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arXiv:cond-mat/0105544 · Physical Review Letters

Quantum Critical Points with the Coulomb Interaction and the Dynamical Exponent: When and Why z=1

Igor F. Herbut

Published Sep 10, 2001• 62 citations
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Abstract

A general scenario that leads to Coulomb quantum criticality with the dynamical critical exponent $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ is proposed. I point out that the long-range Coulomb interaction and quenched disorder have competing effects on $z$, and that balance between the two may lead to charged quantum critical points at which $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ exactly. This is illustrated with the calculation for the Josephson junction array Hamiltonian in dimensions $D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3\ensuremath{-}\ensuremath{\epsilon}$. Precisely in $D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$, however, the above simple result breaks down, and $z>1$. Relation to other studies is discussed.

DOI
10.1103/physrevlett.87.137004

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A general scenario that leads to Coulomb quantum criticality with the dynamical critical exponent z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1 is proposed. I point out that the long-range Coulomb interaction and quenched disorder have competing effects on zzz, and that balance between the two may lead to charged quantum critical points at which z=1z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1z=1 exactly. This is illustrated with the calculation for the Josephson junction array Hamiltonian in dimensions D=3\ensuremath−\ensuremathϵD\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3\ensuremath{-}\ensuremath{\epsilon}D=3\ensuremath−\ensuremathϵ. Precisely in D=3D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3D=3, however, the above simple result breaks down, and z>1z>1z>1. Relation to other studies is discussed.
A general scenario that leads to Coulomb quantum criticality with the dynamical critical exponent z=1 is proposed. I point out that the long-range Coulomb interaction and quenched disorder have competing effects on z, and that the balance between the two may lead to charged quantum critical points at which z=1 exactly. This is illustrated with the calculation for the Josephson junction array Hamiltonian in dimensions D=3-\epsilon. Precisely in D=3, however, the above simple result breaks down, and z>1. Relation to other theoretical studies is discussed.
I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point that describes the proliferation of disorder-induced defects in the superfluid ground state, and that is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension of d=1.d=1.d=1. Within the formulated renormalization procedure near d=1,d=1,d=1, the dynamical critical exponent is obtained exactly and the correlation length critical exponent is calculated as a Laurent series in the parameter $\sqrt{\ensuremath{\epsilon}},$ with $\ensuremath{\epsilon}=d\ensuremath{-}1:$ z=d,z=d,z=d, $\ensuremath{\nu}=1/\sqrt{3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}})$ for the short-range, and z=1,z=1,z=1, $\ensuremath{\nu}=\sqrt{2/3\ensuremath{\epsilon}}+O(\sqrt{\ensuremath{\epsilon}}),$ for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions 1< d< 2,1<~d<~2,1< d< 2, for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons, I find $\ensuremath{\nu}=(\frac{1}{2}\sqrt{\ensuremath{\epsilon}})+\frac{1}{4}+O(\sqrt{\ensuremath{\epsilon}}),$ which yields a result in reasonable agreement with the known XY\mathrm{XY}XY critical exponent in d=2.d=2.d=2. The critical behavior of the superfluid density, phonon velocity, and the compressibility in the system with short-range interactions is discussed.
The deconfined quantum critical point (DQCP) between the N\'eel and valence bond solid order was originally proposed in quantum spin systems with a local Hamiltonian. In the last few years analogs of DQCPs with nonlocal interactions have been explored, which can lead to rich possibilities. The nonlocal interactions can either arise from an instantaneous long-range interaction in the Hamiltonian, or from gapless modes that reside in one higher spatial dimension. Here, we consider another mechanism of generating nonlocal interactions by coupling the DQCP to the ``hot spots'' of a Fermi surface. We demonstrate that at least within a substantial energy window, the physics of the DQCP is controlled by a new fixed point with a dynamical exponent z>1z>1z>1.
Recently, the scaling result z=dz=dz=d for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of zzz. We study a model of quantum bosons at T=0T=0T=0 with disorder in 2D using highly effective worm Monte Carlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are $z=1.8\ifmmode\pm\else\textpm\fi{}0.05$, $\ensuremath{\nu}=1.15\ifmmode\pm\else\textpm\fi{}0.03$, and $\ensuremath{\eta}=\ensuremath{-}0.3\ifmmode\pm\else\textpm\fi{}0.1$, hence suggesting that z=2z=2z=2 is not satisfied.
We present quantum phase transitions and critical phenomena of three-body Coulomb systems with charges (Q,(Q,(Q, q, Q)Q)Q) and masses (M,(M,(M, m, M).M).M). Full numerical results, using the finite-size scaling method, for an arbitrary mass ratio $0<~\ensuremath{\kappa}{=(1+m/M)}^{\ensuremath{-}1}<~1$ over the range $1<~\ensuremath{\lambda}=|Q/q|<~1.25,$ show that there exists a transition curve ${\ensuremath{\lambda}}_{c}(\ensuremath{\kappa})$ through which all systems undergo a first-order phase transition from stable to unstable. Particularly, ${\ensuremath{\lambda}}_{c}(\ensuremath{\kappa})$ has a minimum at ${\ensuremath{\kappa}}_{m}=0.35,$ which leads to a new proposed classification of the three-body Coulomb systems: moleculelike systems, $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{m},$ such as ${\mathrm{Ps}}^{\ensuremath{-}}$ $(\ensuremath{\kappa}=0.5)$ and atomlike systems, $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{m},$ such as p‾p‾d\overline{p}\overline{p}dp​p​d $(\ensuremath{\kappa}=0.33).$
We study the quantum melting of stripe phases in models with competing short-range and long-range interactions decaying with distance as $1/{r}^{\ensuremath{\sigma}}$ in two space dimensions. At zero temperature we find a two step disordering of the stripe phases with the growth of quantum fluctuations. A quantum critical point separating a phase with long-range positional order from a phase with long-range orientational order is found when $\ensuremath{\sigma}\ensuremath{\le}4/3$, which includes the Coulomb interaction case $\ensuremath{\sigma}=1$. For $\ensuremath{\sigma}>4/3$ the transition is first order, which includes the dipolar case $\ensuremath{\sigma}=3$. Another quantum critical point separates the orientationally ordered (nematic) phase from a quantum disordered phase for any value of $\ensuremath{\sigma}$. Critical exponents as a function of $\ensuremath{\sigma}$ are computed at one loop order in an $\ensuremath{\epsilon}$ expansion and, whenever available, compared with known results. For finite temperatures it is found that for $\ensuremath{\sigma}\ensuremath{\ge}2$ orientational order decays algebraically with distance until a critical Kosterlitz-Thouless line. Nevertheless, for $\ensuremath{\sigma}<2$ it is found that long-range orientational order can exist at finite temperatures until a critical line which terminates at the quantum critical point at T=0T=0T=0. The temperature dependence of the critical line near the quantum critical point is determined as a function of $\ensuremath{\sigma}$.