Phase space methods for Majorana fermions

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Dr Ria Rushin Joseph

Deakin University

School of IT

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Thanks to Distinguished Professor Dr Peter D. Drummond

1. Introduction�2. Phase space formulation in quantum mechanics�3. Shock wave Dynamics�4.Entropy, purity and fidelity�5.The Majorana Hubbard model�6. Summary���

Majorana fermions�

• Majorana particles quasiparticles in experiments.
• Quasi particle concept simplify quantum many body problems.
• Alternative basis for Fermi operators

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Ettore Majorana-hypothesized Majorana fermions in 1937 �

• Equal combinations of Fermi creation and annihilation operators

Anticommutation relations for fermions

are extended creation and annihilation operators.

• Applications of Majorana fermions

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1. Topological Quantum computation,
2. the quantum Hall effect,
3. superconductivity,
4. supersymmetry,
5. exotic states of ordinary matter and
6. dark matter searches

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• solid state physics equal superpositions of an electron and a hole.
• Majorana fermions have zero charge topological quantum computers with reduced dissipation.
• Majorana particles new approaches to handle them effectively.

Phase space formulation in quantum mechanics

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• Formulations in quantum mechanics for many body systems.

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Basic idea of phase space methods

• to obtain the probability density for observables.
• It is obtained from the stochastic differential equations(SDE) that are the solutions of Fokker-Planck equations(FPE).
• To get FPE one needs differential identities.

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Examples of quasiprobability distributions

Q-function

P-function

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Fermions

Fermionic P function

(complete and positive)

Fermionic Q function. (complete and positive)

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Fermionic phase space

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Gaussian operator basis

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Covariance matrices

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• Method extended to Majorana phase space

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Advantages:

• to study the dynamics of ultra cold Fermi and Bose systems
• alternative for quantum dynamical simulations of many-body systems.
• overcompleteness of the basis distribution can be chosen positive.
• The Majorana representation together with Quantum Monte-Carlo simulations alternative sign free approach that can be considered.

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• Phase space dimension grows quadratically, not exponentially with the number of modes.
• Includes information about entire density matrix obtain information about higher order correlations as well.
• simulate a dynamical system for a large number of modes and particles by sampling stochastic equations probabilistically.
• can be used to treat dissipation and losses.

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Exponential complexity

Majorana phase space methods

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1. Normalized Majorana Gaussian operator
2. Majorana Q-function
3. Majorana P-function
4. Majorana unordered normalized differential identities

Normalized Majorana Gaussian operator

2M X 2M real antisymmetric matrix

M number of modes.

normalization constant

The Majorana Q-function & P-function

• make use of Gaussian operator basis.
• positive probability distributions.
• used to study correlated fermion systems.
• defined in the space of real antisymmetric matrices

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is a constant real antisymmetric matrix.

is an even, positive scaling function.

• Physical interpretation: relative probability for observing a given Gaussian density matrix
• P-function is the complement of the Q-function.

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The Majorana Q- function

• Majorana Q-function is defined as the inner product of the density operator with a Majorana Gaussian operator.

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Single mode Majorana Q-function

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In single mode case-only one independent variable

The three main properties of the Q function:

1. It exists uniquely for any quantum density-matrix.

2. It is a positive probability distribution.

3. Observables are moments of the distribution.

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Resolution of identity

where the integration is over the bounded real classical domain of antisymmetric matrices

Observables

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• Starting with Majorana correlation function given by

• The above is a general hermitian observable, which includes occupation number since

• Now the expectation value of the observable is defined as:

• Utilizing resolution of identity,

• It expands the density matrix in terms of the Gaussian basis.

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The Majorana P- function

• The distribution is normalized as:

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• Expectation value of any hermitian quantum operator is

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Majorana differential identities

Example: Fermi shockwave dynamics

• The study of finite temperature dynamics of one dimensional Fermi gases is a challenging problem.
• Exhibits different phenomena:

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1. Propagation of shock waves & collective oscillations.
2. Formation of vortices and solitons.

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Dynamical evolution

Number-conserving Hamiltonian,

i, j are summed over the M system modes.

the time evolution equation of the density operator is given by:

Utilizing the expansion of density operator in terms of

Majorana P function,

use the following differential identity,

After performing an integration by parts and assuming that the

boundary terms are zero,

The method of characteristics allows one to solve the above equation,

Runge- Kutta 4-5 adaptive algorithm for simulation

Dispersion relation:

• The chemical potential of the original 1D Fermi gas confined in a box at zero temperature is given by:

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This is equal to:

• For finite temperatures, T, the chemical potential is:

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Initial conditions

Treat Fermi systems in a grand-canonical ensemble at finite temperature. The unnormalized quantum density matrix is then:

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Hamiltonian

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at temperature T

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number operator

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chemical potential

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mean particle number

• This implies that we use the Fermi Dirac distribution:

Special case:

If�

, this simplifies further to give:�

Hamiltonian of 1D Fermi gas in momentum space

• Consider a one-dimensional non-interacting Fermi gas, which is trapped in a box of length L=2l , extending from -l to l ,

After performing the integration,

identical form to the number conserving Hamiltonian

same SDE for simulation

Perturbed flat potential

A flat potential with a time-varying Gaussian perturbation caused by an external dipole coupled laser input:

This implies that the frequency matrix is:

After performing the integration,

Harmonic trap: Quench Dynamics

• For investigating the collective oscillation quench, we consider a one dimensional Fermi gas trapped in a harmonic potential of the form :

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Zero temperature results:

Finite temperature results:

Comparison between zero temperature and finite temperature at a time t=161:

Interacting fermion dynamics

in Majorana phase space

• Using Q-function and the Majorana unordered differential identities to discuss the interacting Fermi models and interacting Majorana systems.
• the Hamiltonian of the model and its dynamical evolution.
• the derivation of the Fokker Planck equation
• properties of the Diffusion term.

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Definitions:

We note that from the above relations,

The Hamiltonian of the model consists of non-interacting linear term and an interacting term. This interaction Hamiltonian consists of a four-Majorana interaction:

One can arrive at the following Fokker Planck equation by making use of the definition of Q-function and the differential identities.

provided,

is the diffusion term and

is the drift term.

• The diffusion term is not positive definite, however it is symmetric and with a zero trace.
• This is consistent with a forward-backward stochastic evolution, that has been found for bosonic and spin Q functions.

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• The two-level atom or spin qubit
• The non-interacting Fermi gas
• The Fermi-Hubbard model
• The Kitaev model
• The Thirring model
• The Sachdev–Ye model

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Nonlinear Hamiltonians:

Entropy, purity and fidelity�

• Majorana phase space methods to calculate the fundamental information- related quantities of entropy, purity and fidelity in Fermi systems.
• obtained expressions for the Renyi entropy, purity and fidelity in Majorana representations.
• the Majorana P representation is utilized here.

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Geometry of the real homogeneous space

Dimensionality of real homogeneous space

Dimensionality of the pure states

Poles

North pole-positive parity pure state

South pole-negative parity pure state

Calculating the trace of the product of two unit trace Majorana gaussian operators is the first step.

are real if we choose Hermitian gaussian pure states, else would be complex

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Renyi Entropy

• Entropy is used to quantify disorder and entanglement.
• The Renyi or linear entropy of a density matrix generalizes the Shannon entropy so that it is possible to treat in finite systems.

Utilizing the Majorana P-representation,

This result is expressed in terms of the inner product of two gaussian operators.

Applications

1. velocity distributions
2. threshold selection
3. cryptography

Sampled entropy

The double integral - summation of samples of orthogonal transformation of pure states.

i.i.d samples from real positive distributions

random orthogonal matrices chosen with a Haar measure.�

initial pure state

Purity

• Purity signifies whether a pure state has undergone decoherence.

• Purity calculations are essential in the studies of quantum information processing

In Majorana P-representation,

• The purity of a state can be determined from the entropy results as well

pure state

mixed state

• Fidelity is another fundamental quantity in quantum information

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Fidelity

two mixed states�

norm based fidelity,

utilizing the Majorana P-function,

Utilizing inner product result of two gaussian operators,

Sampled fidelity

Applications

1. phase transitions
2. quantum teleportation
3. quantum metrology,
4. evolution of open quantum systems ;
5. non-adiabaticity measurements
6. quantum chemistry
7. and quantum chaos .

Inferences from the plot

Two kinds of orthogonal transformation

• half of the fidelities being very small
• half of the transformations are parity changing, and lead to inner products that are zero,

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• the average fidelities do not vanish, but they are reduced as the space dimension is increased.

• Derived novel identities for Majorana Gaussian states
• Used identities to calculate dynamics of density waves
• Method for treating shock waves at finite temperatures.
• Entropy, purity and fidelity calculations.
• Interacting fermions in Majorana phase space.

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Summary