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

1. Mapping Quantum to Classical – (2) 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

1. Mapping Quantum to Classical – (3) Quantum Ising Model – Simple Cases

1. Mapping Quantum to Classical – (4) 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!

1. Mapping Quantum to Classical – (5) Trotter Decomposition – There is NO basis diagonalizing Hamiltonian since it consists of non-commuting parts. – Decompose exponential part in smaller pieces! – Zassenhaus Formula

1. Mapping Quantum to Classical – (6) Trotter Decomposition – For δτ small enough, we can approximate it as following – Now, we insert completeness identity into each exponential terms

1. Mapping Quantum to Classical – (7) Trotter Decomposition 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

1. Mapping Quantum to Classical – (7) Trotter Decomposition 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

1. Mapping Quantum to Classical – (8) 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 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

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.

2. Quantum Phase Transition – (2) Quantum Phase Transition? – 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!

2. Quantum Phase Transition – (3) Quantum Phase Transition? – 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 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

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 Z 2 spin liquids (mutual exchange phase π)

3. Topological Phases Simplest Example of Exactly solvable Topological Phases: Kitaev’s Toric Code It is equivalent to Z 2 X Z 2 spin liquid model Two distinct quasi-particle excitations each has Z 2 symmetry Z 2 charge (spinon) and Z 2 flux (vison) Spinons and visons have mutual π statistics

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 and Results 6.Conclusion

Encodes vortex configuration of loop variables 4. U(1) X U(1) model with mutual statistics Inspired by Z 2 X Z 2 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 4. U(1) X U(1) model with mutual statistics Inspired by Z 2 X Z 2 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!

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

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

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 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, C 22 goes zero at phase (0) and phase (IV), while it diverges at phase (III). Also, C 12 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 (J1, J2) are variables representing quasi-particle excitations. In terms of physical variables (Q1, Q2), phase diagram can be named as follows θ = 4π/5

5. Nature of Multi-critical points [1] Scott D. Geraedts and Olexei I. Motrunich, Phys. Rev. B 86, (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 [1] Scott D. Geraedts and Olexei I. Motrunich, Phys. Rev. B 86, (2012) Properties? Possible Scenario Inconclusive Data from previous research Weird behavior of correlation unlike multi-critical point between phase (III) and (IV)

Result – (1) Point t= 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 (t 1,t 2 ) – 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 doesn’t move much (0.339->0.340), crossing of 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, t 2 =t 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 t 1 =t 2 =0.34 is multi-critical point with two relevant directions.

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 t 1 =t 2 =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

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 (J1, J2) are variables representing quasi-particle excitations. In terms of physical variables (Q1, Q2), phase diagram can be named as follows 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) t1t1 t2t2

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 Z n Toric code (J 1, J 2 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 [1] Scott D. Geraedts and Olexei I. Motrunich, Phys. Rev. B 86, (2012) Properties? Possible Scenario Inconclusive Data from previous research Weird behavior of correlation unlike multi-critical point between phase (III) and (IV)

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 1 st order? 2 nd order? – Critical exponents? For θ=2π/n, how does critical exponents change with n?