SOVIET PHYSICS JETP VOLUME 36(9), NUMBER 6 DECEMBER, 1959

MICROSCOPIC DERIVATION OF THE GINZBURG-LANDAU EQUATIONS IN THE THEORY OF SUPERCONDUCTIVITY

L. P. GOR'KOV

Institute for Physical Problems, Academy of Sciences, U.S.S.R.

Submitted to JETP editor February 3, 1959

J. Exptl. Theoret. Phys. (U.S.S.R.) 36, 1918-1923 (June, 1959)

It is shown that the phenomenological Ginzburg-Landau equations follow from the theory of superconductivity in the London temperature region in the neighborhood of Tc. In these equa-tions there occurs, however, twice the electronic charge; this is related to the physical mean-ing of >l!(x) as the wave function for Cooper pairs. The constant K turns out to be small. The problem of the surface energy for the boundary between the normal and superconducting phases in the neighborhood of T c is discussed.

IT is well known that the behavior of superconduc-tors in a magnetic field in the neighborhood of the critical temperature is qualitatively well described by Ginzburg and Landau's phenomenological theory. 1

We shall show in the present paper that equations of the same type as those in Ginzburg and Landau's theory follow indeed from the theory of supercon-ductivity.

We shall start from the equations for the ther-modynamic Green functions3•4 which we obtained earlier.2 These equations are in a magnetic field of the form

{- _?__ + -1 (_?__- ieA (r) \)2 + G (x x') a-. 2m ar ,

+ 11 (r) p+ (x, x') = o (x- x'),

+ 2!1 + ieA (r)Y + !L} p+ (x, x')

- 11* (r) G (x, x') = 0; (1)

where

11* (r) = gf+ ('•. r; '• r). (1')

The last relation expresses in terms of the coupling constant and the value of the function F+(x, x') for equal arguments. If there is no field,

= const. The quantity is in that case the en-ergy gap in the spectrum. F+(x, x') tends, in fact, logarithmically to infinity as x = x'; this corre-sponds to the well known logarithmic divergence in the equation determining the gap when one integrates in momentum space. In the papers by Bardeen, Cooper, and Schrieffer5 and Bogolyubov6 this diver-gence was removed by cutting-off the energy of the interacting electrons at distances of the order of the Debye frequency w from the Fermi surface; we

shall do the same in the following. One verifies easily that such a cut-off corresponds in the co-ordinate representation to a "spread out" in Eq. (1') over a distance of the order tiv /w .

Let us go over to the Fourier components of the Green functions G(x, x') and F+ (x, x') with respect to the difference of the variables4 u = T -T', for in-stance,

n +!IT

m ( ') - 1 \ '"'n" G ( '. ) d ..v., r, r - 2 ) e r, r, u u, -l!T

where wn = 7r(2n + 1)T (n = ... -1, 0, 1, ... ) . Equa-tions (1) go over into the following equations for the Fourier components ®w(r, r') and & (r, r'):

{iwn + (fr- ie A (r) Y + :-"} @l., (r, r')

+ 11 (r)l};i"(r, r') o (r-r').

f . ' 1 (. a . . A ( l)2 . } q:+ ( r') IWn 1 2m Fr, te r , 0 ., r,

-/1*(r)@lo, (r, r') = 0,

and condition (1') can be written in the form

j.* (r) = T lJ &t (r, r). n

(2)

(2')

We introduce, finally, <liw (r, r'), the Fourier component of the Green function for an electron in the normal metal in a magnetic field. The equation satisfied by @iw(r, r') can be written in two ways:

or

1364

Striped Ground States in the Hubbard model

• Brief Intro to the Hubbard model• Four powerful simulation methods• Results for U=8, 1/8 doping:

- Consensus on the phase of the ground state- Stripes with nearly degenerate wavelengths

• Some additional DMRG results for pairing on 4 and 5 leg cylinders

H = �X

hiji,�

tij(c†i�cj� + c

†j�ci�) + U

X

i

ni"ni#.

0 0.1 0.2 0.3 0.4 0.5T/t

-0.76

-0.72

-0.68

-0.64

E/t

DMETDMET - TLUCCSDFNFN - TLDMRGDCADCA - TLAFQMC - TL

-0.77

-0.76

-0.75

-0.74

2x2

4x2

8x2

4x4

TL

4

6

202TL

122

n=0.875, U=8

T=0

162

Energies vs T, 1/8 doping, U=8

We studied a wide range of doping, U/t, temperature—in thermodynamic limit. For most of parameter space, there was good agreement in energies and other properties.

For parameters relevant to the cuprates (U/t ~ 8)—the toughest to simulate—the methods disagreed on the nature of the ground state.

A smaller group of us then decided to focus on this toughest region to see if we could resolve differences

Organized by the Simons Collaboration on the Many Electron Problem

T=0, U=8, 1/8 doping

maximum uncertainty in the phase, maximum inhomogeneity

What is the ground state phase?

• Density matrix renormalization group (DMRG)

- Weakness: finite cylinder size• Density Matrix Embedding theory (DMET)

- Weakness: finite cluster size• Infinite projected entangled pair states (IPEPS)

- Weakness: finite bond dimension and extrapolation• Constrained Path Monte Carlo (CPMC, AFQMC)

- Weakness: Constraint based on trial wavefunction

A key aspect of the work is that the weaknesses are very different. Thus, if multiple methods agree, we can have high confidence we have the right answer.

Dec ’17 Science

By focusing one one point in the phase diagram, we were all able to improve our results substantially.

Energy extrapolated to thermodynamic limit

Overall uncertainty almost an order of magnitude reduced from previous benchmark

Error bars neglect systematic errors—that is what we need the comparison for.

Uniform state versus stripes• DMET and iPEPS both can be forced to give

uniform states:• DMET has a cluster size. For a 2x2 cluster, no

stripe patterns can form

• iPEPS similarly has a cluster that is repeated to infinity. A 2x2 cluster cannot have stripes

• DMRG always gives stripes. Currently no way to force a uniform state. CPMC also gives stripes as lowest energy state.

uniform d-wave

Both methods’ uniform states show d-wave pairing.

Each is higher in energy than a striped state.

0 0.05 0.1 0.15 0.2 0.25 0.3−0.77

−0.76

−0.75

−0.74

−0.73

−0.72

−0.71

−0.7

−0.69E

1/D or w

E(1/D): uniformE(w): uniformE(1/D): W5 stripeE(w): W5 stripeE(1/D): diagonal stripeE(w): diagonal stripe

0 0.05 0.1 0.15 0.2 0.25 0.3−0.77

−0.76

−0.75

−0.74

−0.73

−0.72

−0.71

−0.7

−0.69

E

1/D or w

E(1/D): W5 stripeE(w): W5 stripeE(1/D): W7 stripeE(w): W7 stripeE(1/D): W8E(w): W8

New iPEPS Energy Extrapolation method (Corboz)

Uniform vs diagonal stripes vs. vertical stripes

Vertical stripes with different fillings/spacings

Vertical stripes: filling, wavelengthFilled Stripe

f = 1 λ = 8

Half filled Stripe

f = 1/2 λ = 4

Filled stripes were found with Hartree Fock in late 80’s—but HF may not be not accurate

Half-filled stripes were found in some cuprates in the mid 90’s. A few years later, DMRG on the t-J model showed half-filled stripes (White & Scalapino) DMRG on the t-J model—

formation of two half-filled stripes

The magnetic wavelength is 2 λ

Zaanen

Tranquada

Vertical stripes: Energy versus wavelength

We find a remarkable near-degeneracy for states with different stripe wavelengths, with λ=8 very slightly lower in energy, and λ=4 significantly higher.

The near degeneracy likely points towards disordered stripes and/or fluctuating stripes.

Pairing and stripes “intertwined” on Lx4 cylinders (DMRG)

position along cylinder

density

pairing

Chia-Min Chung

0.05 0.029

0.1

(c)Response to a uniform d-wave pairing field

D=0.05

Red numbers are stripe fillings on a

16x4 system

Pairing and stripes “intertwined” on Lx4 cylinders (DMRG)

Chia-Min Chung

0.25 0.021

0.3

(c)

The pairing response is much weaker on a single long stripe than at the boundary

between different stripe fillings

To force a lengthwise stripe, chemical potential on bottom leg

and tx = 1.2 ty

Conclusions• We are closing in on solutions to Hubbard models with numerical

simulations!

• Using four different methods with very different uncontrolled errors, we have converged to a consistent general picture of the Hubbard ground state at U=8, near 1/8 doping

• The system exhibits d-wave pairing but intertwined with stripes. The biggest response occurs at the transition between different stripe fillings. (Disordered or fluctuating stripes…)