Generalization to d-dimensions of a fermionic path integral for exact enumeration of polygons on hypercubic lattices
The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectu...
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| Abstract | The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable—or, to be more precise, that there are no differentiably finite (
D
-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on
d
-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in
d
dimensions, the Grassmann action contains terms of degree
2
(
d
-
1
)
, so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. |
|---|---|
| AbstractList | The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable—or, to be more precise, that there are no differentiably finite (
D
-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on
d
-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in
d
dimensions, the Grassmann action contains terms of degree
2
(
d
-
1
)
, so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable-or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree 2 ( d - 1 ) , so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible.The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable-or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree 2 ( d - 1 ) , so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable—or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree 2(d-1), so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable-or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree , so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. Abstract The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable—or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree $$2(d-1)$$ 2 ( d - 1 ) , so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. The generating function for polygons on the square lattice has been known for many decades and is closely related to the path integral formulation of a free fermion model. On the cubic and hypercubic lattices the generating function is still unknown and the problem remains open. It has been conjectured that the three-dimensional (3D) and higher dimensional problems are not solvable—or, to be more precise, that there are no differentiably finite (D-finite) solutions. In this context, very recently a Berezin integral of an exponentiated Grassmann action was found for the polygon generating function on the cubic lattice, making explicit the connection between 3D polygons and a model of interacting fermions. Here we address the problem of how to generalize the 3D result to higher dimensions. We derive a Grassmann representation in terms of a Berezin integral for the generating function of polygons on d-dimensional hypercubic lattices. On the one hand, this new result admittedly brings us no closer to the problem of finding an explicit analytic expression for the desired generating function for polygons. On the other hand, however, the significant advance reported here precisely quantifies the remarkable mathematical difficulty of finding the explicit generating function. Indeed, the non-quadratic functional form of the Grassmann action that we derive here provides a very clear picture of the formidable mathematical obstruction that would need to be overcome. Specifically, in d dimensions, the Grassmann action contains terms of degree $$2(d-1)$$ 2(d-1), so the model describes interacting rather than free fermions. It is an open problem whether or not these models of interacting fermions can in principle be free fermionized through some still undiscovered algebraic method, but it is widely believed that this goal is mathematically impossible. |
| ArticleNumber | 22375 |
| Author | Bezerra, C. G. Ostilli, M. Viswanathan, G. M. Rocha, G. W. C. |
| Author_xml | – sequence: 1 givenname: M. surname: Ostilli fullname: Ostilli, M. email: massimo.ostilli@ufba.br organization: Institute of Physics, Federal University of Bahia – sequence: 2 givenname: G. W. C. surname: Rocha fullname: Rocha, G. W. C. email: gabrielwendell@fisica.ufrn.br organization: Physics Department, Federal University of Rio Grande do Norte – sequence: 3 givenname: C. G. surname: Bezerra fullname: Bezerra, C. G. email: cbezerra@fisica.ufrn.br organization: Physics Department, Federal University of Rio Grande do Norte – sequence: 4 givenname: G. M. surname: Viswanathan fullname: Viswanathan, G. M. email: gandhi@fisica.ufrn.br organization: Physics Department, Federal University of Rio Grande do Norte, National Institute of Science and Technology of Complex Systems, Federal University of Rio Grande do Norte |
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Phys. A20163128n2916450442016IJMPA..3145044G1:CAS:528:DC%2BC28Xhs1ylur%2FO10.1142/S0217751X16450445 PlechkoVNSimple solution of two-dimensional Ising model on a torus in terms of Grassmann integralsTheor. Math. Phys.19856474881510310.1007/BF01017042 SchultzTDMattisDCLiebEHTwo-dimensional Ising model as a soluble problem of many fermionsRev. Mod. Phys.1964368561964RvMP...36..856S18027410.1103/RevModPhys.36.856 MoessnerRSondhiSLResonating valence bond phase in the triangular lattice quantum dimer modelPhys. Rev. Lett.20018618812001PhRvL..86.1881M1:CAS:528:DC%2BD3MXhsVWlur0%3D10.1103/PhysRevLett.86.188111290272 PolletLKiselevMNProkof’evNVSvistunovBVGrassmannization of classical modelsNew J. Phys.2016181130252016NJPh...18k3025P10.1088/1367-2630/18/11/113025 PolyakovAMString representations and hidden symmetries for gauge fieldsPhys. Lett. B1979822471979PhLB...82..247P10.1016/0370-2693(79)90747-0 ShankarRExact critical-behavior of a random-bond two dimensional Ising-modelPhys. Rev. 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Surv.1969243125200510.1070/RM1969v024n03ABEH001346 Balasubramanian, S., Galitski, V. & Vishwanath, A. Classical vertex model dualities in a family of two-dimensional frustrated quantum antiferromagnets, Phys. Rev. B106 (2022). ArdonneFFendleyPFradkinETopological order and conformal quantum critical pointsAnn. Phys.20043104932004AnPhy.310..493A20447441:CAS:528:DC%2BD2cXitVOhurg%3D10.1016/j.aop.2004.01.004 KaufmanBCrystal statistics. II. Partition function evaluated by spinor analysisPhys. Rev.19497612321949PhRv...76.1232K10.1103/PhysRev.76.1232 SamuelSThe use of anticommuting variable integrals in statistical mechanics. III. Unsolved modelsJ. Math. Phys.198021282028331980JMP....21.2820S59760210.1063/1.524406 OnsagerLCrystal statistics. I. A two-dimensional model with an order-disorder transitionPhys. Rev.1944651171944PhRv...65..117O103151:CAS:528:DyaH2cXhtlaqtg%3D%3D10.1103/PhysRev.65.117 MilaFLow-energy sector of the S=1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S = 1/2$$\end{document} Kagome antiferromagnetPhys. Rev. Lett.19988123561998PhRvL..81.2356M1:CAS:528:DyaK1cXmtValsrs%3D10.1103/PhysRevLett.81.2356 MatsumotoNKawabataKAshidaYFurukawaSUedaMContinuous phase transition without gap closing in non-Hermitian quantum many-body systemsPhys. Rev. Lett.20201252606012020PhRvL.125z0601M41979341:CAS:528:DC%2BB3MXivVOksb4%3D10.1103/PhysRevLett.125.26060133449745 Dotsenko, V.l.S. 3D Ising model as a free fermion string theory: An approach to the thermal critical index calculation, Nucl. Phys. B285, 45 (1987). Berezin, F. A. The Method of Second Quantization (Academic Press, 1966). 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R Moessner (71809_CR17) 2001; 86 C Itzykson (71809_CR11) 1982; 210 L Onsager (71809_CR2) 1944; 65 R Shankar (71809_CR13) 1987; 58 G Siudem (71809_CR30) 2016; 6 71809_CR32 71809_CR33 N Matsumoto (71809_CR22) 2020; 125 CR Gattringer (71809_CR26) 1999; 14 D Gaiotto (71809_CR28) 2016; 31 VN Plechko (71809_CR12) 1985; 64 AM Polyakov (71809_CR31) 1979; 82 GM Viswanathan (71809_CR24) 2023; 108 AM Polyakov (71809_CR14) 1987 L Pollet (71809_CR19) 2016; 18 S Samuel (71809_CR8) 1980; 21 B Kaufman (71809_CR4) 1949; 76 TD Schultz (71809_CR5) 1964; 36 F Mila (71809_CR16) 1998; 81 71809_CR29 71809_CR23 AJ Guttmann (71809_CR1) 2009 F Ardonne (71809_CR18) 2004; 310 B Cipra (71809_CR25) 2000; 288 RP Feynman (71809_CR27) 1998 C Itzykson (71809_CR15) 1991 B Dittrich (71809_CR21) 2019; 938 71809_CR7 GM Viswanathan (71809_CR3) 2021; 2021 S Samuel (71809_CR10) 1980; 21 A Smerald (71809_CR20) 2018; 5 FA Berezin (71809_CR6) 1969; 24 S Samuel (71809_CR9) 1980; 21 39506026 - Sci Rep. 2024 Nov 6;14(1):26945. doi: 10.1038/s41598-024-77745-4 |
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| Title | Generalization to d-dimensions of a fermionic path integral for exact enumeration of polygons on hypercubic lattices |
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