二维无序阻挫Josephson结阵列平衡态及动力学相变
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摘要
本论文主要利用标度理论数值研究了二维无序阻挫Josephson结阵列的平衡态及动力学相变。借助二维XY模型,我们先分析了弱磁场中(磁通密度为1／25)Josephson结阵列的平衡态超导相变,零温脱钉相变及低温蠕动规律；然后又研究了强磁场中(磁通密度为1／2)弱无序Josephson结阵列的Ising型相变。当系统处于弱磁场中时,我们在耦合强度中引入随机钉扎势,利用电阻分流结(RSJ)动力学方法,在涨落扭转边界条件下,运用涡旋玻璃相变的电流电压动力学标度理论,探讨了系统的平衡态相变；又采用脱钉相变理论和蠕动标度理论,分析了涡旋的零温脱钉运动和低温蠕动规律,得到了超导相变温度、零温临界电流和临界指数随无序强度变化的规律。当系统处于强磁场中时,我们在相位中引进一个随机相移,采用Monte Carlo数值模拟方法,按照临界短时动力学标度理论,探究了系统的Ising型相变,讨论了无序对临界温度、静态临界指数和动态临界指数的影响。最后,我们首次在生长模型中引入一个Binder累计量,在短时动力学标度方法的框架下,通过直接数值积分,研究了一维quenched Mullins-Herring界面生长模型的脱钉相变,给出了临界驱动力和临界指数,并与长时稳态的数值模拟结果进行了比较。
各部分的主要结论如下：
(1)对于弱磁场中(磁通密度为1／25)的二维Josephson结阵列,当无序不存在时,系统从高温线性电阻态到低温超导态发生KTB相变；当无序存在时,系统的超导相变变成了连续性的非KTB型,通过电流电压动力学标度理论分析,系统平衡态时的低温无序相可能是涡旋玻璃相,非KTB型相变可能是涡旋玻璃相变。随着无序强度的增加,相变温度升高,动力学指数减小,静态临界指数在统计误差范围内几乎不变,说明系统的连续性非KTB相变属于同一普适类,不随无序的变化而改变。无论无序是否存在,系统均表现出连续性的脱钉相变和non-Arrhenius型的蠕动规律。当无序增加时,零温临界电流呈现出非单调性的变化,脱钉指数则单调递增,热圆滑指数减小。我们还发现,脱钉指数和热圆滑指数的乘积只有两个特征值2和5／3,说明根据无序的强弱程度,系统的蠕动规律具有两个普适类。
(2)对于完全阻挫Josephson结阵列(磁通密度为1／2),我们先从两个不同的初始态出发,研究了无序为零时系统的Ising型相变,发现短时动力学标度方法可以推广到基态不易得到的无序系统；然后利用这个推广的短时动力学标度方法研究了弱无序存在时二维完全阻挫Josephson结阵列的Ising型相变。结果表明,当无序比较弱时,系统存在Ising型相变,随着无序的增加,临界温度减小,静态临界指数增加,动力学指数稍微有变化。我们还发现,系统具有一个临界无序强度σc：当σ<σc时,系统具有Ising型相变；当σ>σc时,系统的Ising型相变消失。
(3)对于一维quenched Mullins-Herring界面生长模型,我们利用短时动力学标度方法,得到了临界驱动力和临界指数。与长时稳态模拟相比,我们的创新之处有两点：a.假定生长速度、外界驱动力和生长时间所满足的标度关系在宏观短时区域也成立,计算了时间指数δ；b.引进了一个与时间有关的生长速度的Binder累计量,得到了动力学指数z。我们再结合模型固有的动力学标度关系,利用短时动力学标度方法给出了和长时稳态模拟同样好的结果。此外,我们也讨论了有限尺寸效应对临界驱动力和临界指数的影响。本部分内容充分体现了短时动力学标度方法时间短、尺寸小的优点。
In this dissertation, using dynamic scaling theory, we have studied the equilibrium and dynamic phase transitions in two-dimensional frustrated Josephson junction arrays with disorder, which can be described by two-dimensional XY model. First, we have analyzed the superconducting phase transition, depinning transition at zero temperature and creeping laws at low temperatures when Josephson junction arrays are subjected to a weak magnetic field with the flux density 1/25. Then we have investigated the Ising-like phase transi-tion in weakly disordered Josephson junction arrays with the magnetic flux density 1/2. In the weak magnetic field, the random pinning potential is introduced into the coupling strength. Under the fluctuating twist boundary condition, the equilibrium superconduct-ing phase transition is discussed by means of dynamic scaling analysis of current-voltage characteristics. We also study the motion of vortices at zero temperature and creep at fi-nite temperatures by depinning and creeping scaling theory. The effects of disorder on the critical temperature, the critical current at zero temperature and the critical exponents are also discussed. As for the Josephson junction arrays in the strong magnetic field with the flux density 1/2, a random phase shift is added to the phase difference. Based on Monte Carlo simulations, the Ising-like phase transition is investigated in the short-time dynamic scaling scheme. The critical temperature, the static and dynamic critical exponents are also evaluated for different disorders. Finally, we have applied a short-time dynamic scaling method to the depinning phase transition of the quenched Mullins-Herring equation. For the first time we introduce a time-dependent Binder cumulant of the order parameter in the surface growth model and obtain the critical driving force and the critical exponents, which are consistent with those extracted from the long-time steady-state simulations.
The main results of each part are as follows:
(1) For the two-dimensional Josephson junction arrays exposed to the weak magnetic field with the flux density 1/25, the system undergoes a KTB phase transition in the ab- sence of disorder from a normal phase with linear resistivity to a superconducting state. When the disorder is introduced into the system, the phase transition would be driven into a non-KTB type, where a possible vortex glass transition is suggested by means of dynamic scaling analysis of current-voltage characteristics. It is found that the critical tem-perature increases with the disorder, but the dynamic exponent decreases. The values of static critical exponents are roughly the same within statistical errors, illustrating the same universality class in these continuous phase transitions with different disorder strengths. Whether the disorder is present or not, a continuous depinning transition is found at zero temperature and a non-Arrhenius creeping law is observed at low temperatures. The critical current changes nonmonotonously with the strength of the bond disorder. The depinning exponentβincreases and the thermal rounding exponent 8 decreases with the rise of dis-order. Interestingly, their product only has two characteristic values 2 and 5/3, depending on the strength of the disorder. These two characteristic values imply the existence of two universality classes in this model.
(2) For the two-dimensional fully frustrated Josephson junction arrays in a strong magnetic field with the flux density 1/2, we first investigate the Ising-like phase transition in the system without disorder in a short-time dynamic method from two different initial states. It follows that the short-time dynamic scheme can be extended to such a disordered system, where the ground state is hard to obtain. Whereafter we have used this approach to study a possible Ising-like phase transition in the fully frustrated Josephson junction arrays with weak disorder. It is shown that the critical temperature decreases with disorder, and the static critical exponent increases, while the dynamic exponent varies slightly. More-over, there exists a critical disorder strengthσc above which the Ising-like phase transition disappears.
(3) For the one-dimensional quenched Mullins-Herring surface growth model, the critical driving force and the critical exponents can be extracted by a short-time dynamic scaling approach. Compared with the long-time steady-state simulations, there exist two new points:(a). It is assumed that a universal dynamic scaling relation among the velocity, force and time is still valid in the macroscopic short-time regime. In this way, the time exponent 8 can be calculated in the short-time dynamic method; (b). In order to get the dynamic exponent independently, we introduce a time-dependent Binder cumulant of the order parameter growth velocity. Together with other scaling relations in this model, the present short-time dynamic approach can give equally good results as steady-state simula-tions. In addition, the finite size effects on the critical driving force and the exponents are also discussed. It is found that the finite size effects can well be controlled in the short-time dynamics, which highlights an advantage of this method.
各部分的主要结论如下：
(1)对于弱磁场中(磁通密度为1／25)的二维Josephson结阵列,当无序不存在时,系统从高温线性电阻态到低温超导态发生KTB相变；当无序存在时,系统的超导相变变成了连续性的非KTB型,通过电流电压动力学标度理论分析,系统平衡态时的低温无序相可能是涡旋玻璃相,非KTB型相变可能是涡旋玻璃相变。随着无序强度的增加,相变温度升高,动力学指数减小,静态临界指数在统计误差范围内几乎不变,说明系统的连续性非KTB相变属于同一普适类,不随无序的变化而改变。无论无序是否存在,系统均表现出连续性的脱钉相变和non-Arrhenius型的蠕动规律。当无序增加时,零温临界电流呈现出非单调性的变化,脱钉指数则单调递增,热圆滑指数减小。我们还发现,脱钉指数和热圆滑指数的乘积只有两个特征值2和5／3,说明根据无序的强弱程度,系统的蠕动规律具有两个普适类。
(2)对于完全阻挫Josephson结阵列(磁通密度为1／2),我们先从两个不同的初始态出发,研究了无序为零时系统的Ising型相变,发现短时动力学标度方法可以推广到基态不易得到的无序系统；然后利用这个推广的短时动力学标度方法研究了弱无序存在时二维完全阻挫Josephson结阵列的Ising型相变。结果表明,当无序比较弱时,系统存在Ising型相变,随着无序的增加,临界温度减小,静态临界指数增加,动力学指数稍微有变化。我们还发现,系统具有一个临界无序强度σc：当σ<σc时,系统具有Ising型相变；当σ>σc时,系统的Ising型相变消失。
(3)对于一维quenched Mullins-Herring界面生长模型,我们利用短时动力学标度方法,得到了临界驱动力和临界指数。与长时稳态模拟相比,我们的创新之处有两点：a.假定生长速度、外界驱动力和生长时间所满足的标度关系在宏观短时区域也成立,计算了时间指数δ；b.引进了一个与时间有关的生长速度的Binder累计量,得到了动力学指数z。我们再结合模型固有的动力学标度关系,利用短时动力学标度方法给出了和长时稳态模拟同样好的结果。此外,我们也讨论了有限尺寸效应对临界驱动力和临界指数的影响。本部分内容充分体现了短时动力学标度方法时间短、尺寸小的优点。
In this dissertation, using dynamic scaling theory, we have studied the equilibrium and dynamic phase transitions in two-dimensional frustrated Josephson junction arrays with disorder, which can be described by two-dimensional XY model. First, we have analyzed the superconducting phase transition, depinning transition at zero temperature and creeping laws at low temperatures when Josephson junction arrays are subjected to a weak magnetic field with the flux density 1/25. Then we have investigated the Ising-like phase transi-tion in weakly disordered Josephson junction arrays with the magnetic flux density 1/2. In the weak magnetic field, the random pinning potential is introduced into the coupling strength. Under the fluctuating twist boundary condition, the equilibrium superconduct-ing phase transition is discussed by means of dynamic scaling analysis of current-voltage characteristics. We also study the motion of vortices at zero temperature and creep at fi-nite temperatures by depinning and creeping scaling theory. The effects of disorder on the critical temperature, the critical current at zero temperature and the critical exponents are also discussed. As for the Josephson junction arrays in the strong magnetic field with the flux density 1/2, a random phase shift is added to the phase difference. Based on Monte Carlo simulations, the Ising-like phase transition is investigated in the short-time dynamic scaling scheme. The critical temperature, the static and dynamic critical exponents are also evaluated for different disorders. Finally, we have applied a short-time dynamic scaling method to the depinning phase transition of the quenched Mullins-Herring equation. For the first time we introduce a time-dependent Binder cumulant of the order parameter in the surface growth model and obtain the critical driving force and the critical exponents, which are consistent with those extracted from the long-time steady-state simulations.
The main results of each part are as follows:
(1) For the two-dimensional Josephson junction arrays exposed to the weak magnetic field with the flux density 1/25, the system undergoes a KTB phase transition in the ab- sence of disorder from a normal phase with linear resistivity to a superconducting state. When the disorder is introduced into the system, the phase transition would be driven into a non-KTB type, where a possible vortex glass transition is suggested by means of dynamic scaling analysis of current-voltage characteristics. It is found that the critical tem-perature increases with the disorder, but the dynamic exponent decreases. The values of static critical exponents are roughly the same within statistical errors, illustrating the same universality class in these continuous phase transitions with different disorder strengths. Whether the disorder is present or not, a continuous depinning transition is found at zero temperature and a non-Arrhenius creeping law is observed at low temperatures. The critical current changes nonmonotonously with the strength of the bond disorder. The depinning exponentβincreases and the thermal rounding exponent 8 decreases with the rise of dis-order. Interestingly, their product only has two characteristic values 2 and 5/3, depending on the strength of the disorder. These two characteristic values imply the existence of two universality classes in this model.
(2) For the two-dimensional fully frustrated Josephson junction arrays in a strong magnetic field with the flux density 1/2, we first investigate the Ising-like phase transition in the system without disorder in a short-time dynamic method from two different initial states. It follows that the short-time dynamic scheme can be extended to such a disordered system, where the ground state is hard to obtain. Whereafter we have used this approach to study a possible Ising-like phase transition in the fully frustrated Josephson junction arrays with weak disorder. It is shown that the critical temperature decreases with disorder, and the static critical exponent increases, while the dynamic exponent varies slightly. More-over, there exists a critical disorder strengthσc above which the Ising-like phase transition disappears.
(3) For the one-dimensional quenched Mullins-Herring surface growth model, the critical driving force and the critical exponents can be extracted by a short-time dynamic scaling approach. Compared with the long-time steady-state simulations, there exist two new points:(a). It is assumed that a universal dynamic scaling relation among the velocity, force and time is still valid in the macroscopic short-time regime. In this way, the time exponent 8 can be calculated in the short-time dynamic method; (b). In order to get the dynamic exponent independently, we introduce a time-dependent Binder cumulant of the order parameter growth velocity. Together with other scaling relations in this model, the present short-time dynamic approach can give equally good results as steady-state simula-tions. In addition, the finite size effects on the critical driving force and the exponents are also discussed. It is found that the finite size effects can well be controlled in the short-time dynamics, which highlights an advantage of this method.
引文
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