1. Introduction
It is known that the resistivity
of the normal states of cuprate superconductors obeys the following relationship [
1,
2,
3,
4]
where
is the residual resistivity,
A is a coefficient independent of temperature,
T is temperature.
These phenomena of linear temperature dependence of resistivity are found in numerous strongly correlated electron systems, such as the heavy fermion compounds [
1,
3], transition metal oxides [
1,
4,
5], iron pnictides [
1], magic angle twisted bilayer graphene, organic metals [
1] and conventional metals [
1], often in connection with unconventional superconductivity. Sometimes this linear-in-temperature resistivity is called Planckian resistivity [
2]. When superconductivity is destroyed by a high magnetic field, the recovered normal state still obeys this law of linear-in-temperature resistivity in the low temperature region [
4]. In most of the heavy fermion materials, the linear-in-temperature resistivity appears when they have been tuned by some external parameter to create a low-temperature continuous phase transition which is referred to a quantum critical point (QCP) [
1]. Thus, the linear temperature dependence of resistivity are often associated with quantum criticality. The linear-in-temperature resistivity of
with different gradients, different doping dependencies and different origins appears not only at high temperature but also at low temperature [
6].
Strange metal behavior refers to a linear temperature dependence of the electrical resistivity [
1,
3]. A unified theory of this scaling law (
1) in different strange metals is still an open problem [
2,
3].
Before the discovery of quantum mechanics, a successful formula of resistivity of metals is proposed in the Drude model ([
7], p. 7). Shortly after the discovery of quantum mechanics, Sommerfeld improved the Drude model. In the Sommerfeld model, the following Drude formula of resistivity of metals can be derived approximately based on quantum theoty ([
7], p. 251)
where
n is the number dendity of electrons,
e is the electric charge of an electron,
is the effective mass of an electron,
is the relaxation time of an electron.
If the transport scattering rate
is linear-in-temperature and is the only temperature-dependent quantity in Equation (
2), then the scaling law (
1) of resistivity can be derived directly. Thus, a clue to study the scaling law (
1) is to investigate the relaxation time
in the Drude formula (
2).
The Drude formula (
2) is valid only for charge carriers which obey the Fermi-Dirac distribution. Experiments have shown that the dominant charge carriers in
(YBCO) film are Cooper pairs (CPs) [
8]. Since CPs are not Fermions, the Drude formula (
2) may be not valid in the normal states of cuprate superconductors. Thus, an interesting question is that whether a similar formula for the resistivity of the normal states of cuprate superconductors exists.
According to the Heisenberg uncertainty principle, a local equilibration time of any many-body quantum system cannot be faster than the following Planckian time
[
3]
where
h is the Plank constant,
,
is the Boltzmann constant.
This timescale
is associated with quantum criticality and known to bound the validity of a Boltzmann description of transport [
9].
is suggested to be the lower bound of the phase coherence time in quantum critical systems [
9].
is also known to control the electronic dynamics of the cuprate strange metal [
9]. Thus, an idea is that the relaxation time
of CP in cuprate superconductors may be proportional to the Planckian time
, i.e.,
, where
is a dimensionless parameter. Indeed, experiments have shown that the scattering rate
in the region of the temperature-linear resistivity of a wide range of metals, including heavy fermion, oxide [
4,
5,
8], pnictide, organic metals and conventional metals, can be written as [
1]
where
.
Equation (
4) shows that the relaxation time
is approximately equal to the Planckian time
, i.e.,
. The case of
is referred to the Planckian dissipation [
3]. It is surprising that the linear-in-temperature scattering rate
and the behaviors of Planckian dissipation in these materials (except the conventional metals) can be seen down to low temperatures with appropriate tuning by magnetic field, chemical composition or hydrostatic pressure [
1]. It is suggested that there may be a fundamental principle governing the transport of CPs [
8].
If Equation (
4) and Equation (
2) are valid in the normal states of cuprate superconductors, then Equation (
1) may be derived. In this manuscript we focus on this clue and try to derive the scaling law (
1).
2. Stochastic Mechanics of a Cooper Pair in Two Dimensional Condensed Matter
In order to explain the energy quantization of atoms, E. Schrödinger proposes the following equation for a non-relativistic particle moving in a potential [
10]
where
t is time,
is a point in space,
is the wave function,
m is the mass of the particle,
is the potential,
h is the Plank constant,
,
is the Laplace operator in a Cartesian coordinate
.
The Schrödinger equation (
5) is a fundamental assumption in non-relativistic quantum mechanics [
10]. Although the Schrödinger equation can be used to describe some non-relativistic quantum phenomena, the origin of quantum phenomena remains an unsolved problem in physics for more than 100 years [
11,
12]. Although the axiomatic system of quantum mechanics was firmly established, the interpretation of quantum mechanics is still open [
11,
12]. There exist some paradoxes in quantum mechanics [
13,
14,
15,
16], for instance, the paradox of reduction of a wave packet and the paradox of the Schrödinger cat.
Fényes proposed an interpretation of quantum mechanics based on a Markov process. Fényes’ work was developed by Weizel and discussed by Kershaw [
17]. According to Luis de Broglie [
18], the success of the probabilistic interpretation of
inspired Einstein to speculate that the probability
is generated by a kind of hidden Brownian motions of particles. This kind of hidden motions was called quasi-Brownian motions by Luis de Broglie [
18].
If the quantum phenomena stem from the stochastic motions of particles, then we may establish a more fundamental and more powerful theory of quantum phenomena other than quantum mechanics. The Schrödinger equation may no longer be a basic assumption and may be derived in this new theory. Indeed, E. Nelson [
19] derived the Schrödinger equation by means of theory of stochastic processes based on the assumption that every particle with mass
m in vacuum is subject to Brownian motion with diffusion constant
.
Inspired by Nelson’s stochastic mechanics [
19,
20,
21,
22,
23,
24,
25,
26,
27,
28], we propose a theoretical derivation of the Schrödinger equation based on Newton’s second law and a mechanical model of vacuum [
29].
Recently, monolayer crystals of the high-temperature superconductor
(Bi-2212) was obtained by a fabrication process [
30]. The superconductivity, the pseudogap, charge order and the Mott state at various doping concentrations of the monolayer Bi-2212 reveals that the phases are indistinguishable from those in the bulk [
30]. Monolayer Bi-2212 displays the fundamental physics of cuprate superconductors [
30]. Therefore, monolayer copper oxides is a platform for studying high-temperature superconductivity in two dimensions. Thus, we focus on two dimensional condensed matters.
Modern experiments, for instance, the Casimir effect [
31,
32], have shown that vacuum is not empty. Thus, we suppose that there is a damping force exerted on each particle by vacuum [
29]. For a microscopic particle moving in vacuum, we have the following relation [
29]
where
is a constant similar to the Boltzmann constant
,
is the temperature of the
substratum in the vicinity of the particle in vacuum [
33],
is a damping coefficient related to vacuum.
It is known that a CP in condensed matter may be scattered by ions, electrons, phonon, etc. In the Drude theory of metals, the effect of individual electron collisions is approximately treated by introducing a damping force into the equation of motion of an electron ([
7], p. 11). Following the Drude theory, we suppose that a CP in a condensed matter will experiences not only a damping force exerted by vacuum but also an additional damping force exerted by the condensed matter. We introduce a two dimensional Cartesian coordinate system
which is attached to the condensed matter. We suppose that the two dimensional velocity
of the CP exists. Applying Newton’s second law, the motion of a CP may be described by the following Langevin equation [
34]
where
is the mass of the CP,
is a damping coefficient related to the condensed matter,
is a quasi-inertial force coefficient,
is a two dimensional random force and
is a two dimensional external force field.
We introduce the following definitions
where
is the damping mass of the CP,
is the quasi-inertial mass of the CP.
Equation (
8) and Equation (9) can be written as
Using Equation (
8) and Equation (9), Equation (
7) can be written as
Let be the ith component of the random force , i.e., .
Assumption A1.
Assume that the force field is a continuous function of and t. Inspired by the Ornstein-Uhlenbeck theory [35,36] of Brownian motion, we suppose that the random force ξ exerted on the CP by the condensed matter is a two-dimensional Gaussian white noise [37,38,39,40] and the variance of the ith component of ξ is [29]
where , , is a parameter similar to the Boltzmann constant, is the temperature of the substratum [33] in the location of the condensed matter.
For convenience, we introduce the following notation
The Gaussian white noise
is the generalized derivative of a Wiener process
[
37,
39]. We can write formally [
37,
38,
39,
40]
where
is a two-dimensional Wiener process with a diffusion constant
.
The mathematically rigorous form of Equation (
12) is the following stochastic differential equations based on Itô stochastic integral [
36]
where
,
,
and
are stochastic processes on a probability space
,
are independent of all of the
,
, with
,
,
.
The microstate of the CP at time
t is defined by the random vector
[
29].
Proposition 1.
Suppose that Equations (13) are valid and the force field satisfy a global Lipschitz condition, that is, for some constant ,
for all and in , where denotes the two-dimensional Descartes space, denotes the set of positive real numbers. Then, at a time scale of an observer very large compare to the relaxation time ,
the solution of the Langevin equation Equation (16) converges to the solution of the following Smoluchowski equation Equation (20) with probability one uniformly for t in compact subintervals of for all , i.e.,
where is the solution of the following Smoluchowski equation
where , is a two-dimensional Wiener process with a diffusion constant defined by
A proof of Proposition 1 can be found in the Appendix A. Following similar methods in Ref. [
29], a Schrödinger like equation (
A56) and Equation (
A22) can be derived, refers to Appendix B.
Putting Equation (
A22) into Equation (
A56), we have the following result.
Proposition 2.
The Schrödinger like equation Equation (A56) reduces to the following Schrödinger like equation
where is the wave mass defined by Equation (A21).
From Equation (
10), in vacuum the damping mass
reduces to the mass
of a CP. In vacuum,
reduces to
. Therefore, in vacuum the wave mass
defined by Equation (
A21) reduces to
. Thus, the Schrödinger like equation (
22) in the condensed matter is a generalization of the Schrödinger equation (
5) in vacuum.
3. Calculation of Direct Current (DC) Electrical Conductivity
If there is no external magnetic field and the external electric field
is a constant vector field, then the Langevin equation (
12) can be written as
where
is the electric charge of the CP.
If the mean velocity
of the CP is high enough such that
, then we call this velocity as drift velocity and denotes it as
. Thus, if the observer look at the CP for a time long enough comparing to the relaxation time
, then he will observe the long time averaged quantities of the Langevin equation (
23). Since
and
, the long time averaged form of the Langevin equation (
23) can be written as ([
7], p. 7; [
41], p. 16)
The current density
corresponding to the drift velocity
is ([
7], p. 7; [
41], p. 16)
where
is the number density of CPs.
Putting Equation (
24) and Equation (
25), we have
Equation (
25) can be written as ([
7], p. 7)
where
is the
ith component of the current density
,
is the
jth component of the electric field
,
is the conductivity tensor which can be written as
where
is the Kronecker symbol
It is known that the resistivity of the normal states of cuprate superconductors exhibits strong anisotropy ([
42], p. 190). Thus, Equation (
28) may be only valid for the plane conductivity
of two dimensional cuprate superconductors and not valid for bulk cuprate. Noticing
and Equation (
29), we have
Using Equation (
18), Equation (
30) can also be written as