A Map between Time-Dependent and Time-Independent Quantum Many-Body Hamiltonians

Abstract

Given a time-independent Hamiltonian $\widetilde H$, one can construct a time-dependent Hamiltonian $H_t$ by means of the gauge transformation $H_t=U_t\kern1pt \widetilde H \kern1pt U^\dagger_t-i\kern1pt U_t\kern1pt\partial_t U_t^\dagger$. Here $U_t$ is the unitary transformation that relates the solutions of the corresponding Schrödinger equations. In the many-body case one is usually interested in Hamiltonians with few-body (often, at most two-body) interactions. We refer to such Hamiltonians as physical. We formulate sufficient conditions on $U_t$ ensuring that $H_t$ is physical as long as $\widetilde H$ is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians $H_t$ and $\widetilde H$ such that the driven many-body dynamics governed by $H_t$ can be reduced to the quench dynamics due to the time-independent $\widetilde H$. We apply this method to a number of many-body systems. First we review the mapping of a spin system with isotropic Heisenberg interaction and arbitrary time-dependent magnetic field to a time-independent system without a magnetic field [F. Yan, L. Yang, and B. Li, Phys. Lett. A 251, 289–293; 259, 207–211 (1999)]. Then we demonstrate that essentially the same gauge transformation eliminates an arbitrary time-dependent magnetic field from a system of interacting fermions. Further, we apply the method to the quantum Ising spin system and a spin coupled to a bosonic environment. We also discuss a more general situation where $\widetilde H = \widetilde H_t$ is time-dependent but dynamically integrable.

Appendix A. Eliminating magnetic field: covariant parametrization

Here we outline the derivation of equations (3.4). First we use a formula for the derivative of the exponential map [3] to obtain an integral representation of $W_t$:

$$ W_t=i \kern1pt e^{i \kern.5pt {\mathbf K} _t \kern.5pt {\mathbf S} _{\textrm{tot}} }\frac{d}{dt} e^{-i \kern.5pt {\mathbf K} _t \kern.5pt {\mathbf S} _{\textrm{tot}} } = \intop_0^1 dx\, e^{i \kern.5pt x \kern.5pt {\mathbf K} _t \kern.5pt {\mathbf S} _{\textrm{tot}} } (\dot{ {\mathbf K} }_t \kern1pt {\mathbf S} _{\textrm{tot}} ) \kern1pt e^{-i \kern.5pt x \kern.5pt {\mathbf K} _t \kern.5pt {\mathbf S} _{\textrm{tot}} }.$$

(A.1)

To proceed further, we use

$$ e^{i \kern1pt {\mathbf a} {\mathbf S} } ( {\mathbf b} {\mathbf S} ) \kern1pt e^{-i \kern.5pt {\mathbf a} \kern.5pt {\mathbf S} } = \frac1{a^2} ( {\mathbf a} {\mathbf b} ) ( {\mathbf a} {\mathbf S} )- \frac1{a} ( {\mathbf a} {\mathbf b} {\mathbf S} ) \sin a +\biggl( ( {\mathbf b} {\mathbf S} )- \frac1{a^2} ( {\mathbf a} {\mathbf b} ) ( {\mathbf a} {\mathbf S} ) \biggr) \cos a$$

(A.2)

valid for arbitrary vectors ${\mathbf a}$ and ${\mathbf b}$ and arbitrary spin ${\mathbf S}$. Here $( {\mathbf a} {\mathbf b} )$ denotes the scalar product and $( {\mathbf a} {\mathbf b} {\mathbf S} )$ denotes the scalar triple product. With the help of this formula, (A.1) can be explicitly integrated. Taking into account that $U_t H_{\textrm{H}} U_t^ \dagger = H_{\textrm{H}}$ and, consequently, ${\mathbf B} _t \kern1pt {\mathbf S} _{\textrm{tot}} =W_t$, according to (2.3) and (3.1)–(3.3) we obtain the equation

$$ {\mathbf B} _t = \frac{\sin K}{K} \dot{ {\mathbf K} }_t -\frac{1-\cos K}{K^2} {\mathbf K} _t\times\dot{ {\mathbf K} }_t +\frac1{K^2}\biggl(1-\frac{\sin K}{K} \biggr)( {\mathbf K} _t\dot{ {\mathbf K} }_t) {\mathbf K} _t.$$

(A.3)

Introducing ${\mathbf K} _t=K_t {\mathbf n} _t$, where ${\mathbf n} _t$ is a unit vector, one reduces (A.3) to

$$ {\mathbf B} _t = \dot K_t \kern1pt {\mathbf n} _t+\sin K_t \,\dot{ {\mathbf n} }_t -(1-\cos K_t) ( {\mathbf n} _t\times\dot{ {\mathbf n} }_t).$$

(A.4)

By performing a scalar (vector) multiplication of this equation by ${\mathbf n} _t$ (and doing some additional algebra in the second case), one obtains the first (second) equation in (3.4). Note that equations (A.3) and (A.4) can be more suitable for numerical integration than (3.4).

Appendix B. Eliminating magnetic field: Gauss parametrization

The Gauss parametrization of $U_t$ reads [39]

$$ U_t=\exp(\xi^+_t S_{\textrm{tot}} ^+) \exp(\xi^z_t S_{\textrm{tot}} ^z) \exp(\xi^-_t S_{\textrm{tot}} ^-),$$

(B.1)

where $S_{\textrm{tot}} ^\pm= S_{\textrm{tot}} ^x \pm i S_{\textrm{tot}} ^y$. This operator is unitary whenever

$$ \xi^+ =-(\xi^-)^* e^{i\operatorname{Im} \xi^z}\qquad\text{and}\qquad |\xi^-|^2+1=e^{\operatorname{Re} \xi^z}.$$

(B.2)

Note that the first condition above implies $|\xi^+| =|\xi^-|$.

Equation (2.3) leads to the following differential equations for the functions $\xi^\pm_t$ and $\xi^z_t$:

$$ \begin{aligned} \, i \dot \xi^+_t &= B^-_t(\xi^+_t)^2-B_t^z \kern1pt \xi^+_t - B^+_t, \\[4pt] i \dot \xi^z_t &= 2 B^-_t \kern1pt \xi^+_t -B_t^z, \\[4pt] i \dot \xi^-_t &= - B^-_t \kern1pt \exp(\xi^z_t), \end{aligned}$$

(B.3)

where $B^\pm_t\equiv(B^x_t\mp iB^y_t)/2$ (this definition implies ${\mathbf B} _t \kern1pt {\mathbf S} _{\textrm{tot}} = B_t^+ S_{\textrm{tot}} ^+ + B_t^- S_{\textrm{tot}} ^- + B_t^z S_{\textrm{tot}} ^z$). The initial condition is $\xi^\pm_0=\xi^z_0=0$. In a somewhat different context, the system of equations (B.3) was derived in [39] following the lines of the earlier work [46]. It can be verified that these equations are consistent with conditions (B.2). Note that the first equation in (B.3) is a Riccati equation with a single variable, and the other two are trivially integrated when the solution of the first one is plugged in. These equations are somewhat simpler than the equivalent system (3.4). In addition, the equations in (B.3) have nonsingular right-hand sides, which makes it clear that the solution exists for an arbitrary $B_t$.