Spin-fermion mappings for even Hamiltonian operators

We show how to map local fermionic problems onto local spin problems on a lattice in any dimension. The main idea is to introduce auxiliary degrees of freedom, represented by Majorana fermions, which allow us to extend the Jordan–Wigner transformation to dimensions higher than one. We also discuss the implications of our results in the numerical investigation of fermionic systems.

arXiv:cond-mat/0508353Journal of Statistical Mechanics Theory and ExperimentSep 1, 2005196 citations

Spin-fermion mappings for even Hamiltonian operators

Alberto Anfossi, Arianna Montorsi

Published Jan 1, 2005• 4 citations

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Abstract

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Spin–fermion mappings for even Hamiltonian operators

Alberto Anfossi, Arianna Montorsi

We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.

arXiv:cond-mat/0412533Journal of Physics A Mathematical and GeneralMay 10, 20055 citations

Local spin-fermion mapping for even Hamiltonian operators

Alberto Anfossi, Arianna Montorsi

We revisit the Jordan-Wigner transformation, showing that-rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators-it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan-Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energiesWe revisit the Jordan-Wigner transformation, showing that-rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators-it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan-Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies

Jan 1, 20050 citations

Fermionic mean-field theory as a tool for studying spin Hamiltonians

Thomas M. Henderson, Brent Harrison, Ilias Magoulas +5 more

The Jordan-Wigner transformation permits one to convert spin 1/2 operators into spinless fermion ones, or vice versa. In some cases, it transforms an interacting spin Hamiltonian into a noninteracting fermionic one, which is exactly solved at the mean-field level. Even when the resulting fermionic Hamiltonian is interacting, its mean-field solution can provide surprisingly accurate energies and correlation functions. Jordan-Wigner is, however, only one possible means of interconverting spin and fermionic degrees of freedom. Here, we apply several such techniques to the XXZ and J1-J2 Heisenberg models, as well as to the pairing or reduced Bardeen-Cooper-Schrieffer Hamiltonian, with the aim of discovering which of these mappings is most useful in applying fermionic mean-field theory to the study of spin Hamiltonians.The Jordan-Wigner transformation permits one to convert spin 1/2 operators into spinless fermion ones, or vice versa. In some cases, it transforms an interacting spin Hamiltonian into a noninteracting fermionic one, which is exactly solved at the mean-field level. Even when the resulting fermionic Hamiltonian is interacting, its mean-field solution can provide surprisingly accurate energies and correlation functions. Jordan-Wigner is, however, only one possible means of interconverting spin and fermionic degrees of freedom. Here, we apply several such techniques to the XXZ and J1-J2 Heisenberg models, as well as to the pairing or reduced Bardeen-Cooper-Schrieffer Hamiltonian, with the aim of discovering which of these mappings is most useful in applying fermionic mean-field theory to the study of spin Hamiltonians.

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Fermionization transform for certain higher-dimensional quantum spin models

Victor M. Galitski

Proposed is a generalization of Jordan-Wigner transform that allows to exactly fermionize a large family of quantum spin Hamiltonians in dimensions higher than one. The key steps are to enlarge the Hilbert space of the original model by adding to it a collection of stand-alone free spins and to use a combination of these auxiliary operators and the lattice spins to construct a proper fermion representation of the physical Hamiltonian. The transform is especially useful for lattice spin Hamiltonians, where two-spin interactions of $XY$ type are either absent or exist only within one-dimensional chains and where the chains are coupled via two-spin interactions of Ising type, ring-exchange terms, or more general multispin interactions that involve an even number of spin operators from each chain. Using the proposed fermionization method we provide a simple argument suggesting that a spin Hamiltonian closely related to the ring-exchange model proposed by Paramekanti et al., [Phys. Rev. B 66, 054526 (2002)] indeed realizes a spin-liquid state.Proposed is a generalization of Jordan-Wigner transform that allows to exactly fermionize a large family of quantum spin Hamiltonians in dimensions higher than one. The key steps are to enlarge the Hilbert space of the original model by adding to it a collection of stand-alone free spins and to use a combination of these auxiliary operators and the lattice spins to construct a proper fermion representation of the physical Hamiltonian. The transform is especially useful for lattice spin Hamiltonians, where two-spin interactions of $XY$ type are either absent or exist only within one-dimensional chains and where the chains are coupled via two-spin interactions of Ising type, ring-exchange terms, or more general multispin interactions that involve an even number of spin operators from each chain. Using the proposed fermionization method we provide a simple argument suggesting that a spin Hamiltonian closely related to the ring-exchange model proposed by Paramekanti et al., [Phys. Rev. B 66, 054526 (2002)] indeed realizes a spin-liquid state.

arXiv:1003.3874Physical Review BAug 27, 201011 citations

Local spin description of fermions on a lattice

Adam Wyrzykowski

A local transformation from fermionic operators to spin matrices is proposed and studied in this work. For this purpose, a system of fermions on a lattice is considered and one applies the scheme to replace the fermionic variables with spin matrices, while the transformation relates only those fermionic/spin operators which are assigned to nearby lattice sites. In one dimension, this proposal yields the same result as the well-known Jordan-Wigner transformation, while not being restricted to $d=1$ dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes $3\times3$ , $4\times3$ and $4\times4$ . Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.A local transformation from fermionic operators to spin matrices is proposed and studied in this work. For this purpose, a system of fermions on a lattice is considered and one applies the scheme to replace the fermionic variables with spin matrices, while the transformation relates only those fermionic/spin operators which are assigned to nearby lattice sites. In one dimension, this proposal yields the same result as the well-known Jordan-Wigner transformation, while not being restricted to $d=1$ dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes $3\times3$, $4\times3$ and $4\times4$. Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.

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Mapping local Hamiltonians of fermions to local Hamiltonians of spins

F Verstraete, J I Cirac

arXiv:cond-mat/0508353Journal of Statistical Mechanics Theory and ExperimentSep 1, 2005196 citations

XY

type are either absent or exist only within one-dimensional chains and where the chains are coupled via two-spin interactions of Ising type, ring-exchange terms, or more general multispin interactions that involve an even number of spin operators from each chain. Using the proposed fermionization method we provide a simple argument suggesting that a spin Hamiltonian closely related to the ring-exchange model proposed by Paramekanti et al., [Phys. Rev. B 66, 054526 (2002)] indeed realizes a spin-liquid state.

d=1

dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes

3\times3

4\times3

and

4\times4

. Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.