Sudden Quench of the Local Transverse Magnetisation in the TFIM

This page documents the closed-form derivation of $\langle\sigma^x_i\rangle(t)$ exposed by MagnetizationXLocal{:quench} for the transverse-field Ising model (TFIM). The quench setup follows Calabrese–Essler–Fagotti (J. Stat. Mech. P07016 (2012)); the underlying Bogoliubov machinery dates to Barouch–McCoy–Dresden (Phys. Rev. A 2 (1970) 1075).

Setup

The Hamiltonian

\[H(h) \;=\; -J\sum_i \sigma^z_i \sigma^z_{i+1} \;-\; h \sum_i \sigma^x_i\]

is parameterised by the transverse field $h$ at fixed $J > 0$. A sudden quench prepares the system in the ground state $|\psi_0\rangle$ of $H(h_0)$ and lets it evolve under $H(h_f)$ for $t > 0$. The observable of interest is

\[\boxed{\; \langle\sigma^x_i\rangle(t) \;=\; \langle \psi_0 | e^{i H(h_f) t}\, \sigma^x_i\, e^{-i H(h_f) t} | \psi_0 \rangle. \;}\]

Open-boundary route — Majorana covariance evolution

After the Jordan–Wigner mapping discussed in jw-tfim-bdg, $H(h)$ is quadratic in Majorana operators

\[\gamma_{2i-1} = c_i + c_i^\dagger,\qquad \gamma_{2i} = i (c_i^\dagger - c_i), \qquad H(h) \;=\; \tfrac{i}{4}\,\sum_{ab}\, [h]_{ab}\, \gamma_a \gamma_b,\]

where the $2N \times 2N$ real antisymmetric matrix $[h]$ is built by _majorana_ham(N, J, h). Its only non-zero entries are $[h]_{2i-1,2i} = 2h$ and $[h]_{2i,2i+1} = 2J$. In this representation $\sigma^x_i = -i\,\gamma_{2i-1}\gamma_{2i}$.

Define the ground-state Majorana covariance

\[\Sigma_{ab}^{(0)} \;\equiv\; -i\,\bigl(\langle \gamma_a \gamma_b\rangle_{\rm GS} - \delta_{ab}\bigr) \;=\; -i\,\operatorname{sign}(i\,[h_0]),\]

provided by _majorana_covariance_gs(_majorana_ham(N, J, h_0)). Since $[h_f]$ is real antisymmetric, time evolution is the orthogonal rotation

\[\gamma_a(t) \;=\; R(t)_{ab}\,\gamma_b, \qquad R(t) \;=\; \exp([h_f]\,t) \in \mathrm{SO}(2N).\]

Combining these gives the time-evolved covariance

\[\boxed{\; \Sigma(t) \;=\; R(t)\,\Sigma^{(0)}\,R(t)^{\!T}, \;}\]

and reading off the local magnetisation:

\[\boxed{\; \langle\sigma^x_i\rangle(t) \;=\; -i\,\langle \gamma_{2i-1}(t)\,\gamma_{2i}(t)\rangle \;=\; \Sigma(t)_{2i-1,\,2i}. \;}\]

This is what _tfim_sigma_x_quench_obc returns: one $2N \times 2N$ eigendecomposition for $\Sigma^{(0)}$ plus one matrix exponential for $R(t)$ per call. The answer is exact at every finite $N$.

Infinite-volume route — closed-form $k$-integral

In the thermodynamic limit translation invariance decouples the $(k, -k)$ mode pairs and the per-mode Bogoliubov rotation can be performed analytically. With

\[\varepsilon_k(h) \;=\; h - J \cos k, \qquad \Delta_k \;=\; J \sin k, \qquad \Lambda_k(h) \;=\; 2\sqrt{\varepsilon_k(h)^2 + \Delta_k^2},\]

the Bogoliubov angle $\theta_k(h)$ is fixed (modulo a quadrant choice we resolve via atan2) by

\[2\,\theta_k(h) \;=\; \operatorname{atan2}\bigl(2\Delta_k,\ 2\varepsilon_k(h)\bigr),\]

so that $\cos(2\theta_k) = 2\varepsilon_k/\Lambda_k$ and $\sin(2\theta_k) = 2\Delta_k/\Lambda_k$. In equilibrium at $T = 0$ the ground-state transverse magnetisation is

\[\langle\sigma^x\rangle_{\rm GS}(h) \;=\; \frac{1}{\pi}\int_0^\pi \cos\bigl(2\theta_k(h)\bigr)\,dk,\]

which agrees with the formula already coded as MagnetizationX, Infinite.

The post-quench state has, in the post-quench Bogoliubov basis, an excitation density per mode pair $\sin^2(\Delta\theta_k)$ where

\[\Delta\theta_k \;\equiv\; \theta_k(h_f) - \theta_k(h_0),\]

and the time-dependent expectation of $\sigma^x$ within that pair is the standard two-level Rabi-like oscillation. Summing over $k$ gives

\[\boxed{\; \langle\sigma^x\rangle(t) \;=\; \frac{1}{\pi}\int_0^\pi dk\;\Bigl[\, \cos\bigl(2\theta_k^f\bigr)\,\cos\bigl(2\Delta\theta_k\bigr) \;+\; \sin\bigl(2\theta_k^f\bigr)\,\sin\bigl(2\Delta\theta_k\bigr)\, \cos\bigl(2\,\Lambda_k^f\,t\bigr) \,\Bigr], \;}\]

with $\theta_k^f \equiv \theta_k(h_f)$ and $\Lambda_k^f \equiv \Lambda_k(h_f)$. This is _tfim_sigma_x_quench_infinite, evaluated by adaptive Gauss–Kronrod quadrature.

Sanity checks

The OBC route converges to the infinite-volume integral while the light cone from each open boundary has not yet reached the observation site $i$. The relevant condition at site $i$ is

\[t \, v_{\max} \,<\, d(i),\qquad d(i) \,=\, \min\bigl(i, \, N - i + 1\bigr),\]

where $d(i)$ is the distance to the nearer boundary and $v_{\max} \le 2\,\max(J, h)$ is the maximal group velocity in either phase of the TFIM. For the central site $i = N/2$ the bound becomes $t\,v_{\max} < N/2$, which is the convention quoted in the verification tests. Outside this window the OBC value contains boundary-reflection contributions not present in the infinite-volume integral and a finite (controllable) deviation appears.

API

struct MagnetizationXLocal{M} <: AbstractQuantity end          # M ∈ {:equilibrium, :quench}
MagnetizationXLocal()              = MagnetizationXLocal{:equilibrium}()
MagnetizationXLocal(:quench)       = MagnetizationXLocal{:quench}()

# OBC, single (i, t) point
fetch(model_f::TFIM, ::MagnetizationXLocal{:quench}, ::OBC;
      initial::TFIM, i::Int, t::Real, kwargs...) -> Float64

# Infinite, single t point
fetch(model_f::TFIM, ::MagnetizationXLocal{:quench}, ::Infinite;
      initial::TFIM, t::Real, kwargs...) -> Float64

The pre-quench TFIM initial and the post-quench model_f must share J; a $J \to J'$ jump is rejected with ArgumentError. Sweep callers should hoist the Majorana eigendecomposition / quadrature setup themselves; the public fetch recomputes per call.

References

• E. Barouch, B. McCoy, M. Dresden, Statistical Mechanics of the XY Model. I, Phys. Rev. A 2, 1075 (1970).
• P. Calabrese, F. H. L. Essler, M. Fagotti, Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators, J. Stat. Mech. P07016 (2012).
• I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36, L205 (2003).