvable), and thus offer insight into physics beyond what can be learned from [perturbation theory](https://bliptext.com/articles/perturbation-theory).\nLattice models are also ideal for study by the methods of [computational physics](https://bliptext.com/articles/computational-physics), as the discretization of any continuum model automatically turns it into a lattice model.\nThe exact solution to many of these models (when they are solvable) includes the presence of [solitons](https://bliptext.com/articles/soliton).\nTechniques for solving these include the [inverse scattering transform](https://bliptext.com/articles/inverse-scattering-transform) and the method of [Lax pairs](https://bliptext.com/articles/lax-pair), the [Yang–Baxter equation](https://bliptext.com/articles/yang-baxter-equation) and [quantum groups](https://bliptext.com/articles/quantum-group).\nThe solution of these models has given insights into the nature of [phase transitions](https://bliptext.com/articles/phase-transitions), [magnetization](https://bliptext.com/articles/magnetization) and [scaling behaviour](https://bliptext.com/articles/scaling-behaviour), as well as insights into the nature of [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\nPhysical lattice models frequently occur as an approximation to a continuum theory, either to give an [ultraviolet cutoff](https://bliptext.com/articles/ultraviolet-cutoff) to the theory to prevent divergences or to perform [numerical computations](https://bliptext.com/articles/numerical-analysis).\nAn example of a continuum theory that is widely studied by lattice models is the [QCD lattice model](https://bliptext.com/articles/qcd-lattice-model), a discretization of [quantum chromodynamics](https://bliptext.com/articles/quantum-chromodynamics).\nHowever, [digital physics](https://bliptext.com/articles/digital-physics) considers nature fundamentally discrete at the Planck scale, which imposes [upper limit to the density of information](https://bliptext.com/articles/talk-bekenstein-bound), aka [Holographic principle](https://bliptext.com/articles/holographic-principle).\nMore generally, [lattice gauge theory](https://bliptext.com/articles/lattice-gauge-theory) and [lattice field theory](https://bliptext.com/articles/lattice-field-theory) are areas of study.\nLattice models are also used to simulate the structure and dynamics of polymers.\n# [A three-dimensional lattice filled with two molecules A and B, here shown as black and white spheres. Lattices such as this are used - for example - in the Flory–Huggins solution theory | upload.wikimedia.org/wikipedia/commons/8/86/LatticemodelAB.png]\n\n# Mathematical description\nA number of lattice models can be described by the following data:\n \n A [lattice](https://bliptext.com/articles/lattice-group) \\Lambda, often taken to be a [lattice](https://bliptext.com/articles/lattice-group) in d-dimensional Euclidean space or the d-dimensional torus if the [lattice](https://bliptext.com/articles/lattice-group) is periodic.\nConcretely, \\Lambda is often the [cubic lattice](https://bliptext.com/articles/integer-lattice).\nIf two points on the lattice are considered 'nearest neighbours', then they can be connected by an edge, turning the lattice into a [lattice graph](https://bliptext.com/articles/lattice-graph).\nThe vertices of \\Lambda are sometimes referred to as sites.\n A spin-variable space S. The configuration space \\mathcal{C} of possible system states is then the space of functions.\nFor some models, we might instead consider instead the space of functions where E is the edge set of the graph defined above.\n An energy functional, which might depend on a set of additional parameters or 'coupling constants' \\{g_i\\}.\n \n\n## Examples\nThe [Ising model](https://bliptext.com/articles/ising-model) is given by the usual cubic lattice graph where \\Lambda is an infinite cubic lattice in or a period n cubic lattice in T^d, and E is the edge set of nearest neighbours (the same letter is used for the energy functional but the different usages are distinguishable based on context).\nThe spin-variable space is.\nThe energy functional is\nThe spin-variable space can often be described as a [coset](https://bliptext.com/articles/coset).\nFor example, for the Potts model we have.\nIn the limit, we obtain the XY model which has S = SO(2).\nGeneralising the XY model to higher dimensions gives the n-vector model which has.\n\n## Solvable models\nWe specialise to a lattice with a finite number of points, and a finite spin-variable space.\nThis can be achieved by making the lattice periodic, with period n in d dimensions.\nThen the configuration space \\mathcal{C} is also finite.\nWe can define the [partition function](https://bliptext.com/articles/partition-function-statistical-mechanics)\nand there are no issues of convergence (like those which emerge in field theory) since the sum is finite.\nIn theory, this sum can be computed to obtain an expression which is dependent only on the parameters \\{g_i\\} and \\beta.\nIn practice, this is often difficult due to non-linear interactions between sites.\nModels with a closed-form expression for the partition function are known as [exactly solvable](https://bliptext.com/articles/exactly-solvable).\nExamples of exactly solvable models are the periodic 1D Ising model, and the periodic 2D Ising model with vanishing external magnetic field, H = 0, but for dimension d>2, the Ising model remains unsolved.\n\n## Mean field theory\nDue to the difficulty of deriving exact solutions, in order to obtain analytic results we often must resort to [mean field theory](https://bliptext.com/articles/mean-field-theory).\nThis mean field may be spatially varying, or global.\n\n### Global mean field\nThe configuration space \\mathcal{C} of functions \\sigma is replaced by the [convex hull](https://bliptext.com/articles/convex-hull) of the spin space S, when S has a realisation in terms of a subset of.\nWe'll denote this by.\nThis arises as in going to the mean value of the field, we have.\nAs the number of lattice sites, the possible values of fill out the convex hull of S. By making a suitable approximation, the energy functional becomes a function of the mean field, that is, The partition function then becomes\nAs, that is, in the [thermodynamic limit](https://bliptext.com/articles/thermodynamic-limit), the [saddle point approximation](https://bliptext.com/articles/saddle-point-approximation) tells us the integral is asymptotically dominated by the value at which is minimised:\nwhere is the argument minimising f.\nA simpler, but less mathematically rigorous approach which nevertheless sometimes gives correct results comes from linearising the theory about the mean field.\nWriting configurations as, truncating terms of then summing over configurations allows computation of the partition function.\nSuch an approach to the periodic Ising model in d dimensions provides insight into [phase transitions](https://bliptext.com/articles/phase-transitions).\n\n### Spatially varying mean field\nSuppose the [continuum limit](https://bliptext.com/articles/continuum-limit) of the lattice \\Lambda is.\nInstead of averaging over all of \\Lambda, we average over neighbourhoods of.\nThis gives a spatially varying mean field.\nWe relabel with \\phi to bring the notation closer to field theory.\nThis allows the partition function to be written as a [path integral](https://bliptext.com/articles/path-integral-formulation)\nwhere the free energy F[\\phi] is a [Wick rotated](https://bliptext.com/articles/wick-rotated) version of the action in [quantum field theory](https://bliptext.com/articles/quantum-field-theory).\n\n# Examples\n\n## Condensed matter physics\n\n## Polymer physics\n \n [Bond fluctuation model](https://bliptext.com/articles/bond-fluctuation-model) \n [2nd model](https://bliptext.com/articles/2nd-model) \n \n\n## High energy physics\n \n [QCD lattice model](https://bliptext.com/articles/qcd-lattice-model) \n ","revisionCount":0},"hasMore":false,"nextCursor":null}

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