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arXiv:1304.2555 · Journal of Statistical Mechanics Theory and Experiment

Quantum critical scaling of fidelity in BCS-like model

Mariusz Adamski, Janusz Jȩdrzejewski, Taras Krokhmalskii

Published Jul 4, 2013• 4 citations
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Abstract

We study scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinfull fermions - a BCS-like model. Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the ground-state fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them. Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article. Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multi-critical point and critical points in its proximity where anomalous scaling behavior is found. We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied. Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski in similar studies of a quantum spin chain - an anisotropic XY model in transverse magnetic field.

DOI
10.1088/1742-5468/2013/07/p07001

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We study scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinfull fermions - a BCS-like model. Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the ground-state fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them. Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article. Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multi-critical point and critical points in its proximity where anomalous scaling behavior is found. We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied. Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski in similar studies of a quantum spin chain - an anisotropic XY model in transverse magnetic field.
We study scaling of the ground-state fidelity in neighborhoods of quantum critical points in a model of interacting spinful fermions -a BCS-like model.Due to the exact diagonalizability of the model, in one and higher dimensions, scaling of the groundstate fidelity can be analyzed numerically with great accuracy, not only for small systems but also for macroscopic ones, together with the crossover region between them.Additionally, in one-dimensional case we have been able to derive a number of analytical formulae for fidelity and show that they accurately fit our numerical results; these results are reported in the article.Besides regular critical points and their neighborhoods, where well-known scaling laws are obeyed, there is the multicritical point and critical points in its proximity where anomalous scaling behavior is found.We consider also scaling of fidelity in neighborhoods of critical points where fidelity oscillates strongly as the system size or the chemical potential is varied.Our results for a one-dimensional version of a BCS-like model are compared with those obtained by Rams and Damski [11] in similar studies of a quantum spin chain -an anisotropic XY model in transverse magnetic field.
We study ground-state fidelity defined as the overlap between two ground states of the same quantum system obtained for slightly different values of the parameters of its Hamiltonian. We focus on the thermodynamic regime of the XYXYXY model and the neighborhood of its critical points. We describe extensively fidelity when it is dominated by the universal contribution reflecting the quantum criticality of the phase transition. We show that proximity to the multicritical point leads to anomalous scaling of fidelity. We also discuss fidelity in a regime characterized by pronounced oscillations resulting from the change in either the system size or the parameters of the Hamiltonian. Moreover, we show when fidelity is dominated by non-universal contributions, study fidelity in the extended Ising model, and illustrate how our results provide additional insight into dynamics of quantum phase transitions. Special attention is given to studies of fidelity from the momentum space perspective. All our main results are obtained analytically. They are in excellent agreement with numerics.
We study ground-state correlation functions in one- and two-dimensional lattice models of interacting spinful fermions - BCS-like models, which exhibit continuous quantum phase transitions. The considered models originate from a two-dimensional model of d-wave superconductivity proposed by Sachdev. Due to the exact diagonalizability of the considered models in any dimensionality, exact phase diagrams, with several kinds of quantum-critical points, are constructed and closed-form analytic expressions for two-point correlation functions are obtained. In one- and two-dimensional cases we provide analytic expressions for the asymptotic behavior of those correlation functions at large distances and in neighborhoods of quantum-critical points. The novelty of our results is that in two-dimensions explicit expressions for direction-dependent correlation lengths in terms of model parameters and the values of direction-dependent universal critical indices ννν, that characterize the divergence of correlation lengths on approaching critical points, are determined. Moreover, specific scaling properties of correlation functions with respect to parameters of underlying Hamiltonians are revealed. Besides enriching the knowledge of properties of lattice fermion systems exhibiting continuous quantum phase transitions, especially in two dimensions, our results open new possibilities of testing unconventional methods of studying quantum phase transitions, as the promising fidelity approach or the entanglement approach, beyond one-dimension and beyond the realm of paradigmatic XY and Ising chains in transverse magnetic fields.
We analyze the fidelity per lattice site for two different ground states of the one-dimensional quantum Ising model in a transverse field near the critical point. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength.
Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If g denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility χF can detect a QCP provided that the correlation length exponent satisfies ν< 2. We then show that χF can be used to locate a QCP even if ν 2 if we introduce boundary conditions labeled by a twist angle Nθ ,w hereN is the system size. If the QCP lies at g = 0, we find that if N is kept constant, χF has a scaling form given by χF ∼ θ −2/ν f(g/θ 1/ν )i fθ � 2π/N. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude h and period 2q in which ν = q, and in a XY spin-1/2 chain in which ν = 2. Finally we show that when q is very large, the model has two additional QCPs at h =± 2 which cannot be detected by studying the energy spectrum but are clearly detected by χF. The peak value and width of χF seem to scale as nontrivial powers of q at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.
Recently it has been shown that the fidelity of the ground state of a quantum many-body system can be used to detect its quantum critical points (QCPs). If ggg denotes the parameter in the Hamiltonian with respect to which the fidelity is computed, we find that for one-dimensional models with large but finite size, the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ can detect a QCP provided that the correlation length exponent satisfies $\ensuremath{\nu}<2$. We then show that ${\ensuremath{\chi}}_{F}$ can be used to locate a QCP even if $\ensuremath{\nu}\ensuremath{\ge}2$ if we introduce boundary conditions labeled by a twist angle $N\ensuremath{\theta}$, where NNN is the system size. If the QCP lies at g=0g=0g=0, we find that if NNN is kept constant, ${\ensuremath{\chi}}_{F}$ has a scaling form given by ${\ensuremath{\chi}}_{F}\ensuremath{\sim}{\ensuremath{\theta}}^{\ensuremath{-}2/\ensuremath{\nu}}f(g/{\ensuremath{\theta}}^{1/\ensuremath{\nu}})$ if $\ensuremath{\theta}\ensuremath{\ll}2\ensuremath{\pi}/N$. We illustrate this both in a tight-binding model of fermions with a spatially varying chemical potential with amplitude hhh and period 2q2q2q in which $\ensuremath{\nu}=q$, and in a XYXYXY spin-1/2 chain in which $\ensuremath{\nu}=2$. Finally we show that when qqq is very large, the model has two additional QCPs at $h=\ifmmode\pm\else\textpm\fi{}2$ which cannot be detected by studying the energy spectrum but are clearly detected by ${\ensuremath{\chi}}_{F}$. The peak value and width of ${\ensuremath{\chi}}_{F}$ seem to scale as nontrivial powers of qqq at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.