Nonlinear renormalization-group recursion relations are used to show that the thermal conductivity of $^{4}\mathrm{He}$ has a crossover between a weak-coupling regime for ${t}_{c}\ensuremath{\ll}t<1$ and a scaling regime for $t\ensuremath{\ll}{t}_{c}$, where $t=\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}$ and ${t}_{c}\ensuremath{\approx}{10}^{\ensuremath{-}3}$. The smallness of ${t}_{c}$ ---an essential feature---is explained by a small bare dynamic coupling constant. The asymmetric spin model ( $F$ ) provides a quantitative fit to experiment, whereas the symmetric model ( $E$ ) is only semiquantitative. The border-line dimension ${d}^{*}$ , below which dynamic scaling breaks down, is shown to satisfy ${d}^{*}\ensuremath{\lesssim}2.54$.Nonlinear renormalization-group recursion relations are used to show that the thermal conductivity of $^{4}\mathrm{He}$ has a crossover between a weak-coupling regime for ${t}_{c}\ensuremath{\ll}t<1$ and a scaling regime for $t\ensuremath{\ll}{t}_{c}$, where $t=\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}$ and ${t}_{c}\ensuremath{\approx}{10}^{\ensuremath{-}3}$. The smallness of ${t}_{c}$---an essential feature---is explained by a small bare dynamic coupling constant. The asymmetric spin model ($F$) provides a quantitative fit to experiment, whereas the symmetric model ($E$) is only semiquantitative. The border-line dimension ${d}^{*}$, below which dynamic scaling breaks down, is shown to satisfy ${d}^{*}\ensuremath{\lesssim}2.54$.

Physical Review LettersFeb 16, 198126 citations

One-Dimensional Models

Lev Deych

Springer eBooksJan 1, 20180 citations

Delocalization border and onset of chaos in a model of quantum computation

G. P. Berman, F. Borgonovi, Felix M. Izrailev +1 more

We study the properties of spectra and eigenfunctions for a chain of 1/2 spins (qubits) in an external time-dependent magnetic field and under the conditions of nonselective excitation (when the amplitude of the magnetic field is large). This model is known as a possible candidate for experimental realization of quantum computation. We present the theory for finding delocalization transitions and show that for the interaction between nearest qubits, the transition is very different from that in quantum chaos. We explain this phenomena by showing that in the considered region of parameters our model is close to an integrable one. According to a general opinion, the threshold for the onset of quantum chaos due to the interqubit interaction decreases with an increase of the number of qubits. Contrary to this expectation, for a magnetic field with constant gradient we have found that chaos border does not depend on the number of qubits. We give analytical estimates that explain this effect, together with numerical data supporting our analysis. Random models with long-range interactions have been studied as well. In particular, we show that in this case the delocalization and quantum chaos borders coincide.We study the properties of spectra and eigenfunctions for a chain of 1/2 spins (qubits) in an external time-dependent magnetic field and under the conditions of nonselective excitation (when the amplitude of the magnetic field is large). This model is known as a possible candidate for experimental realization of quantum computation. We present the theory for finding delocalization transitions and show that for the interaction between nearest qubits, the transition is very different from that in quantum chaos. We explain this phenomena by showing that in the considered region of parameters our model is close to an integrable one. According to a general opinion, the threshold for the onset of quantum chaos due to the interqubit interaction decreases with an increase of the number of qubits. Contrary to this expectation, for a magnetic field with constant gradient we have found that chaos border does not depend on the number of qubits. We give analytical estimates that explain this effect, together with numerical data supporting our analysis. Random models with long-range interactions have been studied as well. In particular, we show that in this case the delocalization and quantum chaos borders coincide.

arXiv:quant-ph/0104086Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsOct 25, 200133 citations

One-Dimensional Model Problems

Ralph E. Christoffersen

Springer advanced texts in chemistryJan 1, 19890 citations

Anomalously Slow Cross Symmetry Phase Relaxation, Thermalized Non-Equilibrated Matter and Quantum Computing Beyond the Quantum Chaos Border

Marc Bienert

Thermalization in highly excited quantum many-body system does not necessarily mean a complete memory loss of the way the system was formed.This effect may pave a way for a quantum computing, with a large number of qubits n 100-1000, far beyond the quantum chaos border.One of the manifestations of such a thermalized nonequilibrated matter is revealed by a strong asymmetry around 90 • c.m. of evaporating proton yield in the Bi(γ,p) photonuclear reaction.The effect is described in terms of anomalously slow cross symmetry phase relaxation in highly excited quantum many-body systems with exponentially large Hilbert space dimensions.In the above reaction this phase relaxation is about eight orders of magnitude slower than energy relaxation (thermalization).Thermalization in highly excited quantum many-body system does not necessarily mean a complete memory loss of the way the system was formed.This effect may pave a way for a quantum computing, with a large number of qubits n 100-1000, far beyond the quantum chaos border.One of the manifestations of such a thermalized nonequilibrated matter is revealed by a strong asymmetry around 90 • c.m. of evaporating proton yield in the Bi(γ,p) photonuclear reaction.The effect is described in terms of anomalously slow cross symmetry phase relaxation in highly excited quantum many-body systems with exponentially large Hilbert space dimensions.In the above reaction this phase relaxation is about eight orders of magnitude slower than energy relaxation (thermalization).

arXiv:quant-ph/0602224Symmetry Integrability and Geometry Methods and ApplicationsFeb 27, 20060 citations

Boundary ABF models

Tetsuji Miwa, Robert Weston

arXiv:hep-th/9610094Nuclear Physics BFeb 1, 199729 citations

Nonlinear renormalization-group recursion relations are used to show that the thermal conductivity of

^{4}\mathrm{He}

has a crossover between a weak-coupling regime for ${t}_{c}\ensuremath{\ll}t<1$ and a scaling regime for $t\ensuremath{\ll}{t}_{c}$, where $t=\frac{(T\ensuremath{-}{T}_{\ensuremath{\lambda}})}{{T}_{\ensuremath{\lambda}}}$ and ${t}_{c}\ensuremath{\approx}{10}^{\ensuremath{-}3}$. The smallness of

{t}_{c}

---an essential feature---is explained by a small bare dynamic coupling constant. The asymmetric spin model (

F

) provides a quantitative fit to experiment, whereas the symmetric model (

E

) is only semiquantitative. The border-line dimension

{d}^{*}

, below which dynamic scaling breaks down, is shown to satisfy ${d}^{*}\ensuremath{\lesssim}2.54$.