Lattice Field Theories| 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].
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