Order, Disorder and Phase Transitions in Quantum Many Body Systems

Giuliani, Alessandro

doi:10.4081/scie.2016.576

Cited by 3 publications

(4 citation statements)

References 62 publications

(104 reference statements)

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“…1 L 2 x¨y R β,L , and the expression rJ j , X i s must be understood as explained in the footnote 1 above. This definition can be obtained via a formal 'Wick rotation' of the time variable, t Ñ´it, starting from the original definition of the Kubo conductivity, (2.15), see, e.g., [21]. A posteriori, we will see that in our context the two definitions coincide, see Section 5 below.…”

Section: Euclidean Formalism and Ward Identitiesmentioning

confidence: 96%

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Quantization of the Interacting Hall Conductivity in the Critical Regime

2019

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The Haldane model is a paradigmatic 2d lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a 'topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified ('dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods. arXiv:1803.11213v1 [cond-mat.str-el]

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Section: Euclidean Formalism and Ward Identitiesmentioning

confidence: 96%

“…t " 0 an adiabatic external field of the form e ηt E¨ X , see e.g. [21] for a formal derivation, and [8,9,36,38] for a rigorous derivation in a slightly different setting.…”

Section: Lattice Currents and Linear Reponse Theorymentioning

confidence: 99%

Quantization of the Interacting Hall Conductivity in the Critical Regime

2019

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“…The mathematical tool that allows to prove a bound for the cumulants that grows only as n !, and that is uniform in the size of the system, is the Brydges-Battle-Federbush-Kennedy (BBFK) formula [ 11 , 17 – 19 ], for the connected expectations, or cumulants, of a fermionic theory. See [ 23 ] for a review of recent applications to transport problems in condensed matter systems. Let us review its application to the problem at hand.…”

Section: On the Switch Functionsmentioning

confidence: 99%

Adiabatic Evolution of Low-Temperature Many-Body Systems

Greenblatt,

Lange,

Marcelli

et al. 2024

Commun. Math. Phys.

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We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.

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“…Eq. (2.25) describes the linear response of the system at the time t " 0 after introducing an external perturbation´e ηt E¨ X, for t ď 0 (see [22] for a formal derivation).…”

Section: Linear Response Theorymentioning

confidence: 99%

Universal Edge Transport in Interacting Hall Systems

Antinucci

Mastropietro

Porta

2018

Commun. Math. Phys.

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We study the edge transport properties of 2d interacting Hall systems, displaying singlemode chiral edge currents. For this class of many-body lattice models, including for instance the interacting Haldane model, we prove the quantization of the edge charge conductance and the bulk-edge correspondence. Instead, the edge Drude weight and the edge susceptibility are interaction-dependent; nevertheless, they satisfy exact universal scaling relations, in agreement with the chiral Luttinger liquid theory. Moreover, charge and spin excitations differ in their velocities, giving rise to the spin-charge separation phenomenon. The analysis is based on exact renormalization group methods, and on a combination of lattice and emergent Ward identities. The invariance of the emergent chiral anomaly under the renormalization group flow plays a crucial role in the proof. phases [35]. We prove the exact quantization of the edge conductance, for weak interactions: all interaction corrections cancel out. Combined with [27] and with the noninteracting bulk-edge correspondence [39,51,16], this result provides the first proof of the bulk-edge correspondence for an interacting many-body quantum system. Moreover, we also consider the edge Drude weight and the edge susceptibility, both for charge and spin degrees of freedom; we find explicit expressions for these quantities, which turn out to be nonuniversal in the coupling strength. Nevertheless, the Drude weight D and the susceptibility κ satisfy the universal scaling relation D " κv 2 c , as in the Luttinger model. Finally, we compute the two-point function, and we show that it exhibits spin-charge separation.Notice that our analysis does not extend in a straightforward way to the case of multi-edge currents. The reason being the scattering between different edge modes. In the renormalization group terminology, the edge states scattering is a marginal process; our method allows to control the scattering between edge states with the same velocity, thanks to the comparison with the chiral Luttinger model (see below), but does not allow to control the scattering of edge states with different velocities. We leave the generalization to multi-edge channels Hall systems as a very interesting open problem, on which we plan to come back in the future.The paper is organized as follows. In Section 2 we introduce the class of interacting lattice models we will consider, and we define bulk and edge transport coefficients, in the linear response regime. In Section 3 we recall some known facts about noninteracting Hall systems. Then, in Section 4 we present our main result, Theorem 4.1. In the rest of the paper, we discuss the proof of Theorem 4.1. In Section 5 we introduce a functional integral representation for fermionic lattice models, and in particular in Section 5.2 we derive a rigorous relationship between the model of interest and an interacting one-dimensional quantum field theory. This result actually applies to models with a general number of edge states. Starting from Section 6, we restr...

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Order, Disorder and Phase Transitions in Quantum Many Body Systems

Cited by 3 publications

References 62 publications

Quantization of the Interacting Hall Conductivity in the Critical Regime

Quantization of the Interacting Hall Conductivity in the Critical Regime

Adiabatic Evolution of Low-Temperature Many-Body Systems

Universal Edge Transport in Interacting Hall Systems

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