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arXiv:2204.12335

Linear-response approach to critical quantum many-body systems

Ricardo Puebla, Alessio Belenchia, Giulio Gasbarri, Eric Lutz, Mauro Paternostro

Published Apr 26, 2022• 0 citations
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Abstract

The characterization of quantum critical phenomena is pivotal for the understanding and harnessing of quantum many-body physics. However, their complexity makes the inference of such fundamental processes difficult. Thus, efficient and experimentally non-demanding methods for their diagnosis are strongly desired. Here, we introduce a general scheme, based on the combination of finite-size scaling and the linear response of a given observable to a time-dependent perturbation, to efficiently extract the energy gaps to the lowest excited states of the system, and thus infer its dynamical critical exponents. Remarkably, the scheme is able to tackle both integrable and non-integrable models, prepared away from their ground states. It thus holds the potential to embody a valuable diagnostic tool for experimentally significant problems in quantum many-body physics.

DOI
10.48550/arxiv.2204.12335

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The characterization of quantum critical phenomena is pivotal for the understanding and harnessing of quantum many-body physics. However, their complexity makes the inference of such fundamental processes difficult. Thus, efficient and experimentally non-demanding methods for their diagnosis are strongly desired. Here, we introduce a general scheme, based on the combination of finite-size scaling and the linear response of a given observable to a time-dependent perturbation, to efficiently extract the energy gaps to the lowest excited states of the system, and thus infer its dynamical critical exponents. Remarkably, the scheme is able to tackle both integrable and non-integrable models, prepared away from their ground states. It thus holds the potential to embody a valuable diagnostic tool for experimentally significant problems in quantum many-body physics.
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Characterizing criticality in quantum many-body systems of dimension ≥2\ge 2≥2 is one of the most important challenges of the contemporary physics. In principle, there is no generally valid theoretical method that could solve this problem. In this work, we propose an efficient approach to identify the criticality of quantum systems in higher dimensions. Departing from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and a one-dimensional (1D) quantum state defined in the imaginary time direction in terms of the so-called time matrix product state (tMPS). We show that the criticality of the targeted model can be faithfully identified by the tMPS, using the mature scaling schemes of correlation length and entanglement entropy in 1D quantum theories. We benchmark our proposal with the results obtained for the Heisenberg anti-ferromagnet on honeycomb lattice. We demonstrate critical scaling relation of the tMPS for the gapless case, and a trivial scaling for the gapped case with spatial anisotropy. The critical scaling behaviors are insensitive to the system size, suggesting the criticality can be identified in small systems. Our tMPS scheme for critical scaling shows clearly that the spin-1/2 kagomé Heisenberg antiferromagnet has a gapless ground state. More generally, the present study indicates that the 1D conformal field theories in imaginary time provide a very useful tool to characterize the criticality of higher dimensional quantum systems.
We propose a method to study dynamical response of a quantum system by evolving it with an imaginary-time-dependent Hamiltonian. The leading nonadiabatic response of the system driven to a quantum-critical point is universal and characterized by the same exponents in real and imaginary time. For a linear quench protocol, the fidelity susceptibility and the geometric tensor naturally emerge in the response functions. Beyond linear response, we extend the finite-size scaling theory of quantum phase transitions to nonequilibrium setups. This allows, e.g., for studies of quantum phase transitions in systems of fixed finite size by monitoring expectation values as a function of the quench velocity. Nonequilibrium imaginary-time dynamics is also amenable to quantum Monte Carlo (QMC) simulations, with a scheme that we introduce here and apply to quenches of the transverse-field Ising model to quantum-critical points in one and two dimensions. The QMC method is generic and can be applied to a wide range of models and nonequilibrium setups.
The analytical, experimental, and even numerical investigation of quantum many-body systems is pushed to its limits rather rapidly as the dimension of the Hilbert space scales exponentially with the number of subsystems. As a result, tasks such as the exact diagonalization of general quantum many-body Hamiltonians and the analysis of quantum experiments are limited to small system sizes. In this thesis, we address two issues related to the exponential scaling of the Hilbert space dimension: (i) The approximation of the ground state energy of many-body systems with a lower bound and (ii) efficient strategies for quantum state tomography experiments.
Quantum impurities are ubiquitous in condensed matter physics and constitute the most stripped-down realization of many-body problems. While measuring their finite-frequency response could give access to key characteristics such as excitations spectra or dynamical properties, this goal has remained elusive despite over two decades of studies in nanoelectronic quantum dots. Conflicting experimental constraints of very strong coupling and large measurement bandwidths must be met simultaneously. We get around this problem using cQED tools, and build a precisely characterized quantum simulator of the boundary sine-Gordon model, a non-trivial bosonic impurity problem. We succeeded to fully map out the finite frequency linear response of this system. Its reactive part evidences a strong renormalisation of the nonlinearity at the boundary in agreement with non-perturbative calculations. Its dissipative part reveals a striking many-body broadening caused by multi-photon conversion. The experimental results are matched quantitatively to a perturbative calculation based on a microscopically calibrated model. Furthermore, we push the device into a regime where perturbative calculations break down, which calls for more advanced theoretical tools to model many-body quantum circuits. We also critically examine the technological limitations of cQED platforms to reach universal scaling laws. This work opens exciting perspectives for the future such as quantifying quantum entanglement in the vicinity of a quantum critical point or accessing the dynamical properties of non-trivial many-body problems.