Lattice Field Theories  

In General > s.a. cellular automaton; lattice [math notion]; number theory [geometric] / types of matter in lattice theories.
* Motivation: They allow some approximate calculations to be done, and, in the case of lattice spacetimes, they are a trick to introduce a cutoff for non-perturbative regularization; But one can consider the approach as more fundamental than the continuum one, and take more seriously its results, hoping to get them in some approximation from a full quantum gravity theory (> see discrete spacetime); It is natural in the context of (Euclidean) path-integral quantization.
* Remark: It emphasizes the close connection with statistical mechanics; The strong coupling limit is equivalent to a high-T expansion.
@ General references: Kadanoff RMP(77); Drouffe & Itzykson PRP(78); Dashen & Gross PRD(81); Rebbi ed-83; Rebbi SA(83)feb; 't Hooft et al ed-84; Friedberg et al JMP(94); Boozer AJP(10)dec [periodic lattices in 2D Minkowski space]; Preskill a1811-conf [quantum field theory on a quantum computer].
@ Texts, reviews: Kogut RMP(79); Creutz 83; Creutz ed-92; Montvay & Münster 94; DeGrand ht/96-ln [non-perturbative quantum field theory], comment Neuberger hp/04; Smit 02; Di Pierro IJMPA(06)hl/05 [quantum field theory, especially gauge theory and QCD]; DeGrand a1907-ln [physical motivation].
@ General references: Ueltschi in(02)mp/01 [bosonic particles]; Kuzemsky RNC(02) [many-particle].
@ State entanglement: Narnhofer PRA(05)qp/04 [high-T]; Cramer et al PRA(06) [area scaling law]; Unanyan et al PRA(10)-a0910 [dynamics].

Random Lattices > s.a. chaotic systems; dirac fields; ising model; spin models; lattice gauge theory; regge calculus.
* Idea: Build a lattice using the Voronoi construction on a uniformly random set of points; Can be done in Riemannian geometry, but not in Lorentzian geometry.
* And localization: Simulation of diffusion processes shows that geometrical defects, sites with abnormally low or large connectivities, produce localization of eigenmodes.
* And phase transitions: Some phase transitions observed in simulations with regular lattices are softened in random lattices (e.g., the 10-state Potts model on 2D lattices), while in other systems the critical behavior persists.
@ References: Ren NPB(88) [massless fermions]; Eynard & Kristjansen NPB(98) [three-color problem]; Biroli & Monasson JPA(99) [localized state description]; Zinn-Justin EPL(00)cm/99, Kostov NPB(00)ht/99 [6-vertex model]; Orlandini et al JPA(02) [thermodynamic self-averaging]; Sanpera et al PRL(04) [atomic Fermi-Bose mixtures]; Feng & Siegel PRD(06)ht [superstrings]; Borot & Eynard JSM(11)-a0911 [O(n) loop gas model]; Teixeira a1304 [and exterior calculus]; Barghathi & Vojta PRL(14) [phase transitions].
@ Ising model: Pérez Vicente & Coolen JPA(08); Dommers et al JSP(10); Lage-Castellanos et al JPA(13) [replica-symmetric solution]; Giardinà et al a1509 [annealed central limit theorems].

Other Types of Lattices > s.a. cell complex; graph theory; lattice gauge theory [simplicial lattice]; network; regge calculus; Tangle; tiling.
* Non-Abelian lattices: Ones with a non-Abelian symmetry group; For example, C60.
@ Asymmetric lattices: Csikor & Fodor PLB(96) [SU(2)-Higgs].
@ Fractal lattices: Windus & Jensen PhyA(09) [and order of phase transition]; > s.a. fractals.
@ Cell complexes: Jourjine PRD(85) [dimensional phase transitions], PRD(86) [spinors and gauge fields]; Vanderseypen pr(93) [Langevin equation].
@ Supersymmetric lattice: Grosse CMP(97) [field theory]; Catterall a1005-proc.
@ Non-commutative: Bimonte et al JGP(96)ht/95 [continuum limit]; Ercolessi et al RVMP(98)qa/96 [posets and representations of non-commutative algebras], qa/96 [K-theory]; Balachandran et al MPLA(00)ht/99 [fermion doubling]; Häußling AP(02)ht/01; Besnard a1903 [fermion doubling].
@ Related topics: Chandrasekharan & Wiese NPB(97) [quantum links].

Related Concepts and Techniques > s.a. algebraic quantum field theory; crystals [dynamics of lattices]; field theory; wave-function collapse.
* Techniques: One uses methods of statistical and many-body physics, and discrete approximations to differential equations.
* Coordination sequence: The sequence {S(n)} gives the number of nodes n links away from a given node.
@ Renormalization: Balaban CMP(87); Yamamoto LMP(01); Bertini et al CMP(05) [cluster expansion]; Gu & Wen PRB(09) + Sachdev Phy(09) [tensor networks]; Brydges & Slade JSP(15)-a1403, JSP(15)-a1403, ..., JSP(15)-a1403 [lattice field theories involving boson and/or fermion fields, rigorous]; > s.a. renormalization group.
@ Small vs large lattice: Patrascioiu & Seiler PLB(96); Lin et al PRL(99) [1D Fermi-Pasta-Ulam chain].
@ On curved manifolds: Brower et al a1601-conf [quantum finite elements, simplicial lattice]; Arrighi et al a1812 [from discrete-time quantum walks].
@ Continuum limits: de Lyra et al PRD(91) [differentiability and continuity]; Bimonte et al JGP(96)ht/95; Nielsen & Rugh2 cd/96, ht/96-conf [gauge theory]; Delphenich AdP(10)-a1010 [obstructions from defects, and cohomology classes]; Davoudi & Savage PRD(12)-a1204 [recovery of rotational invariance]; Osborne a1901 [general procedure, Hamiltonian]; Radičević a2105, a2105, a2105 [systematic construction].
@ Coordination sequences: Conway & Sloane PRS(97).
@ Monte Carlo method: Creutz et al PRP(83); NS(91)jan5, 41-44; Easther et al hl/03-conf [modified schemes]; Langfeld & Lucini a1606-conf [density-of-states method]; Lawrence a2006-PhD [approaches to the sign problem].
@ Other techniques: García et al PLB(94) [non-probabilistic]; Markopoulou ht/00 [coarse-graining]; Capitani PRP(03) [perturbation theory]; Succi JPA(07) [Boltzmann discretization, 1+1 Klein-Gordon and Dirac fields]; Amador et al a1610 [mean spherical approximation].
@ Related topics: Ben-Av & Solomon MPLA(89) = IJMPA(90); Schmidt & Stamatescu MPLA(03) [matter determinants]; Flach & Gorbach PRP(08) [discrete breathers]; Campos & Tututi PLA(08) [finite-size effects]; Shaposhnikov & Tkachev PLB(09)-a0811 [quantum scale invariance]; Kevrekidis JO(13)-a1302 [beyond nearest-neighbor interactions]; Dittrich & Kamiński a1311 [dynamics of intertwiners]; Arjang & Zapata CQG(14)-a1312 [coarse-graining and continuum limit]; Zapata a1602 [observable currents]; Imaki & Yamamoto PRD(19)-a1906 [with torsion].
> Other topics: see green functions; holography; phase transitions; types of manifolds [discretized].

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at – modified 28 may 2021