Types of Matter in Lattice Field Theories  

Scalar Fields > s.a. klein-gordon fields; wave equations [discrete].
* λφ^4 theory: It becomes the free scalar field theory when the spacing goes to zero; Possible reactions are (1) The theory cannot be considered seriously; (2) The lattice approximation is ok but there just isn't any λφ4 continuum theory (no self-interacting field, except maybe coupled to other fields); (3) Fix the problem [Klauder].
@ λφ^4 theory: Caiani et al JPA(98) [2D, Hamiltonian]; Klauder LMP(03)ht/02 [D > 3, non-trivial limit]; Albeverio et al JMP(04) [regularization and continuum limit]; Butera & Comi PRB(05)hl [high-T expansion]; Wolff PRD(09)-a0902 [Ising limit, non-triviality].
@ Related topics: Felder & Tkachev CPC(08)hp/00 [C++ program for expanding universe]; Adib & Almeida PRE(01)ht [kink dynamics]; Aoki & Kusnezov AP(02) [non-equilibrium statistical mechanics]; Borasoy & Krebs NPB(06) [2-loop renormalization]; Wetterich NPB(13)-a1212 [scalar lattice gauge theories].

Fermions, Spinor Fields > s.a. dirac fields; ising model; spinors and 2-spinors; spin models; {random lattices below}.
* In lattice gauge theory: There is a problem, the absence of a chiral anomaly (no-go Nielsen-Ninomiya theorem); > s.a. spinors.
* Fermion doubling: The appearance of spurious fermion states in lattice theories; > s.a. Wikipedia page.
* Supersymmetric models: The violation of Leibniz rule on a lattice is an obstruction for constructing a lattice supersymmetric model.
@ General references: Anthony et al PLB(82) [Monte Carlo, proposal]; Bullinaria AP(85); Friedberg et al JMP(94); Foster & Jacobson ht/03 [massless, path integral on tetrahedral lattice]; Del Debbio et al JHEP(08) [higher representations]; Herbut et al PRB(09)-a0811 [honeycomb lattice]; Huijse et al NJP(12)-a1112 [triangular lattice].
@ Chiral fermions: Quinn & Weinstein PRD(86); Bornyakov PTP(98)hl-proc, NPPS(99)hl/98; Zenkin PRD(98)hl [no-go]; Chiu ChJP(00)hl/99-conf; Jahn & Pawlowski NPB(02)hl; Kerler IJMPA(03) [rev]; Poppitz & Shang JHEP(07)-a0706 [decoupling of mirror fermions].
@ Fermion doubling: Barnett & Smolin a1507 [in lqg]; > s.a. lattice theories [non-commutative lattices].
@ Supersymmetric models: Fendley et al JPA(03) [N = 2 supersymmetry]; Giedt & Poppitz JHEP(04) [superfields and renormalization]; Santachiara & Schoutens JPA(05); Giedt IJMPA(06), IJMPA(09) [rev and deconstruction approach]; Takimi JHEP(07)-a0705 [relationships]; D'Adda et al PRD(10)-a0907 [and representation of deformed superalgebra]; Bergner JHEP(10)-a0909; Kato et al JHEP(13)-a1303 [cyclic Leibniz rule]; > s.a. lattice gauge theory.
@ Related topics: Jourjine PRD(85) [Dirac-Kähler fermions]; Bietenholz et al NPB(97) [staggered, perfect lattice actions]; Polley qp/01; Master et al PRA(03)qp/02 [energy spectrum quantum algorithm]; Araki & Moriya RVMP(03)mp/02 [statistical mechanics]; Inagaki & Suzuki JHEP(04)hl [Majorana, Majorana-Weyl, various dimensions]; Dias Barreto & Fidaleo JSP(11) [disordered]; > s.a. correlations; yang-mills fields [from fermions].

Other Theories
> Particle theories: see composite and constrained quantum systems; crystals; integrable systems; Lattice Gas; Many-Body Systems; modified quantum mechanics [supersymmetric]; Percolation; quantum oscillators; statistical mechanics.
> Field theories: see lattice gauge theory; lattice gravity; topological field theories; types of field theories.

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