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arXiv:1210.1382 · Physical Review B

Fidelity susceptibility of one-dimensional models with twisted boundary conditions

Manisha Thakurathi, Diptiman Sen, Amit Dutta

Published Dec 21, 2012• 14 citations
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Abstract

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 ${\ensuremath{\chi}}_{F}$ 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 $N$ is the system size. If the QCP lies at $g=0$, we find that if $N$ 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 $h$ and period $2q$ in which $\ensuremath{\nu}=q$, and in a $XY$ spin-1/2 chain in which $\ensuremath{\nu}=2$. Finally we show that when $q$ 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 $q$ at these QCPs. We argue that these QCPs mark a transition between extended and localized states at the Fermi energy.

DOI
10.1103/physrevb.86.245424

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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 \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ can detect a QCP provided that the correlation length exponent satisfies \ensuremathν<2\ensuremath{\nu}<2\ensuremathν<2. We then show that \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ can be used to locate a QCP even if \ensuremathν\ensuremath≥2\ensuremath{\nu}\ensuremath{\ge}2\ensuremathν\ensuremath≥2 if we introduce boundary conditions labeled by a twist angle N\ensuremathθN\ensuremath{\theta}N\ensuremathθ, where NNN is the system size. If the QCP lies at g=0g=0g=0, we find that if NNN is kept constant, \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ has a scaling form given by \ensuremathχF\ensuremath∼\ensuremathθ\ensuremath−2/\ensuremathνf(g/\ensuremathθ1/\ensuremathν){\ensuremath{\chi}}_{F}\ensuremath{\sim}{\ensuremath{\theta}}^{\ensuremath{-}2/\ensuremath{\nu}}f(g/{\ensuremath{\theta}}^{1/\ensuremath{\nu}})\ensuremathχF​\ensuremath∼\ensuremathθ\ensuremath−2/\ensuremathνf(g/\ensuremathθ1/\ensuremathν) if \ensuremathθ\ensuremath≪2\ensuremathπ/N\ensuremath{\theta}\ensuremath{\ll}2\ensuremath{\pi}/N\ensuremathθ\ensuremath≪2\ensuremathπ/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 \ensuremathν=q\ensuremath{\nu}=q\ensuremathν=q, and in a XYXYXY spin-1/2 chain in which \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2. Finally we show that when qqq is very large, the model has two additional QCPs at h=\ifmmode±\else\textpm\fi2h=\ifmmode\pm\else\textpm\fi{}2h=\ifmmode±\else\textpm\fi2 which cannot be detected by studying the energy spectrum but are clearly detected by \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​. The peak value and width of \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ 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.
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.
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We study the scaling behavior of fidelity susceptibility density $({\ensuremath{\chi}}_{\mathrm{f}})$ at or close to an anisotropic quantum critical point (AQCP) characterized by two different correlation length exponents ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ along parallel and perpendicular spatial directions, respectively. Our studies show that the response of the system due to a small change in the Hamiltonian near an AQCP is different from that seen near an isotropic QCP. In particular, for a finite system with linear dimension L∣∣{L}_{||}L∣∣​ (${L}_{\ensuremath{\perp}}$) in the parallel (perpendicular) directions, the maximum value of ${\ensuremath{\chi}}_{\mathrm{f}}$ is found to increase in a power-law fashion with L∣∣{L}_{||}L∣∣​ for small L∣∣{L}_{||}L∣∣​, with an exponent depending on both ${\ensuremath{\nu}}_{||}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and eventually crosses over to a scaling with ${L}_{\ensuremath{\perp}}$ for ${L}_{||}^{1/{\ensuremath{\nu}}_{||}}\ensuremath{\gtrsim}{L}_{\ensuremath{\perp}}^{1/{\ensuremath{\nu}}_{\ensuremath{\perp}}}$. We also propose scaling relations of heat density and defect density generated following a quench starting from an AQCP and connect them to a generalized fidelity susceptibility. These predictions are verified exactly both analytically and numerically taking the example of a Hamiltonian showing a semi-Dirac band-crossing point.
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