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arXiv:1108.1734 · Physical Review B

Exact zero modes in closed systems of interacting fermions

Garry Goldstein, Claudio Chamon

Published Sep 14, 2012• 40 citations
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Abstract

We show that for closed finite-sized systems with an odd number of real fermionic modes, even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator, as shown by Akhmerov [Phys. Rev. B 82, 020509 (2010)], as well as additional linearly independent zero modes, one of which is (1) continuously connected to the Majorana mode solution in the noninteracting limit, and (2) less prone to decoherence when the system is opened to contact with an infinite bath. We also show that in the idealized situation where there are two or more well-separated zero modes each associated with a finite number of interacting fermions at a localized vortex, these modes have non-Abelian Ising statistics under braiding. Furthermore the algebra of the zero mode operators makes them useful for fermionic quantum computation [S. B. Bravii and A. Y. Kitaev, Ann. Phys. 298, 210 (2002)].

DOI
10.1103/physrevb.86.115122

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We show that for closed finite-sized systems with an odd number of real fermionic modes, even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator, as shown by Akhmerov [Phys. Rev. B 82, 020509 (2010)], as well as additional linearly independent zero modes, one of which is (1) continuously connected to the Majorana mode solution in the noninteracting limit, and (2) less prone to decoherence when the system is opened to contact with an infinite bath. We also show that in the idealized situation where there are two or more well-separated zero modes each associated with a finite number of interacting fermions at a localized vortex, these modes have non-Abelian Ising statistics under braiding. Furthermore the algebra of the zero mode operators makes them useful for fermionic quantum computation [S. B. Bravii and A. Y. Kitaev, Ann. Phys. 298, 210 (2002)].
Abstract For random interacting Majorana models where the only symmetries are the parity P and the time-reversal-symmetry T , various approaches are compared to construct exact even and odd normalized zero modes Γ in finite size, i.e. Hermitian operators that commute with the Hamiltonian, that square to the identity, and that commute (even) or anticommute (odd) with the parity P . Even normalized zero-modes are well known under the name of ‘pseudo-spins’ in the field of many-body-localization or more precisely ‘local integrals of motion’ (LIOMs) in the many-body-localized-phase where the pseudo-spins happens to be spatially localized. Odd normalized zero-modes are popular under the name of ‘Majorana zero modes’ or ‘strong zero modes’. Explicit examples for small systems are described in detail. Applications to real-space renormalization procedures based on blocks containing an odd number of Majorana fermions are also discussed.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are “delocalized” between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the “nearest-neighbor” form of the effective low-energy Hamiltonian.
Majorana fermions often coexist with other low-energy fermionic degrees of freedom. In such a situation, topological quantum computation requires the use of fermionic zero modes of a many-body system. We classify all such modes for interacting fermions and show how to select the mode that maximizes the decoherence time. We find that in a typical interacting system the maximal decoherence time is within one order of magnitude from the decoherence time of a qubit based on the local part of the fermion parity operator.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the "trivial" and "topological" regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes - the nonuniform Ising-Kitaev chain, containing two "topological" regions separated by a "trivial" region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are "delocalized" between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the "nearest-neighbor" form of the effective low-energy Hamiltonian.
We study a system of interacting spinless fermions in one dimension that, in the absence of interactions, reduces to the Kitaev chain [Kitaev, Phys. Usp. 44, 131 (2001)]. In the noninteracting case, a signal of topological order appears as zero-energy modes localized near the edges. We show that the exact ground states can be obtained analytically even in the presence of nearest-neighbor repulsive interactions when the on-site (chemical) potential is tuned to a particular function of the other parameters. As with the noninteracting case, the obtained ground states are twofold degenerate and differ in fermionic parity. We prove the uniqueness of the obtained ground states and show that they can be continuously deformed to the ground states of the noninteracting Kitaev chain without gap closing. We also demonstrate explicitly that there exists a set of operators each of which maps one of the ground states to the other with opposite fermionic parity. These operators can be thought of as an interacting generalization of Majorana edge zero modes.
The one-dimensional (1D) Kitaev model in the topological phase, with open boundary conditions, hosts Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators are enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.