Result and Conclusion
6.2 TDVP Result
The most interesting subject is a quench across the phase boundary. For instance, from the KT phase to the other phase. Even in the same phase, it is still interesting to investigate the dynamics of the Thirring model.
The most important observable for the real time evolution is return probability (Loschmidt echo) [18] defined by
L(t) =|⟨ψ(0)|ψ(t)⟩|2 = ⟨ψ(0)|e−iHt|ψ(0)⟩ 2. (6.2) However, for an infinite 1D system, we cannot calculate this observable exactly since it will always be zero for t > 0. Instead, we use the norm square of the dominant eigenvalue of the transfer matrix, E(t) =∑
sA(t)s⊗ ¯As(0), arising from the overlap between the initial state and the time-evolved state at time t to represent the return probability (denoted as P (t)). Another important observable is the return rate function defined by:
g(t) =− lim
N→∞
1
N ln L(t), (6.3)
which is well-defined even in the thermodynamic limit, and L(t)≈ PN(t)
⇒g(t) = − lim
N→∞
1
NN ln P (t) =− ln P (t). (6.4) We can represent the return-rate function with the negative logarithm of the domi-nant eigenvalue of the transfer matrix.
In this thesis, we investigate the dynamics of the Thirring model by TDVP algorithm with bond dimension D=100 without a penalty term. Note that⟨Sz⟩ and
⟨H⟩ are conserved quantities since [H, H] = [Sz, H] = 0. We can see that ⟨Sz⟩ is conserved in our simulation (Fig. 6.11) even though we turned off the penalty term.
We found that the simulation is very unstable if we turn on the penalty term.
Now let us examine the case starting from the ground state with the parameters (∆,m)=(-0.8, 0.2) and use the Hamiltonian with parameters (∆,m)=(0.5, 0.2) to evolve the state. In Fig. 6.11, we observe that the entanglement entropy saturates after t≈ 15 since the quantum state has evolved to a state far away from the uMPS manifold. So we can only trust the result before t = 15.
doi:10.6342/NTU201802766
(a) (b)
(c) (d)
Figure 6.11: The Thirring model evolving from (∆,m)=(-0.8, 0.2) to (∆,m)=(0.5, 0.2).
(a) (b)
Figure 6.12: The Thirring model evolving from (∆,m)=(0.5, 0.2) to (∆,m)=(-0.8, 0.2).
We can present results of a quench on ∆− m plane and we show the dynamics of the chiral condensate, the return probability and the fermion correlator. More
numerical results are shown in the Appendix. We observe that when quenching from the KT phase with the same mass (m ̸= 0), there exist several non-analytic cusps (see Fig. 6.11) of the return-rate function which indicates that there may be a dynamical phase transition. Otherwise, the return-rate function evolving with the same mass will be very smooth (see Fig. 6.12). We obtain numerical results for the quench dynamics of the Thirring model. It may exist dynamical phase transition for some cases. Further exploration is necessary.
doi:10.6342/NTU201802766
Chapter 7 Summary
Tensor network methods are a powerful tool for studying many-body systems and there are several efficient algorithms available for 1D systems. After discretizing the Thirring model on a 1D infinite lattice, we represent it as a spin-1/2 representation.
We use a tensor network method to find its ground state and to study time evolution.
The time-dependent vaiational principle algorithm (TDVP) and the variational optimization methods for uniform matrix product state (VUMPS) are very efficient and accurate. In this thesis, we use the VUMPS algorithm to find the ground state of the Thirring model and characterize the phase diagram. We then extend the TDVP algorithm in MPO form such that we can deal with the Thirring model problem and investigate the quench dynamics. We can see that TDVP algorithm preserves conserved quantities very well.
We want to ask whether the Thirring model exists the dynamical phase transi-tion. So we use TDVP algorithm to do real-time evolution and see the return-rate function. And we found the existence of non-analytic cusps in the return-rate func-tion which suggest the existence of the dynamical phase transifunc-tion. We still wonder what is the physical meaning of the dynamics of the fermion correlator for the Thirring model and it is worthwhile to research it in the future.
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