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Physics Procedia

Variational Study of Mott Transition by Means of Drude Weight and Superfluid Density

Shun Tamura, H. Yokoyama

Published Jan 1, 2013• 6 citations
Publisher

Abstract

To distinguish a metal from an insulator, the Drude weight (D, zero-frequency component of conductivity) is a useful measure, as Kohn pointed out long ago: D > 0(=0) for a metal (insulator). Later, Millis and Coppersmith showed that a variational wave function ΨQ in which the key ingredient for a Mott transition (a doublon-holon binding effect) is introduced exhibits D > 0 (metallic) even for sufficiently large correlation strength, namely, a Mott transition is absent from. ΨQ In contrast, variational Monte Carlo studies using ΨQ confirmed, by studying relevant quantities such as doublon density d and a superconducting correlation function, Pd∞ that ΨQ undoubtedly raises a Mott transition. This contradiction has been a long-standing perplexing problem. We definitely settle this problem by adding ΨQ to a configuration-dependent phase factor Pθ, which has been hitherto overlooked. This factor appropriately picks out a negative counterpart in D for insulators, so that D vanishes. Because Pθ does not affect the quantities such as d and Pd∞, the previous results of ΨQ on the Mott transition remain intact for PθΨQ.

DOI
10.1016/j.phpro.2013.04.038

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To distinguish a metal from an insulator, the Drude weight (D, zero-frequency component of conductivity) is a useful measure, as Kohn pointed out long ago: D > 0(=0) for a metal (insulator). Later, Millis and Coppersmith showed that a variational wave function ΨQ in which the key ingredient for a Mott transition (a doublon-holon binding effect) is introduced exhibits D > 0 (metallic) even for sufficiently large correlation strength, namely, a Mott transition is absent from. ΨQ In contrast, variational Monte Carlo studies using ΨQ confirmed, by studying relevant quantities such as doublon density d and a superconducting correlation function, Pd∞ that ΨQ undoubtedly raises a Mott transition. This contradiction has been a long-standing perplexing problem. We definitely settle this problem by adding ΨQ to a configuration-dependent phase factor Pθ, which has been hitherto overlooked. This factor appropriately picks out a negative counterpart in D for insulators, so that D vanishes. Because Pθ does not affect the quantities such as d and Pd∞, the previous results of ΨQ on the Mott transition remain intact for PθΨQ.
Drude weight (DDD) is a useful measure to distinguish a metal from an insulator. However, DDD has not been justifiably estimated by the variation theory for long, since Millis and Coppersmith [Phys. Rev. B 43 (1991) 13770] pointed out that a variational wave function ΨQΨ_QΨQ​, which includes the key ingredient (doublon-holon binding effect) for a Mott transition, yields a positive DDD (namely metallic) even in the Mott-insulating regime. We argue that, to obtain a correct DDD, an imaginary part must exist in the wave function. By introducing a configuration-dependent phase factor Pθ{\cal P}_θPθ​ to ΨQΨ_QΨQ​, Mott transitions are successfully represented by DDD (D=0D=0D=0 for U>UcU>U_{\rm c}U>Uc​) for a normal and ddd-wave pairing states; thereby, the problem of Millis and Coppersmith is settled. Generally, Pθ{\cal P}_θPθ​ plays a pivotal role in describing current-carrying states in regimes of Mott physics. On the other hand, we show using a perturbation theory, the one-body (mean-field) part of the wave function should be complex for band insulators such as antiferromagnetic states in hypercubic lattices.
To understand the mechanism of Mott transitions in case of no magnetic influence, superfluid-insulator (Mott) transitions in the S=0 Bose Hubbard model at unit filling are studied on the square and triangular lattices, using a variational Monte Carlo method. In trial many-body wave functions, we introduce various types of attractive correlation factors between a doubly-occupied site (doublon, D) and an empty site (holon, H), which play a central role for Mott transitions, in addition to the onsite repulsive (Gutzwiller) factor. By optimizing distance-dependent parameters, we study various properties of this type of wave functions. With a hint from the Mott transition arising in a completely D-H bound state, we propose an improved picture of Mott transitions, by introducing two characteristic length scales, the D-H binding length ξdhξ_{\rm dh}ξdh​ and the minimum D-D exclusion length ξddξ_{\rm dd}ξdd​. Generally, a Mott transition occurs when ξdhξ_{\rm dh}ξdh​ becomes comparable to ξddξ_{\rm dd}ξdd​. In the conductive (superfluid) state, domains of D-H pairs overlap with each other (ξdh>ξddξ_{\rm dh}>ξ_{\rm dd}ξdh​>ξdd​); thereby D and H can propagate independently as density carriers by successively exchanging the partners. In contrast, intersite repulsive Jastrow (D-D and H-H) factors have little importance for the Mott transition.
The mechanism of Mott transition in the Hubbard model on the square lattice is studied without explicit introduction of magnetic and superconducting correlations, using a variational Monte Carlo method. In the trial wave functions, we consider various types of binding factors between a doubly-occupied site (doublon, D) and an empty site (holon, H), like a long-range type as well as a conventional nearest-neighbor type, and add independent long-range D-D (H-H) factors. It is found that a wide choice of D-H binding factor leads to Mott transitions at critical values near the band width. We renew the D-H binding picture of Mott transitions by introducing two characteristic length scales, the D-H binding length l_{DH} and the minimum D-D distance l_{DD}, which we appropriately estimate. A Mott transition takes place at l_{DH}=l_{DD}. In the metallic regime (l_{DH}>l_{DD}), the domains of D-H pairs overlap with one another, thereby doublons and holons can move independently by exchanging the partners one after another. In contrast, the D-D factors give only a minor contribution to the Mott transition.
We study the Mott transition occurring for bosonic Hubbard models in one, two, and three spatial dimensions by means of a variational wave function benchmarked by Green's function Monte Carlo calculations. We show that a very accurate variational wave function, which is constructed by applying a long-range Jastrow factor to the noninteracting boson ground state, can describe the superfluid-insulator transition in any dimensionality. Moreover, by mapping the quantum averages over such a wave function into the partition function of a classical model, important insights into the insulating phase are uncovered. Finally, the evidence in favor of anomalous scenarios for the Mott transition in two dimensions is reported whenever additional long-range repulsive interactions are added to the Hamiltonian.
We prove that a large class of generalized Gutzwiller wave functions, previously believed to be useful in the study of the Mott metal-insulator transition, are always metallic in the sense that the response to an electric field includes a \ensuremath{\delta} function at zero frequency (corresponding to free acceleration of carriers in a dc field). We also show that modifying these wave functions to enforce binding of empty and doubly occupied sites is insufficient to produce insulating behavior. We derive a criterion which an insulating wave function must satisfy.