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arXiv:1301.0347 · Physical Review B

Reply to “Comment on ‘ Z2 -slave-spin theory for strongly correlated fermions’”

Andreas Rüegg, Sebastian D. Huber, Manfred Sigrist

Published Jan 2, 2013• 0 citations
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Abstract

We show that the physical subspace in the ${\mathbb{Z}}_{2}$-slave-spin theory is conserved under the time evolution of the system. Thus, when restricted to the physical subspace, this representation gives a complete and consistent description of the original problem. In addition, we review two known examples from the existing literature in which the projection onto the physical subspace can be relaxed: (i) the noninteracting limit in any dimension at half-filling and (ii) the interacting model in the infinite-dimensional limit at half-filling. In both cases, physical observables are correctly obtained without explicit treatment of the constraints which define the physical subspace. In these examples, correct results are obtained, despite the fact that unphysical states enter the solution.

DOI
10.1103/physrevb.87.037102

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We show that the physical subspace in the Z2{\mathbb{Z}}_{2}Z2​-slave-spin theory is conserved under the time evolution of the system. Thus, when restricted to the physical subspace, this representation gives a complete and consistent description of the original problem. In addition, we review two known examples from the existing literature in which the projection onto the physical subspace can be relaxed: (i) the noninteracting limit in any dimension at half-filling and (ii) the interacting model in the infinite-dimensional limit at half-filling. In both cases, physical observables are correctly obtained without explicit treatment of the constraints which define the physical subspace. In these examples, correct results are obtained, despite the fact that unphysical states enter the solution.
The authors of Ref. 1 introduced a new slave-spin (SS) \nmean-field (MF) approach to the Hubbard model of strongly \ncorrelated electrons in terms of Ising spins and standard \nfermion operators by constructing an enlarged Hilbert space \nwith both physical and unphysical states. The purpose of \nthis construction is clear: this formulation involves a much \nsmaller number of slave fields, compared with more standard \nslave particle frameworks. In this way, the complexity of the \noriginal model is reduced to a more readily tractable system \nof Ising spins in a transverse field coupled to free fermions. \nIn this Comment, we show that the unphysical states are not \nproperly excluded, and despite the claimed good agreement \nwith previous results, the consistency of their MF theory is \nunjustified
Slave-particle method is a powerful tool to tackle the correlation effect in quantum many-body physics. Although it has been successfully used to comprehend various intriguing problems, such as Mott metal–insulator transition and Kondo effect, there is still no convincing theory so far on the availability and limitation of this method. The abuse of slave-particle method may lead to wrong physics. As the simplest slave-particle method, Z 2 slave spin, which is widely applied to many strongly correlated problems, is highly accessible and researchable. In this work, we will uncover the nature of the Z 2 slave-spin method by studying a two-site Hubbard model. After exploring aspects of properties of this toy model, we make a comparative analysis of the results obtained by three methods: (i) slave-spin method on mean-field level, (ii) slave-spin method with gauge constraint, and (iii) the exact solution as a benchmark. We find that, protected by the particle–hole symmetry, the slave-spin mean-field method can recover the static properties of ground state exactly at half filling. Furthermore, in the parameter space where both U and T are small enough, the slave-spin mean-field method is also reliable in calculating the dynamic and thermal dynamic properties. However, when U or T is considerably large, the mean-field approximation gives ill-defined behaviors, which result from the unphysical states in the enlarged Hilbert space. These findings lead to our conclusion that the accuracy of slave particle can be guaranteed if we can exclude all unphysical states by enforcing gauge constraints. Our work demonstrates the promising prospect of slave-particle method in studying complex strongly correlated models with specific symmetry or in certain parameter space.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous Z2Z_2Z2​ symmetry breaking that can be identified as the Mott insulator-to-metal transition.
The infinite-dimensional half-filled Hubbard model can be mapped exactly with no additional constraint onto a model of free fermions coupled in a Z2{Z}_{2}Z2​ gauge-invariant manner to auxiliary Ising spins in a transverse field. In this slave-spin representation, the zero-temperature insulator-to-metal transition translates into spontaneous breaking of the local Z2{Z}_{2}Z2​ gauge symmetry, which is not forbidden in infinite dimensions, thus endowing the Mott transition of an order parameter that is otherwise elusive in the original fermion representation. We demonstrate this interesting scenario by exactly solving the effective spin-fermion model by dynamical mean-field theory both at zero and at finite temperature.
Slave bosons are commonly introduced in order to implement an infinite Hubbard U by means of a local constraint. The usual starting point for investigations within this scheme is a mean-field theory in which the constraint is taken to be global. This approximate treatment of the constraint is studied in the context of a two-band Hubbard model, and it is shown that (i) the ground state has a significant number of doubly occupied sites, despite the infinite on-site repulsion in the original model, and (ii) there is an unphysical tendency for pairing. However, it is found that if the local constraint is retained for the insulator at half filling, then mean-field theory gives the correct result that the double occupancy is zero.