Microscopic Derivation of GinzburgLandau Theory and the BCS Critical Temperature Shift in the Presence of Weak Macroscopic External Fields

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Microscopic Derivation of Ginzburg–Landau Theory

and the BCS Critical Temperature Shift

in the Presence of Weak Macroscopic External Fields

Andreas Deuchert, Christian Hainzl, Marcel Maier (born Schaub)

February 14, 2023

Abstract

We consider the Bardeen–Cooper–Schrieﬀer (BCS) free energy functional with

weak and macroscopic external electric and magnetic ﬁelds and derive the Ginzburg–

Landau functional. We also provide an asymptotic formula for the BCS critical tem-

perature as a function of the external ﬁelds. This extends our previous results in [16]

for the constant magnetic ﬁeld to general magnetic ﬁelds with a nonzero magnetic ﬂux

through the unit cell.

Contents

1 Introduction and Main Results 2

1.1 Introduction.................................... 2

1.2 Gauge-periodic samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 TheBCSfunctional ............................... 4

1.4 The translation-invariant BCS functional . . . . . . . . . . . . . . . . . . . 6

1.5 The Ginzburg–Landau functional . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Mainresults.................................... 8

1.7 Organization of the paper and strategy of proof . . . . . . . . . . . . . . . . 10

1.8 Heuristic computation of the terms in the Ginzburg–Landau functional . . . 12

2 Preliminaries 13

2.1 Schattenclasses.................................. 13

2.2 Gauge-periodic Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Trial States and their BCS Energy 16

3.1 The Gibbs states Γ∆............................... 17

3.2 The BCS energy of the states Γ∆........................ 18

3.3 The upper bound on (1.22) and proof of Theorem 2 (a) . . . . . . . . . . . 21

4 Proofs of the Results in Section 3 21

4.1 Schatten norm estimates for operators given by product kernels . . . . . . . 21

4.2 Magnetic resolvent estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 ProofofLemma3.1 ............................... 27

4.4 ProofofLemma3.4 ............................... 28

4.5 ProofofTheorem3.6............................... 29

4.6 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Proof of Proposition 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

arXiv:2210.09356v3 [math-ph] 11 Feb 2023

5 The Structure of Low-Energy States 65

5.1 A lower bound for the BCS functional . . . . . . . . . . . . . . . . . . . . . 67

5.2 ProofofTheorem5.1............................... 69

6 The Lower Bound on (1.22) and Proof of Theorem 2 (b) 69

6.1 The BCS energy of low-energy states . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Estimate on the relative entropy . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Conclusion .................................... 70

6.4 Proof of the equivalent of [16, Lemma 6.2] in our setting . . . . . . . . . . . 70

A Gauge-Invariant Perturbation Theory for KTc,A−V71

A.1 Preparatorylemmas ............................... 72

A.2 Proof of Proposition A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

1 Introduction and Main Results

1.1 Introduction

Ginzburg–Landau (GL) theory has been introduced as the ﬁrst macroscopic and phe-

nomenlogical description of superconductivity in 1950 [29]. The theory comprises a sys-

tem of partial diﬀerential equations for a complex-valued function, the order parameter,

and an eﬀective magnetic ﬁeld. Ginzburg–Landau theory has been highly inﬂuencial and

investigated in numerous works, among which are [7, 8, 46, 45, 10, 11, 13, 12, 1] and

references therein.

Bardeen–Cooper–Schrieﬀer (BCS) theory of superconductivity is the ﬁrst commonly

accepted and Nobel prize awarded microscopic theory of superconductivity [2]. As a major

breakthrough, the theory features a pairing mechanism between the electrons below a

certain critical temperature, which causes the electrical resistance in the system to drop

to zero in the superconducting phase. This eﬀect is due to an eﬀective attraction between

the electrons, which arises as a consequence of the phonon vibrations of the lattice ions in

the superconductor.

One way to formulate BCS theory mathematically is via the BCS free energy func-

tional or BCS functional for short. As Leggett pointed out in [38], the BCS functional can

be obtained from a full quantum mechanical description of the system by restricting at-

tention to quasi-free states, see also [28]. Such states are determined by their one-particle

density matrix and the Cooper pair wave function. The BCS functional has been studied

intensively from a mathematical point of view in the absence of external ﬁelds in [32, 20,

34, 35, 26, 4, 23, 15] and in the presence of external ﬁelds in [36, 5, 24, 14, 6]. The BCS

gap equation arises as the Euler–Lagrange equation of the BCS functional and its solution

is used to compute the spectral gap of an eﬀective Hamiltonian, which is open in the

superconducting phase. BCS theory from the point of view of its gap equation is studied

in [43, 3, 50, 51, 41, 52].

The present article continues a series of works, in which the macroscopic GL theory is

derived from the microscopic BCS theory in a regime close to the critical temperature and

for weak external ﬁelds. This endeavor has been initiated by Gor’kov in 1959 [30]. The

ﬁrst mathematically rigorous derivation of the GL functional from the BCS functional has

been provided by Frank, Hainzl, Seiringer, and Solovej for periodic external electric and

magnetic ﬁelds in 2012 in [21]. An important assumption of this work is that the ﬂux

of the external magnetic ﬁeld through the unit cell of periodicity of the system vanishes.

This excludes for example a homogeneous magnetic ﬁeld. The techniques from this GL

February 14, 2023 2 Deuchert, Hainzl, Maier

1.1 Introduction

derivation have been further developed in [22] to compute the BCS critical temperature

shift caused by the external ﬁelds. The ﬁrst important step towards overcoming the zero

magnetic ﬂux restriction in [21, 19] has been made by Frank, Hainzl, and Langmann,

who considered in [19] the problem of computing the BCS critical temperature shift for

systems exposed to a homogeneous magnetic ﬁeld within the framework of linearized BCS

theory. Recently, the derivation of the GL functional and the computation of BCS critical

temperature shift (for the full nonlinear model) could be extended to the case of a constant

magnetic ﬁeld by Deuchert, Hainzl, and Maier in [16]. The goal of the present work is to

further extend the results in [16] to the case of general external magnetic ﬁelds with an

arbitrary ﬂux through the unit cell.

GL theory arises from BCS theory when the temperature is suﬃciently close to the

critical temperature and when the external ﬁelds are weak and slowly varying. More

precisely, if 0< h 1denotes the ratio between the microscopic and the macroscopic

length scale, then the external electric ﬁeld Wand the magnetic vector potential Aare

given by h2W(hx)and hA(hx), respectively. Furthermore, the temperature regime is such

that T−Tc=−TcDh2for some constant D > 0, where Tcis the critical temperature in

absence of external ﬁelds. When this scaling is in eﬀect, it is shown in [21] and [16] that

the Cooper pair wave function α(x, y)is given by

α(x, y) = h α∗(x−y)ψ h(x+y)

2!(1.1)

to leading order in h. Here, α∗is the microscopic Cooper pair wave function in the absence

of external ﬁelds and ψis the GL order parameter.

Moreover, the inﬂuence of the external ﬁelds causes a shift in the critical temperature

of the BCS model, which is described by linearized GL theory in the same scaling regime.

More precisely, it has been shown in [22, 19], and [16] that the critical temperature shift

in BCS theory is given by

Tc(h) = Tc(1 −Dch2)(1.2)

to leading order, where Dcdenotes a critical parameter that can be computed using

linearized GL theory.

The present work is an extension of the paper [16], where the case of a constant

magnetic ﬁeld was considered. In this article, we incorporate periodic electric ﬁelds Wand

general vector potentials Athat give rise to periodic magnetic ﬁelds. This, in particular,

generalizes the results in [21, 22] to the case of general external magnetic ﬁelds with non-

zero ﬂux through the unit cell. We show that within the scaling introduced above, the

Ginzburg–Landau energy arises as leading order correction on the order h4. Furthermore,

we show that the Cooper pair wave function admits the leading order term (1.1) and

that the critical temperature shift is given by (1.2) to leading order. The main technical

novelty of this article is a further development of the phase approximation method, which

has been pioneered in the framework of BCS theory for the case of the constant magnetic

ﬁeld in [19] and [16]. It allows us to compute the BCS energy of a class of trial states

(Gibbs states) in a controlled way. This trial state analysis is later used in the proofs

of the upper and of the lower bound for the BCS free energy. The proof of our lower

bound additionally uses a priori bounds for certain low-energy BCS states that include

the magnetic ﬁeld and have been established in [16].

February 14, 2023 3 Deuchert, Hainzl, Maier

1.2 Gauge-periodic samples

Our objective is to study a system of three-dimensional fermionic particles that is subject

to weak and slowly varying external electromagnetic ﬁelds within the framework of BCS

theory. Let us deﬁne the magnetic ﬁeld B:=h2e3. It can be written in terms of the vector

potential AB(x):=1

2B∧x, where x∧ydenotes the cross product of two vectors x, y ∈R3,

as B= curl AB. To the vector potential ABwe associate the magnetic translations

T(v)f(x):= eiB

2·(v∧x)f(x+v), v ∈R3,(1.3)

which commute with the magnetic momentum operator −i∇+AB. The family {T(v)}v∈R3

satisﬁes T(v+w)=eiB

2·(v∧w)T(v)T(w)and is therefore a unitary representation of the

Heisenberg group. We assume that our system is periodic with respect to the Bravais

lattice Λh:=√2π h−1Z3with fundamental cell

Qh:=h0,√2π h−1i3⊆R3.(1.4)

Let bi=√2π h−1eidenote the basis vectors that span Λh. The magnetic ﬂux through

the face of the unit cell spanned spanned by b1and b2equals 2π, and hence the abelian

subgroup {T(λ)}λ∈Λhis a unitary representation of the lattice group.

Our system is subject to an external electric ﬁeld Wh(x) = h2W(hx)with a ﬁxed

function W:R3→R, as well as a magnetic ﬁeld deﬁned in terms of the vector potential

Ah(x) = hA(hx), which admits the form A:=Ae3+Awith A:R3→R3and Ae3as

deﬁned above. We assume that Aand Ware periodic with respect to Λ1. The ﬂux of

the magnetic ﬁeld curl Ahthrough all faces of the unit cell Qhvanishes because Ahis a

periodic function. Accordingly, the magnetic ﬁeld curl Ahhas the same ﬂuxes through the

faces of the unit cell as B.

The above representation of Ahis general in the sense that any periodic magnetic ﬁeld

ﬁeld B(x)that satisﬁes the Maxwell equation div B= 0 can be written as the curl of a

vector potential ABof the form AB(x) = 1

2b∧x+Aper(x), where bdenotes the vector with

components given by the average magnetic ﬂux of Bthrough the faces of Qhand Aper is

a periodic vector potential. For more information concerning this decomposition we refer

to [40, Chapter 4]. For a treatment of the two-dimensional case, see [49].

1.3 The BCS functional

In BCS theory a state is conveniently described by its generalized one-particle density

matrix, that is, by a self-adjoint operator Γon L2(R3)⊕L2(R3), which obeys 06Γ61

and is of the form

Γ = γ α

α1−γ!.(1.5)

Here, αdenotes the operator αwith the complex conjugate integral kernel in the position

space representation. Since Γis self-adjoint we know that γis self-adjoint and that αis

symmetric in the sense that its integral kernel satisﬁes α(x, y) = α(y, x). This symmetry

is related to the fact that we exclude spin degrees of freedom from our description and

assume that all Cooper pairs are in a spin singlet state. The condition 06Γ61implies

that the one-particle density matrix γsatisﬁes 06γ61and that αand γare related

through the inequality

αα∗6γ(1 −γ).(1.6)

February 14, 2023 4 Deuchert, Hainzl, Maier

1.3 The BCS functional

Let us deﬁne the magnetic translations T(λ)on L2(R3)⊕L2(R3)by

T(v):= T(v) 0

0T(v)!, v ∈R3.

We say that a BCS state Γis gauge-periodic provided T(λ) Γ T(λ)∗= Γ holds for any

λ∈Λh. This implies the relations T(λ)γ T (λ)∗=γand T(λ)α T (λ)∗=α, or, in terms

of integral kernels,

γ(x, y)=eiB

2·(λ∧(x−y)) γ(x+λ, y +λ),

α(x, y)=eiB

2·(λ∧(x+y)) α(x+λ, y +λ), λ ∈Λh.(1.7)

We further say that a gauge-periodic BCS state Γis admissible if

Trhγ+ (−i∇+AB)2γi<∞(1.8)

holds. Here Tr[R]denotes the trace per unit volume of an operator Rdeﬁned by

Tr[R]:=1

|Qh|TrL2(Qh)[χRχ],(1.9)

where χdenotes the characteristic function of the cube Qhin (1.4) and TrL2(Qh)[·]is the

usual trace over an operator on L2(Qh). By the condition in (1.8), we mean that χγχ

and χ(−i∇+AB)2γχ are trace-class operators. Eqs. (1.6), (1.8), and the same inequality

with γreplaced by γimply that α,(−i∇+AB)α, and (−i∇+AB)αare locally Hilbert–

Schmidt. We will rephrase this property as a notion of H1-regularity for the kernel of α

in Section 2 below.

Let Γbe an admissible BCS state. We deﬁne the Bardeen–Cooper–Schrieﬀer free

energy functional, or BCS functional for short, at temperature T>0by the formula

FBCS

h,T (Γ) := Trh(−i∇+Ah)2−µ+Whγi−T S(Γ)

−1

|Qh|ˆQh

dXˆR3

dr V (r)|α(X, r)|2,(1.10)

where S(Γ) = −Tr[Γ ln(Γ)] denotes the von Neumann entropy per unit volume and µ∈R

is a chemical potential. The interaction energy is written in terms of the center-of-mass

and relative coordinates X=x+y

2and r=x−y. Throughout this paper, we write, by

a slight abuse of notation, α(x, y)≡α(X, r). That is, we use the same symbol for the

function depending on the original coordinates and for the one depending on Xand r.

The natural space for the interaction potential guaranteeing that the BCS functional

is bounded from below is V∈L3

/2(R3) + L∞

ε(R3), that is, the set of interaction poten-

tials, for which Vis relatively form bounded with respect to the Laplacian. Under these

assumptions it can be shown that the BCS functional satisﬁes the lower bound

FBCS

h,T (Γ) >1

2Trhγ+ (−i∇+AB)2γi−C(1.11)

for some constant C > 0. In other words, the BCS functional is bounded from below and

coercive on the set of admissible BCS states.

The normal state Γ0is the unique minimizer of the BCS functional when restricted to

admissible states with α= 0 and reads

Γ0:= γ00

0 1 −γ0!, γ0:=1

1+e((−i∇+Ah)2+Wh−µ)/T .(1.12)

February 14, 2023 5 Deuchert, Hainzl, Maier

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MicroscopicDerivationofGinzburgLandauTheoryandtheBCSCriticalTemperatureShiftinthePresenceofWeakMacroscopicExternalFieldsAndreasDeuchert,ChristianHainzl,MarcelMaier(bornSchaub)February14,2023AbstractWeconsidertheBardeenCooperSchrieer(BCS)freeenergyfunctionalwithweakandmacroscopicexternalelectrica...

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Microscopic Derivation of GinzburgLandau Theory and the BCS Critical Temperature Shift in the Presence of Weak Macroscopic External Fields

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