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arXiv:2009.13873 · Proceedings of the Steklov Institute of Mathematics

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

Oleksandr Gamayun, Oleg V. Lychkovskiy

Published Jul 1, 2021• 2 citations
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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.

DOI
10.1134/s008154382102005x

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Given a time-independent Hamiltonian H~\widetilde HH , one can construct a time-dependent Hamiltonian HtH_tHt​ by means of the gauge transformation Ht=UtH~Ut†−iUt∂tUt†H_t=U_t\kern1pt \widetilde H \kern1pt U^\dagger_t-i\kern1pt U_t\kern1pt\partial_t U_t^\daggerHt​=Ut​HUt†​−iUt​∂t​Ut†​ . Here UtU_tUt​ 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 UtU_tUt​ ensuring that HtH_tHt​ is physical as long as H~\widetilde HH is physical (and vice versa). This way we obtain a general method for finding pairs of physical Hamiltonians HtH_tHt​ and H~\widetilde HH such that the driven many-body dynamics governed by HtH_tHt​ can be reduced to the quench dynamics due to the time-independent H~\widetilde HH . 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 H~=H~t\widetilde H = \widetilde H_tH=Ht​ is time-dependent but dynamically integrable.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.
We find a relationship between unitary transformations of the dynamics of quantum systems with time-dependent Hamiltonians and gauge theories. In particular, we show that the nonrelativistic dynamics of spin-12\frac{1}{2}21​ particles in a magnetic field Bi(t){B}^{i}(t)Bi(t) can be formulated in a natural way as an SU(2) gauge theory, with the magnetic field Bi(t){B}^{i}(t)Bi(t) playing the role of the gauge potential Ai{A}^{i}Ai. The present approach can also be applied to systems of n levels with time-dependent potentials, U(n)U(n)U(n) being the gauge group. This geometric interpretation provides a powerful method to find exact solutions of the Schr\"odinger equation. The root of the present approach rests in the Hermiticity property of the Hamiltonian operators involved. In addition, the relationship with true gauge symmetries of n-level quantum systems is discussed.
We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.
We study the dynamics of multiparticle Carroll-Schrödinger (CS) quantum systems in 1+11{+}11+1 dimensions, where xxx acts as the evolution variable and ttt as the configuration coordinate. We derive the NNN-body theory on equal-xxx slices as the Carrollian limit of a relativistic multi-time Klein-Gordon model, introducing temporal interactions via minimal coupling to the temporal energy operators. An xxx-dependent gauge transformation maps this to an equivalent description with explicit many-body potentials, illustrated by a temporal coupled-oscillator model that exhibits synchronization. Adopting a complementary spatial viewpoint with a static potential U ⁣tot(x)U_{\!tot}(\mathbf x)Utot​(x), we show that the evolution is driven by the collective force ∑j∂xjU ⁣tot\sum_j\partial_{x_j}U_{\!tot}∑j​∂xj​​Utot​; for any translation-invariant interaction (such as a regularized Coulomb potential), these internal forces cancel, rendering the collective dynamics free and highlighting Carrollian ultralocality. We also construct a coordinate duality mapping separable Schrödinger Hamiltonians to CS generators via Schwarzian derivatives. Exchange symmetry is formulated in the time domain, yielding temporal bunching for bosons and antibunching for fermions via the second-order coherence function g(2)(t,t′)g^{(2)}(t,t')g(2)(t,t′). In second quantization, the contact limit yields a temporal derivative cubic--quintic nonlinear Schrödinger equation with a theoretically fixed nonlinearity coefficient β=−3/16β=-3/16β=−3/16. Finally, by coupling canonical pairs to external scalar and gauge fields, we establish an isomorphism with one-dimensional current-density functional theory, outlining a Carrollian Hohenberg-Kohn mapping and Kohn-Sham scheme.
We introduce a new class of quantum models with time-dependent Hamiltonians of a special scaling form. By using a couple of time-dependent unitary transformations, the time evolution of these models is expressed in terms of related systems with time-independent Hamiltonians. The mapping of dynamics can be performed in any dimension, for an arbitrary number of interacting particles and for any type of the scaling interaction potential. The exact solvability of a "dual" time-independent Hamiltonian automatically means the exact solvability of the original problem with model time-dependence.