Quantum Phase Transitions

1
Quantum Phase Transitions and Non-uniform Phases
in Low-dimensional Superconductors
Wei Zhang and Carlos A. R. Sá de Melo
2005 APS March Meeting Los Angeles, March 2005
2
Magnetic Order V.S. Superconductivity
Copper Oxides
Mutually exclude? OR Coexist?
Strontium Ruthenate
Ferromagnetic Superconductors
Organic Superconductors
Coexisting, inhomogeneous, striped phase
3
Phase Diagram of (TMTSF)2PF6
D. Jérome, et al., Mol. Cryst. Liq. Cryst. 79,
155 (1982).
?
T. Vuletic, et al., Eur.Phys.J. B 25, 319 (2002)
4
Proof of Coexistence
Superconducting phase onsets for different H
Proton spin-lattice relaxation time and local
field (believed characterized SDW) show no
dependence on TS transition.
I. J. Lee, et al., (2005)
5
Upper Critical Magnetic Field

• Hb2 can reach as large as 9T

• Pauli Paramagnetic Limit 2.2T

• Hc2 shows upper curvature behavior from 0.3T

I.J. Lee, et al., Phys. Rev. B 62, R14 669 (2000)
I.J. Lee, et al., Phys. Rev. Lett. 88, 207002
(2002).
6
Proposed AF-TS Coexistence Region

• Coexisting phase is inhomogeneous and can
• exist in a narrow range of pressures.

• Stripe pattern is proposed.

7
Quasi 1-D System
S
D

• Spin density wave order parameter S

• Triplet superconductivity order parameter D

8
Ginzburg-Landau Free Energy

• TS State

• Typical GL free energy

• Extra terms in expansion, which are relevant
  away from critical point

• AF State Analogous

9
Ginzburg-Landau Free Energy (Contd)

• Interface coupling energy

• If , approximate order
  parameters by linear functions,

Where .
10
Ginzburg-Landau Free Energy (Contd)
Positive interface energy
Negative interface energy
Mutually exclude
Coexist
Bulk phase
Stripe phase
11
Variation Method and Results

• Minimize 0-order free energy w.r.t. two order
  parameters

• ,

• TS solutions Analogous

• Stripe length

iAF, TS
12
Pressure Dependence

• From P1 to P2, AF stripes shrink while TS
  stripes expanding

• AF parameters

• TS parameters Analogous

• Interface parameter

13
Continuous Phase Transition
Exponents
Dimensionless parameters
14
Discontinuous Phase Transition
Exponents
Dimensionless parameters
15
Microscopic Description
where
TS
AF
Action
Partition Function
16
Auxiliary Field and Functional Integral
Introducing auxiliary field
TS
AF
Partition Function
where
17
Time Independent Case
where
with
K1 and K2 are linear transforms of TS and AF
order paramenters, respectively.
18
Expansion around saddle point near Tc

• Quadratic Term

• Quartic Term

19
Coupling Term in S4

• Consider anisotropy of the system

AF
TS
Where is the leading term of momenta
expansion of ,
With is the Matsubara summation over all
dependent terms.
Negative or Positive coupling energy
Negative or Positive
20
Conclusions and Future Topics

• Coexistence region of AF and TS phases is
  proposed under the existence of negative AF-TS
  coupling.

• The intermediate phase is inhomogeneous and
  consists of alternating AF and TS stripes.

• By changing pressure dependence of parameters,
  a bicritical, tricritical, or tetracritical point
  can be presented.

• Stripe Pattern

Magnetic field

• Conditions for the negative AF-TS coupling

• Calculate other G-L parameters