熱的な量子純粋状態を用いた統計力学の定式化
Dept. of Basic Science, Univ. Tokyo

Sho Sugiura (杉浦祥), Akira Shimizu (清水明)
SS and A.Shimizu, PRL 108, 240401 (2012)
SS and A.Shimizu, PRL 111, 010401 (2013)Table of Contents
1. Introduction

2. Canonical TPQ State
3. Microcanonical TPQ Stateand Its Relation to Canonical TPQ State
4. Entanglement

Table of Contents
1. Introduction

2. Canonical TPQ State
3. Microcanonical TPQ Stateand Its Relation to Canonical TPQ State
4. Entanglement

Principle ofEqual Weight

BoltzmannformulaMicroscopic View
All the microstates that have energy E

There are huge numberof states

Principle of Equal Weight:
When all the microstates emerge in the same probability,
the average value give the equilibrium value.

How can we justify this principle?

Explanation using the Ergordic Hypothesis
All the microstates that have energy E

Ergordic theory gives the fruitful mathematics
But…
It takes too much time (physically nonsense)
It gets harder as the system size increases

Explanation using the Typicality（田崎晴明 統計力学I）

Almost all the microstate at energy
aremacroscopically indistinguishable!

Explanation ofthe approach to equilibrium（田崎晴明 統計力学I）

Previous Works
Ensemble
Formulation
PreviousWorks

Total System

Sub System

Mixed

Mixed

Pure

Mixed
“Canonical Typicality”+ (exponentially small error)

A.Sugita (2007),
P.Rieman (2008)

S.Popescu et al. (2006)
S.Goldstein et al. (2006)

S.Popescu et al. (2006),
S.Goldstein et al. (2006)

Take a random vector
i.e. the random vector in the specified energy shell: an arbitrary orthonormal basis spanningthe enegy shell: a set of random complex numbers
with.

When we see the subsystem of,the expectation value is very close to
the canonical ensemble average

Subsystem

Trace out

S.Popescu et al. (2006),
S.Goldstein et al. (2006)

When we see the subsystem of,the expectation value is very close to
the canonical ensemble average

Subsystem

Trace out

That is, the deviation ratio satisfies: mechanical variable on the subsystem

Conditon for

can be weaken

A.Sugita (2007),
P.Rieman (2008)

When we see the observables which are
low-degree polynomials of local operators,
the expectation value is very close to
the microcanonical ensemble average.

That is,: mechanical variable on the subsystem(Conditon for

can be weaken even more)Previous Works
Ensemble
Formulation
PreviousWorks

Total System

Sub System

Mixed

MixedPureMixed“Canonical Typicality”+ (exponentially small error)
A.Sugita (2007),
P.Rieman (2008)

S.Popescu et al. (2006)
S.Goldstein et al. (2006)

Principle ofEqual Weight

Boltzmann
formulaPurpose of Our Work
Establish the formulation of statisticalmechanics using a single pure quantum state.

Macroscopic Variables
Mechanical Variables‧ Low-degree polynomials of local operators( i.e. their degree)Ex) Magnetization, Spin-spin correlation function

Genuine Thermodynamic Variables
Ex) Temperature , Entropy‧ Cannot be represented as mechanical variables‧ All genuine thermodynamic variables can be derived
from entropy .

Thermal Pure Quantum (TPQ) StateWhen
we callis generated from some probability measure,a TPQ state if

uniformly for every mechanical variable

as: ensemble average of: convergence in probability

Independent variables
Microcanonical TPQ stateenergy density
Canonical TPQ state.

New Formulation
Total System

Sub System

Ensemble

Mixed

MixedPrevious Works

Pure

Mixed
“CanonicalTypicality”

New Formulation
Total System

Sub System

Ensemble

Mixed

Mixed

Previous Works

Pure

Mixed
“CanonicalTypicality”

New Formulation
Total System

Sub System

Ensemble

Mixed

Mixed

Previous Works

Pure

Mixed

Microcanonical
TPQ state

Canonical
TPQ state“Canonical
Typicality”

Pure
Mixed

New Formulation
Total System

Sub System

Ensemble

Mixed

Mixed

Previous Works

Pure

Mixed

Microcanonical
TPQ state

Canonical
TPQ state“CanonicalTypicality”

Pure
Mixed

New Formulation
Total System

Sub System

Ensemble

Mixed

Mixed

Previous Works

Pure

Mixed

Microcanonical
TPQ state

Canonical
TPQ state“CanonicalTypicality”

PureMixed

Table of Contents
1. Introduction

2. Canonical TPQ State
3. Microcanonical TPQ Stateand Its Relation to Canonical TPQ State
4. Entanglement

SetupSystem:‧ Discrete quantum system composed of N sites,‧ The dimension of the Hilbert space is .
‧ The ensemble formulation gives correct results,
which are consistent with thermodynamics in‧ Assume every mechanical variable is normalized
asTo exclude foolish operators (ex.
: Constant independent ofand‧ We use quantities per site,‧ The spectrum ofisand).

Canonical TPQ StatePRL 111, 010401(2013)Firstly, take a random vector
from the whole Hilbert space.
: an arbitrary orthonormal basis of
the whole Hilbert space: a set of random complex numberswith.

Notice:This random vector
choice of the basis set

is independent of the.

Preparation ofis easy.

Canonical TPQ StatePRL 111, 010401(2013)Firstly, take a random vector
from the whole Hilbert space.
: an arbitrary orthonormal basis of
the whole Hilbert space: a set of random complex numberswith.

Then, calculate

is the canonical TPQ state
at temperature

Canonical TPQ StatePRL 111, 010401(2013)

Mechanical Variables

Genuine Thermodynamic Variables
Free energy

𝜆𝑁 : Dimension of total Hilbert space

Error Probabilisty of Canonical TPQ State: Random average over

𝜆𝑁 : Dimension of total Hilbert space: free energy density

Error Probability of Canonical TPQ State: Random average over

Error Probability of Canonical TPQ State:Variance ofA single realization of a TPQ state gives theequilibrium values of all mechanical variables.

S=1/2 kagome-lattice Heisenberg
antiferromagnet

Second peak vanishesas as?

Error Probability ofCanonical TPQ State

Almost self-validating!

S=1/2 kagome-lattice Heisenberg
antiferromagnet

Error Probability ofCanonical TPQ State

Error Probability ofCanonical TPQ State

Error Probability ofCanonical TPQ State

S=1/2 kagome-lattice Heisenberg
antiferromagnet

0.1J

Free energy density
Temperature T

Δf=0.02J
Free energydensityf

Error Probability ofCanonical TPQ State≒1800

Δf≒0.02J
β=10J
N=30J

S=1/2 kagome-lattice Heisenberg
antiferromagnet

0.1J

Error is less than 1% down to T=0.1J!

Error Probability of Canonical TPQ State

Even when we replaceby the dynamical quantitiese.g.
, the error is still exponentially small,becauseTable of Contents
1. Introduction

2. Canonical TPQ State
3. Microcanonical TPQ Stateand Its Relation to Canonical TPQ State
4. Entanglement

PRL
108.240401
Microcanonical TPQ State(2012)Start from the same state, which is
the random vector in the whole Hilbert space
Then, calculate

for k=1,2,...

: arbitrary constantof O(1) s.t.
.

is the microcanonical TPQ state
at energyPRL
108.240401
Microcanonical TPQ State(2012)

for k=1,2,...

Energydistribution: arbitrary constantof O(1) s.t.
.

Density of statesofEnergy

PRL
108.240401
Microcanonical TPQ State(2012)
Mechanical Variables

Genuine Thermodynamic Variables
Entropy: Unnormalized microcanonical
TPQ state at energyAnalytic Relations
Canonical and microcanonical TPQ states are related
by simple analytic transformations.

Energydistribution

Other’s

Energy density

Advantages for Numerical Method
Many Advantages :・ Free from dimension and structure of Hamiltonian.
Free From Negative Sign Problem
Applicable to Higher Dimensional Systems
・ Finite temperature.
・ Less amount of calculation than a diagonalizationof Hamiltonian.
・ Only 2 vectors (i.e. Computer Memory) are neededTable of Contents
1. Introduction

2. Canonical TPQ State
3. Microcanonical TPQ Stateand Its Relation to Canonical TPQ State
4. EntanglementDifferent Representations of
the Same Equilibrium State
Conventional Formulation

TPQ States Formulation

As far as we see macroscopic quantities,
we cannot distinguish them.

N siteEntanglement -Purity
q siteTrace outlowenergy highMinimum value
Average value ofrandom vector in whole Hilbert space( A.Sugita & A.Shimizu (2005) )TPQ states
N=16q

TPQ states are almost maximally entangledDifferent Representations of
the Same Equilibrium State
Conventional Formulation

At high temperature, they have little entanglement.

TPQ States Formulation
TPQ states have almost maximum entanglement.

Microscopically completely different states
represent the same equilibrium state.

Principle of
TPQ State
Equal Weight

Boltzmann
formulaThank You!