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关联多体系统中的前沿物理问题
Levitate
张广铭
清华大学物理系
Waves
Electromagnetic Interaction: Old Story
Maxwell
1831--1879
Coulomb
1736-1806
Faraday
1791－1867
Ampere
1775-1836
Faraday Law of Induction
Questions:
• What are the fundamental microscopic laws of nature?
• From the microscopic world to the macroscopic one，
what new principles and laws emerge ?
对自然的两种不同看法/研究方法
• 还原论 (Reductionism)：
一切归结为最基本的组成单元和决定单元行
为的最基本规律。“最终目的”是建立包罗万象
的“大统一理论”（Theory of Everything）
• 演生论 (Emergence) 或整体论 (Whoism)：
客观世界是分层次的，每个层次都有自己的基
本规律，重要的是承认客观现实，以它为依据，
找出它的基本规律，理解这些现象是如何“演生”
的。
这两者是对立的，但又是互补的！
Philip W. Anderson:
More is different (1972)
更多是不同的
······将万事万物还原成简单的基本规律，并不蕴含着从这些
规律出发重建宇宙的能力······
面对尺度与复杂性的双重困难，重建论的假定就崩溃了。
不能依据少数粒子的性质作简单外推来解释由大量粒子构成
的复杂集聚体的行为。正好相反，在复杂性的每一个层次会
呈现全新的性质，为理解这些新行为所需要作的研究，就其
基础性而言，与其他研究相比毫不逊色。
Charge-Spin Interaction: Emergent Quantum Phenomena
Copper-oxide high-Tc superconductors
Charge dynamics
Iron-based superconductors
Spin dynamics
Heavy fermion, Kondo effect
Quantum Hall effect
Colossal magnetoresistance
Superfluid-Mott insulator transition
Metal Na
Organic conductors
Giant thermoelectric materials
... …
SmAsFeO0.86-xFx
superconductor
YBa2Cu3O6 magnetic
insulator
Emergent phenomena with characteristic energy scale
新奇量子现象
能量
10-6
原子BEC
10-4
10-2
重费米子超导
3He超流
100
4He超流
102
高温超导
量子霍耳效应 庞磁阻
104
半导体
金属
生物过程
化学过程
物理过程
从简单体系到复杂体系
温度(K)
Quantum Magnets: Haldane Conjecture
Integer Heisenberg spin chain has a finite excitation gap
H   Si  Si 1
Haldane
i
Ni2+ ：S = 1
Energy gap
Y2BaNiO5
Hidden Order
Hidden “antiferromagnetic” order
000000000000000
Is there any order in the

 
 
Haldane spin chain?


S = -1 () , 0, 1 ()
Nd2BaNiO5
Antiferromagneti Neel order
Neel
Nobel 1970
YBa2Cu3O6

Novel Features of Quantum States of Matter
• Linear T resistivity
• Fermi arc
• Dark entropy
• Linear T susceptibility
ab 面内电阻率线性温度依赖
Fermi Arc vs Quantum Oscillation:
But there are quantum oscillations
associated with a closed Fermi
surface!
Fermi surface is not closed!
kb
ka
Fermi surface of underdoped highTc cuprate
Quantum oscillation of resistance
in YBa2Cu3O6.5
Dark Entropy:
High temperature entropy:
S  T
The system is no longer
renormalizable!
熵缺失
Entropy is not conserved! To recover the missing entropy, one needs to go to a
temperature as high as 10000 K
Universal linear T susceptibility in iron-based superconductors
BaFe2As2
Universal linear susceptibility
Cr and Cr1-xRux alloys
Zhang et al., EPL 86, 37006 (2009).
Paramagnetic susceptibility in a metal
Curie Law of magnetic impurity
FeBr(C44H28N4)
Pauli susceptibility
 (T )  0  B2    F 
Pauli
1
  F 
2
C

T
Pierre Curie
1859-1906
E  2 B B
Short range antiferromagnetic fluctuations
 (T ) 
  s 2
J
 0.05190  0.03726 TJ 
Zhang et al., EPL 86, 37006 (2009).
Frontiers in Correlated Matter
Early in August 2004, leading condensed physicists convened for a summit on
"Frontiers in Correlated Matter'' in Colorado, USA.
Eleven big questions of correlated matter
1. What fundamentally new classes of matter await discovery?
2. What is the origin of high temperature superconductivity?
3. What is the nature of strange metals?
4. What new principles of the cosmos can be discovered from a study of condensed matter?
5. Is quantum computation feasible?
6. Why don't glasses flow like liquids?
7. What principles govern the organization of matter away from equilibrium?
8. Can statistical mechanics be applied to a system as complex as the living cell?
9. How do singularities form in collective matter and in space time?
10. What principles govern the flow of granular materials?
11. What are the physical principles of biological self-organization?
1. What fundamentally new classes of matter await discovery?
When matter develops new forms of organization, its properties transform in
unexpected ways. Crystals become solid. Magnets produce an external
magnetic field. Superfluids flow without resistance. Superconductors expel
magnetic fields. These new states of matter were unexpectedly discovered,
mostly in the 20th century.
There are roughly 100 elements, so each time we add one more element to a
crystal, the number of new types of material that become possible grows by at
least a factor of 100. At the turn of the 21st century, scientists have barely
begun to explore the 10,000,000 plus materials that become possible with four
elements. What new states of matter - beyond such recently discovered
examples as high temperature superconductivity and the fractional quantum
Hall effect hide out in this frontier, awaiting discovery?
New materials, New states of matter, New properties.
2. What is the origin of high temperature superconductivity
and is room temperature superconductivity feasible?
New materials have recently been discovered which lose all of their electrical
resistance to become "superconducting" at a transition temperature as as high as
160 degrees absolute. This is a far higher temperature than the previous generation
of superconductors, much higher than ever thought feasible - but still not room
temperature.
Room temperature superconductors - if they are feasible- would be of tremendous
importance for a wide range of applications ranging from the power to the medical
industry. Answering the vital question about the feasibility of room temperature
superconductivity requires that we solve the mechanism of high temperature
superconductivity. Many believe that its ultimate answer will require a
fundamentally new mathematical description of the collective behavior of quantum
matter.
Mechanism of high Tc and the room T superconductors
3. What is the nature of strange metals?
When we tune metals to the brink of instability, they develop strange new
properties and a propensity to develop new forms of order, such as
superconductivity. Inside these materials, electricity can no longer be described
as a fluid of individual electrons, but appears to require a fundamentally new,
collective description.
Nature of non-Fermi liquid metals
4. What new principles of the cosmos can be discovered
from a study of matter in the laboratory?
The principles that govern the properties of matter studied in a laboratory
environment - also govern cosmological matter. Tremendous insight into the
cosmos at large has been gained in the past from laboratory experiments.
Studies of diamond led Einstein to propose the quantum nature of energy inside
matter. Similarly, research into how superconductors levitate provided the
conceptual insights that enabled physicists to understand the weak nuclear
forces that make the sun shine.
Many areas of current activity in condensed matter physics hold promise of
providing new insight into the cosmos at large. For example, the study of
quantum phase transitions inside the cryostat may provide new models for
phase transitions in the early universe. The study of glasses, which have many
stable minima, may have connections to many-valued solutions of string theory
and the study of superfluid Helium -3 may provide insights into quantum
gravity and unexplained smallness of the cosmological constant.
5. Is quantum computation feasible?
Theoretically, quantum mechanics can be used to develop a fundamentally new
type of computation, in which the computing power scales not linearly, but
exponentially with the size of the computer. Such computers could decode the
most sophisticated ciphers, could encode unbreakable codes and may be able to
solve the hardest "NP complete" problems in computer science. But can they be
made to work in practice?
Scientists are only just discovering how to make the simplest element of the
quantum computer - the qubit. To make a fully functioning quantum computer
will be the development of a new quantum technology that can successfully
isolate and protect the coherent quantum computer from the incoherent
classical world of its environment.
Topological quantum computation may be possible.
6. Why don't glasses flow like liquids?
Glasses are liquids that have become frozen in time: window glass requires times
longer than the age of cathedrals to flow. Analogous freezing arise in colloids and
magnetic materials. What are the physical principles governing these various
frozen fluids? In some cases, a flowing state can be restored by exceeding a
threshold of external force. In other words, such a force seemingly creates a
glass where none was before ('jamming'). When, and why, do stressed glasses
melt and freeze?
7. What principles govern the organization of matter away
from equilibrium?
Most 20th century physics describes matter in equilibrium, governed by the
overarching principles of thermodynamics and statistical mechanics. These
directly relate the macroscopic properties of a system to its underlying 'energy
landscape'. Increasingly we want to address systems far from equilibrium, such as
turbulent fluid flow, waves at the sea-shore, electricity driven through an atom,
and even life itself. Each of these systems is driven by an external force, often into
parts of the landscape far from those explored in equilibrium. What are the new
principles that govern the organization of matter in these new, non-equilibrium
situations?
8. Can statistical mechanics be applied to a system as
complex as the living cell?
Statistical mechanics describes the emergent properties of large collections of
atoms and molecules caused by thermal excitations. Such excitations are
certainly present in living cells, but there is much else happening too: the
processes of life drive the system far from a state of thermal equilibrium.
The growing field of nonequilibrium statistical mechanics addresses these
problems, but which aspects of cellular life can it help us to understand? Are
some of these aspects just too complicated for the 'physicists view' -- that
simplification is the first step towards understanding -- to be useful?
9. How do singularities form in collective matter and in
space time?
One basic behavior of matter---especially tenuous matter, is its characteristic of
creating sharp, singular spatial structure spontaneously. The vortex in a draining
sink and a crumpled sheet of paper are two examples. Often these structures are
fundamental consequences matter's essential nature---its local connectivity and
its inertia. What is the range of this singularity-forming behavior, to what degree
does it condense the system's energy into the singularity, and what analogous
singularities might occur in space time?
10. What principles govern the flow of granular materials?
Granular materials include sand in dunes, snow (or rocks) in avalanches,
sediments on the sea floor, and powders used commercially e.g. to make
pharmaceutical pills. As the grains collide with each other they dissipate energy,
causing new and unpredictable behavior.
Granular materials can sustain ripples and waves, including shock waves; can
evolve spontaneous avalanches; and can unmix (rather than mix as fluids would)
when stirred together. Even at rest, the grains can carry subtly imprinted
memories of earlier flows, through the particular set of positions and contacts
that remain. A deeper knowledge of the physical principles of granular flow will
help understand how the earth's surface came to be as it is, and unlock the door
to many other areas of science and technology.
11. What are the physical principles of biological
self-organization?
We only partially know how DNA encodes the structure of proteins, While the
"standard model" works for bacteria and viruses, we now are realizing that, for
multi-cellular organisms the transcription from DNA to mRNA is a highly "edited"
process. What is the machinery that controls this editing process? How does DNA
encode the software and the large-scale organization of an organism? At a higher
level, how do 109 bits of information in the human genome encode the
information for the 1011 neurons and the 1015 synapses of the human brain?
Topological ordered phases &
quasi particles with fractionalized
quantum numbers
Fractional quantum Hall states
as topological ordered phases
Klitzing
1. 不存在明显的对称性自
发破缺，相变点两边的
相都是无序的，低能激
发存在能隙
2. 不存在局域序参量
3. 不同的相可用不同的拓
扑数（如Chern数）来
表征
Tsui
Störmer
Fractional charge
zi    ( zi  z j ) e
m

 zi
2
/4
i
i j
Laughlin
1998
Laughlin wavefunction
 I   qI B
2
Saminadayar et al, PRL 79, 2526 (1997)
Non-Abliean Anyons & Quantum Computation
2D系统不仅存在玻色子、
费米子，还存在满足分数
统计的任意子
Wilczek
量子计算可通过操控非阿贝
尔任意子来实现
Kitaev
• 分数统计和分数电荷是孪生兄弟
• 可能存在非阿贝尔任意子系统
• 12/5, 5/2 分数量子霍耳效应系统
• px + ipy 超导系统
• Kitaev 模型
• ？
退相干是实现量子计算面临的最困难
的问题，拓扑激发（非阿贝尔任意子）
依赖于系统整体的性质，对局部的扰
动不敏感，具有抗退相干性
Novel Quantum Phase Transition
通常的连续相变：
• 相变点一边有序，
一边无序
• 对称性破缺
• 定域序参量
Landau
1936-37
拓扑量子相变：
z z
H= -J  i  j
ij
Lars Onsager
1903-76
拓扑对称是一种离散对称
• 相变点两边都是无序的
• 没有对称性破缺
• 不存在定域序参量
Exact solution of the 2D Ising Model
Quasiparticles with fractionalized quantum numbers
Soliton mechanism:
Soliton excitation in 1D is charge fractionalized
R. Jackiw C Rebbi
Soliton Mechanism of Fractional Charge: 1D
Polyacetylene
energy
1/2 state
Soliton solution
e* = +1/2 (occupied)
1/2 state
k
Energy band
-1/2 (not occupied)
Polyacetylene: Fractional Charge Conducting
Polyacetylene
Wu-Pei Su
Alen Heeger
1979, PRL
soliton conductor
How Does Charge Fractionalized in 2D?
DH Lee, GM Zhang, T Xiang, PRL 99, 196805 (2007)
2D vortex excitation in a topological degenerate ground state is charge fractionalized
1D and 2D charge fractionalization is unified: Jackiw-Rebbi mechanism
2D vortex = Edge soliton (1D)
奇异规范变换
   ~e
i
vortex
 '   ~ei  
总 结
•
关联量子系统蕴含着丰富而又新奇的量子相变和量
子态，是对微观量子理论及计算方法的挑战。
•
研究和解决基本问题，其目的就是要提出和发展一
些新的概念和新方法，建立描述关联量子系统的
“Maxwell方程组”，推动微观量子理论的发展。
谢谢大家！
Kondo effect and its basic physics
Kondo Effect – “golden anomalous” in low T
Simple metals (Pb, Nb, Al)
Nobel metals (Cu, Au)
Alloys with Fe/Co/Ni
Jun Kondo
• Anomalous at low T was first observed in 1930.
• Jun Kondo gave an explanation in 1964.
Resistivity vs T for various Mo-Nb alloys containing 1% Fe
M. Sarachik, et al., Phys. Rev. 135, A1041 (1964).
Subtracted residual and lattice (phonon) contributions
M. Sarachik, et al., Phys. Rev. 135, A1041 (1964).
62
Perturbation theory of a magnetic impurity problem
Resistance Minimum in Dilute Magnetic Alloys
Jun Kondo, Progress of Theoretical Physics, 32, 37-- 49 (1964).
Kondo model:
Electrical resistance and spin magnetic susceptibility
Born approximation
-- classical results
Second order spin-flip scatterings
Electron scattering
Electron-hole pair
involved scattering
Quantum
corrections
Perturbation theory is divergent as T→0K, i.e., the Kondo problem.
Emergence of the Kondo resonance as T → 0K
from RG study of the Anderson impurity model
Emergence of a
singlet bound state
between the impurity
and the conduction
electrons as the energy
scale goes down to the
low energy limit.
The phase shift of
other conduction
electrons off the
bound state is the
maximal one – the
unitary limit.
B. Horvatic, D. Sokcevic, and V. Zlatic, Phys. Rev. B 36, 675 (1987).
A universal crossover from weak to strong couplings
i.e., from lnT to T2 behavior --- the Kondo physics
Kondo spin
screening effect
Tk
↓
Spin fluctuations: Kondo effect
F. T. Hedgecock and C. Rizzuto, Phys. Rev. 163, 517 (1967).
From the Kondo problem to rare earth/actinide compounds
Y. Onuki and T. Komastubara, J. Magnetism & Magnetic Materials, 54, 433 (1986).
Heavy Fermion Metals & Superconductors
History heavy fermions:
• First HF metal, CeAl3, was measured
in 1975.
Effective charge-carrier masses are
hundreds of times the bare electron
mass, and the relevant energy scales
are very small.
• First HF superconductor, CeCu2Si2
was observed in 1979;
Other HF superconductors were also
found: UBe13, UPt3, URu2Si2.
• Typical Kondo lattice systems:
CeCu2Si2 and CeAl3
Z. Fisk, et.al., Nature 320, 124 (1986).
拓扑量子相变及其微观刻画的严格理论
X.Y.Feng, G.M.Zhang and T.Xiang, Physical Review Letters 98，087204（2007）
• 对二维量子自旋1/2并可用于拓扑量子计算的多体系统 - Kitaev模型，借
助于Jordan-Wigner变换将该模型严格约化为一个在静态Z_2规范场下的自由
Majorana费米系统。这是迄今极少有的量子二维可严格求解的多体系统。
H
 J 
i  j  even
 i
1
 ix1, j  J 2 iy1, j iy, j  J 3 iz, j iz, j 1 
x
i, j
 J c
i  j  even
c
1 i , j i 1, j
 J 2ci 1, j ci , j  J 3 Di , j ci , j ci , j 1 
• 为了描述基态拓扑量子相变，
提出在有能隙相存在非局域的拓
扑弦序参量。在对偶变换下，拓
扑弦序参量变为局域的序参量，
从而拓扑量子相变可纳入传统的
Landau对称性破缺的相变理论。
拓扑准粒子激发的统一的微观理论
D.H.Lee, G.M.Zhang and T.Xiang, Physical Review Letters 99，196805 （2007）
• 在Kitaev模型引入三自旋相互作用，
使相图中B相产生能隙。可以证明其中
的拓扑准粒子激发态对应于零能量的
磁通缺陷态，并满足非阿贝尔统计。
借助于边缘态，我们发现缺陷态等同
一维量子系统中带分数电荷的孤立子。
• 证明二维自旋极化的p_x+ip_y
超导磁通心内的满足非阿贝尔统计
的准粒子态也可以用相同的方法获
得，对应一维量子系统中带分数电
荷的缺陷态。
• 提出了凝聚态物理系统中描述分数化准粒子的一个普适原理：二维量子系统
中分数化的低能准粒子，对应于某种特殊形式的零能量的缺陷态；而这些不寻
常的缺陷与一维量子系统中存在的带分数电荷的孤立子相同。从而，统一了一
维和二维量子系统中的量子数分数化的理论，为利用非阿贝尔统计的准粒子实
现可容错的拓扑量子计算提供了新的理论基础。
Topological Quantum Computation
• Error correction & faulttolerance are essential in
the operation of large
scale quantum computer.
• non-Abelian anyons as
topological qubits, are
resistant to local
perturbations
fractional QHE 5/2, 12/5
• Microsoft Station Q set
up in 2003.
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