Study of Quantum Phase Transition in Topological phases in U(1) x U(1) System using Monte-Carlo...

Study of Quantum Phase Transition in Topological phases in U(1) x U(1) System

using Monte-Carlo Simulation

Presenter : Jong Yeon Lee

Overview

1. Mapping Quantum to Classical

2. Quantum Phase Transition

3. Topological Phases and Example: Toric Code

4. U(1) x U(1) Model for bosonic fractionalized phase

5. Simulation Results

6. Conclusion

1. Mapping Quantum to Classical – (1)

• What is Quantumness?

– Superposition Principle

– Heisenberg’s Uncertainty Principle

– If every operators commutes, quantum mechanics would have

been much easier.

– Non-commutability of operators brings about interesting

phenomena!

– Example: Quantum Harmonic Oscillator’s zero-point energy


• Quantum Ising Model

– If operators commute in Hamiltonian, problem would be trivial

– However, since, z-direction spin operator and x-direction spin

operator don’t commute, there arises some ‘quantum dynamics’

Transversal Field


• Quantum Ising Model

– Simple Cases


• Quantum Statistical Mechanics

– There is NO basis diagonalizing Hamiltonian since it consists of

non-commuting parts – How to study ground state property?

– Strategy: study quantum statistical mechanics, and then trying to

send temperature to zero. Since free-energy minimization would

yield lowest energy state, we would end up in studying ground-

state physics!


• Trotter Decomposition

– There is NO basis diagonalizing Hamiltonian since it consists of

non-commuting parts.

– Decompose exponential part in smaller pieces!

– Zassenhaus Formula



– For δτ small enough, we can approximate it as following

– Now, we insert completeness identity into each exponential terms



Now, problem is mapped to Classical stat. mechanics problem with one higher dimension!

Classical problem with one more ‘effective’ dimension!

• With periodic boundary condition for imaginary dimension



Now, problem is mapped to Classical stat. mechanics problem with one higher dimension!

Classical problem with one more ‘effective’ dimension!

• With periodic boundary condition for imaginary dimension

Action in classical field theory!


• Classical Statistical Mechanics in higher dimension!

– If we think about the case where Hamiltonian is diagonalizable,

then set of all basis is equal to set of all possible configuration

since each basis for quantum state is just tensor product of all

possible state of individual sites.

– However, in quantum ising model with transversal field, there is no

basis diagonalizing Hamiltonian due to non-commuting parts.

– This ‘Non-commutativity’ effectively yields ‘imaginary dimension’

– In the limit where temperature goes to zero, we would have

infinite size for imaginary dimension – thermodynamic limit!

Remark

• In such way, every (equilibrium) quantum mechanical

problems can be mapped into classical statistical

mechanics problem in higher dimension

• Ground state properties can be calculated in terms of

corresponding classical variables correlations!

• However, unlike this basic example, there can be complex

phase factors or other complications in this mapping

• Method is called Euclidean space-time path integral

Overview






6. Conclusion

2. Quantum Phase Transition – (1)

• Quantum Phase Transition?

– Unlike classical phase transition tuned by temperature, this

happens at zero temperature.

– Tuning parameters are strength of various interaction terms in

Hamiltonian. Since the system is at zero temperature, Quantum

fluctuation plays significant role in such transition.

– As for a classical second order transition, a quantum second order

transition has a quantum critical point (QCP) where the quantum

fluctuations driving the transition diverge and become scale

invariant in space and time.



– Example: Fractional Quantum Hall Phases

Advanced Epitaxy for Future Electronics, Optics, and Quantum Physics: Seventh Lecture International Science Lecture Series ( 2000 )

• By changing magnetic

field, we change the

ground state properties,

which is called Hall

Conductivity.

• This means that ground

state is undergoing

quantum phase

transitions!



– As we know from the classical continuous phase transitions, it

would have interesting properties like universality

– Especially, universality of Topological Phase Transition is what

physicists don’t know yet exactly

– Since most effective field theoretic approaches break down,

studying quantum phase transition is very hard work.

– It would be valuable if we can study phase transition of topological

phases in direct way. -> Euclidean space-time path integral!

Overview






6. Conclusion

3. Topological Phases

• One of signature of topological phase – excitation carries fractional

quantum numbers (charge) with unusual Anyonic statistics (2D).

• In 2D Topological Phase of matter, non-trivial statistical interactions

between quasi-particle excitations can be found :

– Anyonic Excitation in fractional quantum hall effect (exchange phase π/3)

– Spinons and visons in Z2 spin liquids (mutual exchange phase π)

3. Topological Phases

• Simplest Example of Exactly solvable Topological Phases:

Kitaev’s Toric Code

• It is equivalent to Z2 X Z2 spin liquid model

• Two distinct quasi-particle excitations each has Z2 symmetry

Z2 charge (spinon) and Z2 flux (vison)

• Spinons and visons have mutual π statistics

Overview





5. Simulation and Results

6. Conclusion

Encodes vortex configuration of loop variables

4. U(1) X U(1) model with mutual statistics

• Inspired by Z2 X Z2 model with mutual π statistics , we proposed (2+1)D

classical action which is effective mapping (Trotter decomposition) of

quantum 2D Hamiltonian. We can use this model to study topological

phases of quantum system!

Can be cast in to SIGN FREE form by reformulation!

Encodes vortex configuration of loop variables


• Inspired by Z2 X Z2 model with mutual π statistics , we proposed (2+1)D

classical action which is effective mapping (Trotter decomposition) of

quantum 2D Hamiltonian. We can use this model to study topological

phases of quantum system!


• Original 2D quantum system?

– 2D Quantum system with interpenetrating lattices

– Having U(1) degrees of freedom for each lattice sites, and Hamiltonian has

U(1) X U(1) Symmetry


• Original 2D quantum system

– 2D Quantum system with interpenetrating lattices

– Having U(1) degrees of freedom for each lattice sites, and Hamiltonian has

U(1) X U(1) SymmetryVery Long andMessy Derivation and Reformulation

will give a (2+1)D Classical Action with mutual statistics

Overview





5. Simulation and Results

6. Conclusion

5. Monte-Carlo Simulation

• Once the action S is cast in sign-free form, we can use

Monte-Carlo method to simulate the action

• In Monte-Carlo simulation, we use probabilistic local

update rules to simulate ensemble average

5. Correlation Functions

• Current-Current Correlation Functions

• Using current-current correlation function, we can calculate Hall

conductivity!

5. Correlation Functions

• Behavior of Current-Current Correlation– In thermodynamic limits, C22 goes zero at phase (0) and phase (IV), while it

diverges at phase (III). Also, C12 goes zero at phase (0) and phase (III), it

obtains some fixed value at phase (IV), implying some correlated state of

quasi-particles J1 and J2

5. Phase Diagram of Θ = 2π/n case

(J1, J2) are variables representing quasi-particle excitations.In terms of physical variables (Q1, Q2), phase diagram can be named as follows

5. Phase Diagram of Θ = 4π/5 case


θ = 4π/5

5. Nature of Multi-critical points

[1] Scott D. Geraedts and Olexei I. Motrunich, Phys. Rev. B 86, 045106 (2012)

Properties?

Possible Scenario

• Weird behavior of correlation unlike multi-critical point between phase (III) and (IV)

Phase (IV)

Phase (III)

5. Nature of Multi-critical points


Properties?

Possible Scenario

• Inconclusive Data from previous research


Result – (1)

Point t=0.3395 is minimum critical region

Drift of crossing points Giving Lower Bound!

Result – (1)

• We can get dual variables’ correlations from original variables’ correlations.

• If phase of one variable changes at certain parameter, phase of dual variables must change

• As dual variables have opposite phase behaviors, these crossings of correlations in dual variables will give information about upper bound of the phase transition point!

Point t=0.344 is maximum critical region

Giving Upper Bound!

Result – (2)

• Multi-Critical Point

– Two parameters (t1,t2)

– Symmetry Consideration:

differentiation along

symmetric line (s=t1+t2)

and anti-symmetric line

(a=t1-t2)

s=t1+t2

a=t1-t2

Result – (2)

Cusp Singularity!

Narrow down critical region – upper limit is 0.342!

Narrowing down critical region

• By analyzing data we obtained, we can conclude that range of criticality must be in [0.340, 0.342]

• Though crossing of <J1 J2> doesn’t move much (0.339->0.340), crossing of <Q1Q2> moves a lot (0.355->0.344) with the system size L. Therefore, simulating larger system size will further narrow this range down.

Length < 0.002

Data along the line parallel to symmetric line

For parallel line, t2=t1+0.002

Length < 0.004

Result (1) and (2)

• From these observation, we can conclude scenarios (b) and (c) are hardly plausible– If segment exists in that small region, it doesn’t make much sense

because all the other parameters are order O(1), while that segment must be order O(10-3)

• Phase transition is continous; Critical point t1=t2=0.34 is multi-critical point with two relevant directions.

Extraction of Critical Exponents

• After concluding it is a continuous transition, we assumed scaling hypothesis to get the location of critical point and extract the correlation length critical exponents

Critical Exponents

Future Direction?

• Recently, I discovered that we can find a class of actions

which gives locally same phase diagrams with topological

phases.

• They have different microscopic interactions, and they

don’t result from one another by dual transformation

• This implies that the study of their critical behaviors would

give the insight of universality class in the case of phase

transitions involving topological phases.

6. Conclusion

• We can study quantum ground state by mapping it to the one higher

dimensional classical statistical mechanics problem (“Euclidean space-

time path integral method”)

• Quantum phase transition happens when some essential quality of

ground state is changed. It is very interesting but difficulty problem.

• We can study this phase transition using Euclidean path integral

• Topological phases have interesting properties, like mutual statistics

• We proposed model that realize arbitrary mutual statistics for U(1) X

U(1) model, and studied it using Euclidean path integral

• Obtained interesting phase diagrams and dynamics

Example – θ=4π/5

? -> now potential becomes long-ranged

Behavior of correlations in dual variables

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

<Q1xQ2y>*L

L=12

t1=t2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

<J1xJ2y>*L

L=12

t1=t2

Histogram data of critical point

Studied evolution of energy histogram near the critical region; energy histogram is single-peaked; this is evidence for continous phase transition

Also, heat capacity was not proportional to system volume – which means it is unlikely to be a first order transition

Dual Transformation• Basically, this is “Change of Variable”; However, with some physical

intuition, we can analyze system without actually doing simulation

Using Dual Transformation?

• On-site Interaction?

Short-Ranged Potential!

Finite Size Scaling Method

Simulated for different system sizes and fit the scaling form

Phase Diagram of θ = 2π/n case


θ = 2π/3 case

For n > 2, all phase diagrams have same form as left one. Only difference is position of two multi-critical points

Phase boundaries were identified by examining divergence of heat capacity & crossing points of correlation

(IV)

(0)

(III)

(I)

(II)t1

t2

Studying Fractionalized Phase

• People know lots about fractionalized phases, but much less about

phase transitions involving fractionalized phases.

• We study specifically which can have bosonic fractionalized phase

where each quasi excitation carries separately conserved charge

• In previous research, we confirmed such phase exists, and this is an

example of symmetry enhanced topological phases. It has similar

structure as Zn Toric code (J1, J2 <-> e/m), but as we gave U(1)

symmetry to the excitation, the model in our study obtained

symmetry-enhanced phase, which is fractionalized quantum hall phase

Quantum Phase Transition

• Continuous Phase Transition

– A method to study universality class in phase transition of topological

phases.

– Corresponding Chern-Simon Field theory exists!

– We don’t have any control to study phase of topological field theory;

However with this model, we have completely unbiased method to

study this field theory.

Goal of Research


Properties?

Possible Scenario

• Inconclusive Data from previous research


Goal of Research

• Study Quantum Phase Transition from bosonic fractional hall phase to

Mott Insulating phase using Monte-Carlo Simulation optimized to

those region

• We will use Finite Size Scaling Method

• Identify properties of critical regions in 2π/3 model

– Is it a point?

– Is transition 1st order? 2nd order?

– Critical exponents?

• For θ=2π/n, how does critical exponents change with n?

Study of Quantum Phase Transition in Topological phases in U(1) x U(1) System using Monte-Carlo...

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