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arXiv:1201.4558 · Physical Review A

Completeness of classical φ4 theory on two-dimensional lattices

Vahid Karimipour, Mohammad Hossein Zarei

Published Mar 14, 2012• 10 citations
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Abstract

We formulate a quantum formalism for the statistical mechanical models of discretized field theories on lattices and then show that the discrete version of ${\ensuremath{\varphi}}^{4}$ theory on 2D square lattice is complete in the sense that the partition function of any other discretized scalar field theory on an arbitrary lattice with arbitrary interactions can be realized as a special case of the partition function of this model. To achieve this, we extend the recently proposed quantum formalism for the Ising model [M. Van den Nest, W. Dur, and H. J. Briegel, Phys. Rev. Lett. 98, 117207 (2007)] and its completeness property [M. Van den Nest, W. Dur, and H. J. Briegel, Phys. Rev. Lett. 100, 110501 (2008)] to the continuous variable case.

DOI
10.1103/physreva.85.032316

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We formulate a quantum formalism for the statistical mechanical models of discretized field theories on lattices and then show that the discrete version of \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 theory on 2D square lattice is complete in the sense that the partition function of any other discretized scalar field theory on an arbitrary lattice with arbitrary interactions can be realized as a special case of the partition function of this model. To achieve this, we extend the recently proposed quantum formalism for the Ising model [M. Van den Nest, W. Dur, and H. J. Briegel, Phys. Rev. Lett. 98, 117207 (2007)] and its completeness property [M. Van den Nest, W. Dur, and H. J. Briegel, Phys. Rev. Lett. 100, 110501 (2008)] to the continuous variable case.
Systematic unification of the partition functions of various field theories in a two dimensional classical ϕ4\phi^4ϕ4 model
We show that the partition functions of several field theories with different symmetries and in various dimensions are equal to the partition function of a two-dimensional (2D) classical ϕ4\phi^4ϕ4 field theory model. Although the completeness of 2D ϕ4\phi^4ϕ4 model has been already proved by introducing a quantum formalism for the partition functions, in this paper we study it in a fundamental systematic approach with purely mathematical basis to re-derive such completeness in a general manner. Due to generality, such re-derivation will lead to more clarification of the result for other communities of physics as well as quantum information theorists. Furthermore, our proof of the completeness has an important advantage that is a step-by-step proof which is supported by some graphical transformations. Specifically, we consider discrete version of field theories on lattices with arbitrary dimensions where they are defined as arbitrary polynomial functions of fields with different symmetries. Then we give a set of simple graphical transformations on such field theories to convert them to a classical ϕ4\phi^4ϕ4 field theory on a 2D square lattice with the same partition function.
The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical [Formula: see text] filed theory whose completeness has been proved in [V. Karimipour and M. H. Zarei, Phys. Rev. A 85 (2012) 032316]. We give a new and general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical [Formula: see text] model. To this end, we start from various classical field theories containing models on arbitrary lattices and also [Formula: see text] lattice gauge theories. Then we convert them to a new classical field model on a nonplanar bipartite graph with imaginary kinetic terms. After that, we show that any polynomial function of the field in the corresponding Hamiltonian can approximately be converted to a [Formula: see text] term by adding enough numbers of vertices to the bipartite graph. In the next step, we give a few graphical transformations to convert the final nonplanar graph to a 2D rectangular lattice. We also show that the number of vertices which should be added grows polynomially with the number of vertices in the original model.
We give efficient quantum algorithms to estimate the partition function of (i) the six-vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi-2D square lattice and (iv) the Z 2 lattice gauge theory on a 3D square lattice.Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation.The results are proven for a complex parameter regime of the models.The proofs are based on a mapping relating partition functions to quantum circuits introduced by Van den Nest et al (2009 Phys.Rev. A 80 052334) and extended here.
Systematic method for unification of various field theories in a two-dimensional classical ϕ4\phi^4ϕ4 field theory
Although creating a unified theory in Elementary Particles Physics is still an open problem, there are a lot of attempts for unifying other fields of physics. Following such unifications, we regard a two dimensional (2D) classical Φ4\Phi^{4}Φ4 field theory model to study several field theories with different symmetries in various dimensions. While the completeness of this model has been already proved by a mapping between statistical mechanics and quantum information theory, here, we take into account a fundamental systematic approach with purely mathematical basis to re-derive such completeness in a general manner. Due to simplicity and generality, we believe that our method leads to a general approach which can be understood by other physical communities as well as quantum information theorists. Furthermore, our proof of the completeness is not only a proof-of-principle, but also an interesting algorithmic proof. We consider a discrete version of a general field theory as an arbitrary polynomial function of fields which is defined on a lattice with arbitrary dimension. Then we give a few specific transformations on such field theories to convert them to a classical ϕ4\phi^4ϕ4 field theory on a 2D square lattice with the same Partition function. We also extend the results to the U(1)U(1)U(1) lattice gauge theories in arbitrary dimensions.
We study mappings between distinct classical spin systems that leave the partition function invariant. As recently shown in [Phys. Rev. Lett. 100, 110501 (2008)], the partition function of the 2D square lattice Ising model in the presence of an inhomogeneous magnetic field, can specialize to the partition function of any Ising system on an arbitrary graph. In this sense the 2D Ising model is said to be "complete". However, in order to obtain the above result, the coupling strengths on the 2D lattice must assume complex values, and thus do not allow for a physical interpretation. Here we show how a complete model with real -and, hence, "physical"- couplings can be obtained if the 3D Ising model is considered. We furthermore show how to map general q-state systems with possibly many-body interactions to the 2D Ising model with complex parameters, and give completeness results for these models with real parameters. We also demonstrate that the computational overhead in these constructions is in all relevant cases polynomial. These results are proved by invoking a recently found cross-connection between statistical mechanics and quantum information theory, where partition functions are expressed as quantum mechanical amplitudes. Within this framework, there exists a natural correspondence between many-body quantum states that allow universal quantum computation via local measurements only, and complete classical spin systems.
We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.