Variational Schrieffer-Wolff transformations for quantum many-body dynamics

Building on recent results for adiabatic gauge potentials, we propose a variational approach for computing the generator of Schrieffer-Wolff transformations. These transformations consist of block diagonalizing a Hamiltonian through a unitary rotation, which leads to effective dynamics in a computationally tractable reduced Hilbert space. The generators of these rotations are computed variationally and thus go beyond standard perturbative methods, with error controlled by the locality of the variational ansatz. The method is demonstrated on two models. First, in the attractive Fermi-Hubbard model with onsite disorder, we find indications of a lack of observable many-body localization in the thermodynamic limit due to the inevitable mixture of different spinon sectors. Second, in the low-energy sector of the XY spin model with a broken $\text{U}\left(1\right)$ symmetry, we analyze ground-state response functions by combining the variational Schrieffer-Wolf transformation with the truncated spectrum approach.

Figure 1

A Hamiltonian $H_0$ with equidistant highly degenerate energy levels separated by $\mathrm{\Omega }$. An extra term $\lambda V$ breaks this degeneracy and induces level splitting on the order of $\lambda$, but dynamics within a given subspace $P$ may still be well described via a SW rotation.

Figure 2

Given an initial wave function $|\psi \left(0\right)\rangle$, we wish to find time evolution $|\psi \left(t\right)\rangle$ with respect to Hamiltonian $H$ (top). However, this may be intractable. Instead, after a unitary transformation with respect to $U$ of both the wave function and the Hamiltonian, a projected effective Hamiltonian ${\tilde{H}}_{\text{eff}}$ can be constructed allowing for tractable dynamics within the projective subspace (bottom).

Figure 3

Graphical representation of various transformations depending on the generator. A Hamiltonian $H$ is written in the eigenbasis of $H_0$ (which can be separated in subspaces $P+Q$), and generically has off-diagonal elements. The Schrieffer-Wolff generator $A$ block diagonalizes this Hamiltonian, the Wegner flow generated by the first-order approximation to the gauge potential $[H_0,V]$ band diagonalizes this Hamiltonian by suppressing off-diagonal elements between states with large energy differences, and the exact adiabatic gauge potential $\mathcal{A}$ goes a step further by exactly diagonalizing $H$ in the eigenbasis of $H_0$.

Figure 4

Comparison exact (black dashed line), projected variational (blue solid line), and perturbative SW (red dashed line) dynamics of the Fermi-Hubbard model for a small system of eight fermionic sites, $\mathrm{\Omega }/\lambda =5.0$, and a single disorder realization of strength $\mathrm{\Delta }=2.5$. The initial condition is a Néel state of alternating doubly occupied and nonoccupied sites, with periodic boundary conditions. Inset details how the projected dynamics miss high-frequency oscillations.

Figure 5

Comparison of exact and projected dynamics of the Fermi-Hubbard model for a large system of 18 fermionic sites, beyond the reach of exact dynamics, with $\mathrm{\Omega }/\lambda =5$ and a boundary-wall initial condition. The black dashed line represents exact results for a small system of 8 sites, where finite-size effects occur after a time of order 2. Error at time $t=0$ is due to missing overlap with respect to higher-energy sectors. This can be seen from the green line in the inset, representing the variational results renormalized by the fidelity of the projected initial state as $\langle \tilde{O}\left(t\right)\rangle /\langle \tilde{\mathcal{P}}\rangle$.

Figure 6

Steady-state imbalance for a disordered Fermi-Hubbard model. The initial condition is a Néel state of alternating doubly occupied and nonoccupied sites with periodic boundary conditions, and is computed at time $t=25{\mathrm{\Omega }}^2/\lambda$. The imbalance decreases with decreasing $\mathrm{\Omega }$ and increasing system size, mainly due to fidelity loss. (a) System of 18 fermionic sites and varying disorder and $\mathrm{\Omega }$. Inset is the imbalance rescaled by the fidelity, collapsing all lines to the $\mathrm{\Omega }\to \infty$ result and indicating that the loss is due to overlap with thermalizing finite-spinon sectors. (b) Varying system size at fixed disorder $\mathrm{\Delta }=10$ and $\lambda /\mathrm{\Omega }=7.5$. Small system sizes are consistent with exact results (black).

Figure 7

Illustration of the matrix structure of 14-site original and rotated Hamiltonians in both the eigenbasis of ${\sum }_i{\sigma }_z^i$ (top) and the eigenbasis of $H_0$ (bottom).

Figure 8

Approximate and exact response functions for the $\text{U}\left(1\right)$-broken XY model for a system of 16 sites. Left figures present time-dependent expectation values for $\langle {\sigma }_{\alpha }^0\left(t\right){\sigma }_{\alpha }^0\left(0\right)\rangle$ with $\alpha =x,z$, both for the exact ($2^{16}$ degrees of freedom) and the projected (137 degrees of freedom) system. Observe that exact and projected results are almost indistinguishable. Right figure presents the dynamic structure factor $S\left(\omega \right)$ as Fourier transform of $\langle {\sigma }_x^0\left(t\right){\sigma }_x^0\left(0\right)\rangle$. The rotated Hamiltonian gets both the correct energy eigenvalues and wave-function overlaps. A decoherence width of $0.05J$ has been applied to smoothen the spectrum.

Figure 9

Offset time correlation function $\langle {\sigma }_x^i\left(t\right){\sigma }_x^0\left(0\right)\rangle$ for a system of 144 spin sites and $10441$ reduced degrees of freedom. The panels from top to bottom represent (i) the correlation function $\langle {\sigma }_x^i\left(t\right){\sigma }_x^0\left(0\right)\rangle$, (ii) its spatial- and time-Fourier transform returning the dynamic structure factor $S(k,\omega )$, and (iii) the integrated frequency response $S\left(\omega \right)$. Quasiparticle excitations can be clearly observed in all the figures. At low energies, the response is that of a free particle; at larger energies the response widens, signaling finite-particle lifetime. In the bottom panel a decoherence factor with width of $0.05J$ has again been applied to smoothen the function.

Figure 10

Level spacing statistics $P\left(s\right)$ with $s_n=E_{n+1}-E_n$ for the zero-momentum sector of a 20-spin XY model in a longitudinal field (see text for details). The blue dashed line represents the Wigner-Dyson statistics characteristic of chaotic models, whereas the red line represents the Poisson distribution characterizing integrability. The average value of $r_n=\text{min}(s_n,s_{n+1})/\text{max}(s_n,s_{n+1})$ also returns a value close to the expected Wigner-Dyson value of $\langle r\rangle =0.536$ [54].