From étale P+-representations to G-equivariant sheaves on G/P

Resonant tunneling through quantum dot under a finite bias voltage at zero temperature is investigated by using the adaptive time-dependent density matrix renormalization group (TdDMRG) method. Quantum dot is modeled by the Anderson Hamiltonian with the one-dimensional nearest-neighbor tightbinding leads. Initially the ground state wave function is calculated with the usual DMRG method. Then the time evolution of the wave function due to the slowly changing bias voltage between the two leads is calculated by using the TdDMRG technique. Even though the system size is finite, the expectation values of current operator show steady-like behavior for a finite time interval, in which the system is expected to resemble the real nonequilibrium steady state of the infinitely long system. We show that from the time intervals one can obtain quantitatively correct results for differential conductance in a wide range of bias voltage. Finally we observe an anomalous behavior in the expectation value of the double occupation operator at the dot hn " n # i as a function of bias voltage.

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“…(6) and (7) should be the same in steady states. However for a finite system these currents show oscillations 20) as in Fig. 1.…”

Section: Oscillations In J L ðþ and J R ðþmentioning

confidence: 82%

Time-Dependent DMRG Study on Quantum Dot under a Finite Bias Voltage

et al. 2008

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“…This is certainly not possible in general, as for instance finite dimensional representations are in the kernel of D ∨ ∆ (unless the set ∆ of simple roots is empty). However, using the ideas of [26] we show the following positive results in this direction. For an object M ∈ M ∆ (π H ∆,0 ) we denote by M ∨ ∞ the étale hull of the image M ∨ ∞ of the natural map from π ∨ to the étale…”

Section: By Considering Finitely Generatedmentioning

confidence: 68%

“…Another promising property of D ∨ ∆ is that we can indeed recover successive extensions π of irreducible principal series representations from D ∨ ∆ (π). In other words we show-using the methods of [26] [17] realizing π ∨ as a G-invariant subspace of the global sections of a G-equivariant sheaf on G/B-that D ∨ ∆ is fully faithful on the category SP 0 A of these representations. By the aforementioned work of Breuil and Paškūnas [11] we cannot expect a bijection between smooth Z/p h -representations of G and mod p h Galois representations of Q p .…”

Section: Background and Motivationmentioning

confidence: 87%

“…Here we in fact use the G-action on π in order to construct the sheaf Y π,M unlike in [26] where the operators H g = res(gC 0 ∩ C 0 ) • (g·) for the open cell C 0 := N 0 B ⊂ G/B ∼ = G/B are constructed as a limit. Apparently the formulas defining this limit do not converge in the weak topology of the finitely generated…”

Section: By Considering Finitely Generatedmentioning

confidence: 99%

“…The reason why this is not a formal consequence of Thm. 8.20 in [26] or of Prop. 3.1 in [28] or even of those proofs is that there is no section of the group homomorphism N 0 ։ N ∆,0 in general.…”

Section: The Equivalence Of Categoriesmentioning

confidence: 99%

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Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Zábrádi¹

2016

Sel. Math. New Ser.

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Let G be a Q p -split reductive group with connected centre and Borel subgroup B = T N . We construct a right exact functor D ∨ ∆ from the category of smooth modulo p n representations of B to the category of projective limits of finitely generated étale (ϕ, Γ)-modules over a multivariable (indexed by the set of simple roots) commutative Laurentseries ring. These correspond to representations of a direct power of Gal(Q p /Q p ) via an equivalence of categories. Parabolic induction from a subgroup P = L P N P gives rise to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component L P . D ∨ ∆ is exact and yields finitely generated objects on the category SP A of finite length representations with subquotients of principal series as Jordan-Hölder factors. Lifting the functor D ∨ ∆ to all (noncommuting) variables indexed by the positive roots allows us to construct a G-equivariant sheaf Y π,∆ on G/B and a G-equivariant continuous map from the Pontryagin dual π ∨ of a smooth representation π of G to the global sections Y π,∆ (G/B). We deduce that D ∨ ∆ is fully faithful on the full subcategory of SP A with Jordan-Hölder factors isomorphic to irreducible principal series.

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Conductance of inhomogeneous systems: Real‐time dynamics

2010

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Numerical time evolution of transport states using time dependent Density Matrix Renormalization Group (td-DMRG) methods has turned out to be a powerful tool to calculate the linear and finite bias conductance of interacting impurity systems coupled to non-interacting one-dimensional leads. Several models, including the Interacting Resonant Level Model (IRLM), the Single Impurity Anderson Model (SIAM), as well as models with different multi site structures, have been subject of investigations in this context. In this work we give an overview of the different numerical approaches that have been successfully applied to the problem and go into considerable detail when we comment on the techniques that have been used to obtain the full I-V-characteristics for the IRLM. Using a model of spinless fermions consisting of an extended interacting nanostructure attached to non-interacting leads, we explain the method we use to obtain the current-voltage characteristics and discuss the finite size effects that have to be taken into account. We report results for the linear and finite bias conductance through a seven site structure with weak and strong nearest-neighbor interactions. Comparison with exact diagonalisation results in the non-interacting limit serve as a verification of the accuracy of our approach. Finally we discuss the possibility of effectively enlarging the finite system by applying damped boundaries and give an estimate of the effective system size and accuracy that can be expected in this case.

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From étale P₊-representations to G-equivariant sheaves on G/P

Cited by 43 publications

References 11 publications

Time-Dependent DMRG Study on Quantum Dot under a Finite Bias Voltage

Time-Dependent DMRG Study on Quantum Dot under a Finite Bias Voltage

Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Conductance of inhomogeneous systems: Real‐time dynamics

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