Transcript Mott –Hubbard Transition & Thermodynamic Properties in Nanoscale Clusters. Armen Kocharian (California State University, Northridge, CA) Gayanath Fernando (University of Connecticut, Storrs, CT) Jim Davenport (Brookhaven.

Mott –Hubbard Transition &
Thermodynamic Properties in
Nanoscale Clusters.
Armen Kocharian
(California State University, Northridge, CA)
Gayanath Fernando (University of Connecticut, Storrs, CT)
Jim Davenport (Brookhaven National Laboratories, Upton, NY)
Kalum Palandage (University of Connecticut, Storrs, CT)
Outline
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•
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•
Motivation
Small Hubbard clusters (2-site, 4-site)
Ground state properties
Exact Thermodynamics
– Charge dos and Mott-Hubbard crossover
– Spin dos and AF Nee l crossover
– Phase diagrams
• QMC calculations in small clusters
• Conclusions
Quantum Monte Carlo
h=0
•
Exact analytical
results and QMC
Motivation
Electron Correlations
- Large Thermodynamic System:
• Interplay between charge and spin degrees
• Mott-Hubbard Transition
• AFM-PM (Nee l Transition)
• Magnetic and transport properties
-Nanoscale Clusters:
• Mott-Hubbard crossover?
• Charge and spin degrees?
• AFM-PM crossover?
Finite size Hubbard model
• Simplest lattice model to include correlations:
 Tight binding with one orbital per site
 Repulsion: on-site only
 Nearest neighbor hopping only
Magnetic field
• Bethe ansatz solution [Lieb & Wu. (’67)]
 Ground state but not correlation functions
Finite size Hubbard cluster
Ground state (T=0)
• Weak correlations in 1d systems:
 power law decay (Schulz ’91, Korepin & Frahm ’90)
• Long range order in finite clusters:
 saturated ferromagnetism (Nagaoka’65)
Thermodynamics (T≠0)
• No long range correlations:
no magnetic order in 1d (Mermin & Wagner. ’66, Ghosh ’71)
• Signature of short range correlations:
 weak magnetization (Aizenman & Lieb’90)
correlations decay faster than power law like (Koma & Tasaki ’92)
Large Clusters:
Lieb & Wu. (’67)
• Bethe-ansatz calculations
Jarrell et al. (’70)
• Lanczos
Dagotto et al. (’ 84)
• Monte Carlo
• Numerical diagonalization Canio et al. (’96)
• DMFT
Kotliar. et al. (’97)
Small Clusters:
• Exact analytical diagonalization
• Charge and spin gaps (T=0)
• Pseudogaps (T≠0)
Mott-Hubbard transition:
• Temperature
• Magnetic field
AF-PM Transition:
• Exchange
• Susceptibility
Kotliar (’67)
Canio et al. (’96)
HTSC superconductivity: Schrieffer et al. (’ 90)
• Pseudogap formation
• Chemical potential (n≠1)
Two phase transitions in Hubbard
Model
Mott-Hubbard Phase TMH
M. Cyrot et al. (’70)
From D. Mattis et al.
(’69)
J. R. Schrieffer et al. (’70
Neel Magnetic Phase TN
Mott Hubbard and AF transitions
TN consequence of MottHubbard phase
Anderson (’97)
TMH consequence of Nee l
anti-ferromagnetism
Slater (’51)
Approaching to MH phase
from insulator: T↑,U↓
Hubbard (’64)
Approaching to TMH from
metallic state: U↑, T↓
Brinkman et al. (’70)
Evolution of dos and pseudogaps, TMH and TN
for 2 and 4 site clusters at arbitrary U, T and h
Thermodynamics of small clusters
High temperature peak – MH transition
From Shiba et al., (‘72)
Specific heat
of finite chains
N=2, 3, 4, 5
Low temperature peak – AFM-PM
Focus on 2 and 4-site clusters
Harris et al. (’72)
Kocharian et al. (’ 96)
Shiba et al. (’70)
Shumann (’02)
A single hydrogen molecule acting as a nanowire
Mott-Hubbard Transition
AFM-PM Transition
Driven by h and T
Exact ground state properties
Exact mapping of 2-site Hubbard and Heisenberg
ground states at half filling (A. Kocharian et al. ’91, 96):
e.g., hC=J(U)
hC - critical field of ferromagnetic saturation
Ground state charge gap (N=2)
Half filling 
e.g., h<hC
e.g., h≥hC
•
Gap is monotonic versus U
and non monotonic versus h
Ground state charge gap (N=1, 3)
Quarter and three quarter fillings 
e.g., h<hC
•
e.g., h≥hC
Charge gap versus h and U
is monotonic everywhere
Exact thermodynamics (T≠0)
2 sites: n 24
4 sites: n 44
h=0
•
Number of particles N
at h=0 versus µ and T
•
Sharp step like behavior
only in the limit T  0
N versus chemical potential (T/t=0.01)
h=0
h=0
Real plateaus
exist only T=0
(not shown)
Chemical potential in magnetic field
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•
•
h/t=2.0,
U/t=5.0
Number of particles N
at h/t=2 versus µ and T
Plateau at N=2
decreases with h
Sharp step like behavior
only in limit T  0
•
Plateaus at N=1 and
N=3 increases with h
Magnetic susceptibility χ at half filling
•
•
Susceptibility versus
h at T=.05

As temperature T  0
peaks of χ closely tracks
U dependence of hC (U)
hc(U)/t
4
U/t
Number of electrons vs. μ clusters
h=0
•
Plateaus at integer N exist only at T=0
(not shown in figure)
Charge pseudogap at infinitesimal T≠0
h=0
Charge and spin dos in 2-site cluster
U=6 and h=2
•
Charge dos for general
N has four peaks
•
Spin dos at half filling
has two peaks
Thermodynamic charge
dos and pseudogap
U=0 and h=0
U=5 and h=0
Two peaks merge in
Saddle
one
peak point
saddle
point
TMH
•
Charge dos for general
U≠0 has four peaks
•
Charge pseudogap
disappears at TMH
Charge dos and pseudogap
σ
h=0
•
Charge dos for general
N has four peaks
h=2t
Charge dos for general
N has four peaks
Spin dos and pseudogap
U=6
Saddle point
•
spin pseudogap at TN
disappears (saddle point)
•
Spin dos at half filling
has two peaks
Thermodynamic charge and spin dos
σ
•
Charge dos for general
N has four peaks
•
Spin dos at half filling
has two peaks
Weak singularity in charge dos
MH Transition at half-filling (N=2)
•
True gap at μC=U/2 exists
only at T=0
•
Infinitesimal temperature
smears ρ(μC)≠0 and
results in pseudo gap
•
At TMH, ρ(μC)≠0 and
ρ′(μC)=0 ρ″(μC)>0 . It is
a saddle point
n 1
•
Forth order MH
phase transition
Weak singularity in spin dos
Neel Transition at N=2
•
True gap exists only at
T=0
•
Infinitesimal temperature
smears σ(0)≠0 at h=0 and
results in pseudo gap
•
At TN, σ(0)≠0 and σ′(0)=0
σ ″(0)>0. . It is a saddle
point
n 1
•
Forth order Nee
phase transition
l
Weak singularity in charge dos
TMH versus μ
MH crossover
N=2
Bifurcations at
μ=U/2 &
μ≠U/2
•
Distance between charge peak
positions versus temperature
Weak singularity in spin dos
TN versus h
crossover
•
Distance between spin peak
positions versus temperature
N=2
Spin magnetization
•
Magnetization at quarter
filling (no spin gap)
•
Magnetization at half
filling (spin gap)
Magnetization versus h
h=0
•
No spin gap at N=1 and 3
Zero field spin susceptibility (N=2)
•
TN from
maximum
susceptibility
•
TN from peaks
distance
TN temperature versus U
•
T•N versus
TN from
U spin
(AF dos
gap)peaks
•• TTNF from
maximum
versus
U
spin susceptibility
• of
(Ferro
gap)
Zero field magnetic susceptibility χ
h=0
At U/t»1 χ increases
linearly
N=2
•
At large U magnetic susceptibility ~T
Spin susceptibility
N=2
•
Susceptibility at half
filling (UC/t=6)
h/t=2
N=1
•
Susceptibility at quarter
filling (no spin gap)
Phase diagram, TMH versus U
h=0
•
At t=0
TMFMH =U/2
result at t=0. D.Mattis’69
•
At t=0
TMH = U/2ln2
and ρ(µC )=2ln2/5U
Phase diagram, TMH versus U
TMH
•
h=0
Staggered magnetization i (-1)x+y+z (spin at site i)
Phase Diagram at half filling
T/t N
4
TMH
MH
TN
MH + AF
•
U/t
TMH & TN versus U, at which pseudogap disappears
Mott-Hubbard crossover
N=2
σ
• TMH versus h and U
4-site clusters
h=0
U 2  16t 2
U  16t
2
  arccos
3
2
4  3 t 2U2
U
2
 16t
2

3

s  sz  0
4-site clusters
36t U  U 

  arccos
U  48t 
2
2
3
2
3

s 1
s z  1,0, 1
4-site clusters
h=0
U 2  16t 2
U  16t
2
2

s 1
s z  1,0, 1
N versus μ in 4 site cluster
No gap at
U=0 and N=4
h=0
T/t=0.01
U/t=4.0
•
Plateaus at integer N exist only at T=0
(not shown in figure)
Weak singularity in charge dos
h=0
Bifurcations at N = 2, 4 and 6
Weak singularity in charge dos
h=0
Bifurcations at N = 1, 2, 4, 6 and 7
Thermodynamic dos for 4-site cluster
h=0
Analytical calculations
DMFT calculations
Quantum Monte Carlo
h=0
•
Exact analytical
results and QMC
Quantum Monte Carlo studies
h=0
•
Magnetism and MH
crossovers in rings
and pyramids
QMC studies of small clusters
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
5 sites pyramids
h=0
•
Magnetization versus h
5 sites pyramids
h=0
•
Staggered magnetization versus n
14 sites pyramids
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
14 sites pyramids
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
Conclusions
• Exact mapping Hex~HU in the ground state
• True spin and charge gaps exist only at T=0
ECGap(U)≠ESGap(U ) at U≠0
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Pseudogaps appear at infinitesimal T
Charge dos - MH crossover (TMH >TN )
Spin dos - AFM-PM crossover (TN )
Temperature driven bifurcation – generic feature
1d Hubbard model, UC=0 and true gap in ρ(μC)=0 exists
only at T=0 and n=1
• 2 and 4 site Hubbard clusters reproduces main features of
small and large system
• Evolution of pseudogap versus μ in HTSC
Rigid spin dynamics
h=0
QMC studies of small clusters
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
14 sites pyrmids
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
QMC studies of small clusters
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
14 sites pyrmids
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
14 sites pyrmids
h=0
•
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Staggered magnetization (spin at site i)
Staggered magnetization i (-1)x+y+z (spin at site i)
14 sites pyrmids
h=0
•
Staggered magnetization i (-1)x+y+z (spin at site i)
4-site clusters
s z  1,0, 1
s  sz  0
h=0
s z  1,0, 1

s 1
Ground state charge gap
•
Gap is monotonic versus U
and non monotonic versus h