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arXiv:1101.1713 · Physical Review B

Fidelity susceptibility and general quench near an anisotropic quantum critical point

Victor Mukherjee, Amit Dutta

Published Jun 8, 2011• 20 citations
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Abstract

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 ${L}_{||}$ (${L}_{\ensuremath{\perp}}$) in the parallel (perpendicular) directions, the maximum value of ${\ensuremath{\chi}}_{\mathrm{f}}$ is found to increase in a power-law fashion with ${L}_{||}$ for small ${L}_{||}$, 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.

DOI
10.1103/physrevb.83.214302

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We study scaling behavior of the geometric tensor ${\ensuremath{\chi}}_{\ensuremath{\alpha},\ensuremath{\beta}}({\ensuremath{\lambda}}_{1},{\ensuremath{\lambda}}_{2})$ and the fidelity susceptibility ${\ensuremath{\chi}}_{F}$ in the vicinity of a quantum multicritical point (MCP) using the example of a transverse XY\mathit{XY}XY model. We show that the behavior of the geometric tensor (and thus of ${\ensuremath{\chi}}_{F}$) is drastically different from that seen near a critical point. In particular, we find that it is a highly nonmonotonic function of $\ensuremath{\lambda}$ along the generic direction ${\ensuremath{\lambda}}_{1}~{\ensuremath{\lambda}}_{2}=\ensuremath{\lambda}$ when the system size LLL is bounded by the shorter and longer correlation lengths characterizing the MCP: $1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{1}}\ensuremath{\ll}L\ensuremath{\ll}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, where ${\ensuremath{\nu}}_{1}<{\ensuremath{\nu}}_{2}$ are the two correlation-length exponents characterizing the system. We find that the scaling of the maxima of the components of ${\ensuremath{\chi}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ is associated with the emergence of quasicritical points at $\ensuremath{\lambda}~1/{L}^{1/{\ensuremath{\nu}}_{1}}$, related to the proximity to the critical line of the finite-momentum anisotropic transition. This scaling is different from that in the thermodynamic limit $L\ensuremath{\gg}1/|\ensuremath{\lambda}|{}^{{\ensuremath{\nu}}_{2}}$, which is determined by the conventional critical exponents. We use our results to calculate the defect density following a rapid quench starting from the MCP and show that it exerts a steplike behavior for small quench amplitudes. A study of heat density and diagonal entropy density also shows signatures of quasicritical points.
We study the dynamical response of a system to a sudden change of the tuning parameter $\ensuremath{\lambda}$ starting (or ending) at the quantum critical point. In particular, we analyze the scaling of the excitation probability, number of excited quasiparticles, heat and entropy with the quench amplitude, and the system size. We extend the analysis to quenches with arbitrary power law dependence on time of the tuning parameter, showing a close connection between the scaling behavior of these quantities with the singularities of the adiabatic susceptibilities of order mmm at the quantum critical point, where mmm is related to the power of the quench. Precisely for sudden quenches, the relevant susceptibility of the second order coincides with the fidelity susceptibility. We discuss the generalization of the scaling laws to the finite-temperature quenches and show that the statistics of the low-energy excitations becomes important. We illustrate the relevance of those results for cold-atom experiments.
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.
We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ in the direction along defects and a perpendicular correlation length ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ in the direction perpendicular to the lines. Both ${\ensuremath{\xi}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\xi}}_{\ensuremath{\perp}}$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ take different values. This property is specific for anisotropic scaling and the ratio ${\ensuremath{\nu}}_{\ensuremath{\parallel}}/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ defines the anisotropy exponent $\ensuremath{\theta}$. Until now, estimates of quantitative characteristics of the critical behavior for such systems have been obtained only within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and nonmagnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $\ensuremath{\theta}$ of the system are obtained, as well as an estimate of the susceptibility exponent $\ensuremath{\gamma}$. Our results corroborate the renormalization group predictions obtained earlier.
It has been found via numerical simulations that the geometric phase (GP) and fidelity susceptibility (FS) across the quantum critical points exhibit some universal scaling laws. Here we propose a singular function expansion method to find their exact singular forms and the related coefficients across the critical points. For models where the gaps are closed and reopened at special points (${k}_{0}=0,\ensuremath{\pi}$), scaling laws can be found as a function of the system length NNN and parameter deviation $\ensuremath{\lambda}\ensuremath{-}{\ensuremath{\lambda}}_{c}$, where ${\ensuremath{\lambda}}_{c}$ refers to one of the critical parameters. Although the GP and FS are defined in totally different ways, we find that these two measurements are essentially determined by the same physics, and as a consequence, their coefficients are closely related. Some of these exact relations are found in the anisotropic XY model and extended Ising models. We also show that the constant term in FS may be accompanied by a discontinuous jump across the critical points and, thus, does not have a universal scaling form. These findings should be in contrast to the cases where the gaps are not closed and reopened at the special points, in which some of the above scaling laws may break down as a function of the system length. Finally, we investigate the second-order derivative of GP, which may also exhibit some scaling laws across the critical point. These exact results can greatly enrich our understanding of GP and FS in the characterization of quantum phase transitions and may even find important applications in related physical quantities, such as entanglement, discord, correlation, and quantum Euler numbers, which may also exhibit scaling laws across the critical points.
We study the nonequilibrium slow dynamics for the Kitaev model both in the presence and the absence of disorder. For the case without disorder, we demonstrate, via an exact solution, that the model provides an example of a system with an anisotropic critical point and exhibits unusual scaling of defect density nnn and residual energy QQQ for a slow linear quench. We provide a general expression for the scaling of n(Q)n(Q)n(Q) generated during a slow power-law dynamics, characterized by a rate ${\ensuremath{\tau}}^{\ensuremath{-}1}$ and exponent $\ensuremath{\alpha}$, from a gapped phase to an anisotropic quantum critical point in ddd dimensions, for which the energy gap ${\ensuremath{\Delta}}_{\stackrel{P\vec}{k}}\ensuremath{\sim}{k}_{i}^{z}$ for mmm momentum components (i=1,…,m)(i=1,\dots{},m)(i=1,…,m) and $\ensuremath{\sim}{k}_{i}^{{z}^{\ensuremath{'}}}$ for the rest $d\ensuremath{-}m$ components (i=m+1,…,d)(i=m+1,\dots{},d)(i=m+1,…,d) with $z\ensuremath{\le}{z}^{\ensuremath{'}}$: $n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}[m+(d\ensuremath{-}m)z/{z}^{\ensuremath{'}}]\ensuremath{\nu}\ensuremath{\alpha}/(z\ensuremath{\nu}\ensuremath{\alpha}+1)}(Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}[(m+z)+(d\ensuremath{-}m)z/{z}^{\ensuremath{'}}]\ensuremath{\nu}\ensuremath{\alpha}/(z\ensuremath{\nu}\ensuremath{\alpha}+1)})$. These general expressions reproduce both the corresponding results for the Kitaev model as a special case for $d={z}^{\ensuremath{'}}=2$ and $m=z=\ensuremath{\nu}=1$ and the well-known scaling laws of nnn and QQQ for isotropic critical points for $z={z}^{\ensuremath{'}}$. We also present an exact computation of all nonzero, independent, multispin correlation functions of the Kitaev model for such a quench and discuss their spatial dependence. For the disordered Kitaev model, where the disorder is introduced via random choice of the link variables Dn{D}_{n}Dn​ in the model's fermionic representation, we find that $n\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ and $Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1}$ $(Q\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2})$ for a slow linear quench ending in the gapless (gapped) phase. We provide a qualitative explanation of such scaling.