I'll create a modern navbar component for you using React and Tailwind CSS. This navbar will be responsive, include a logo, navigation links, and a mobile menu.

This modern navbar component includes the following features:

1. Responsive design: It adapts to both desktop and mobile screens.
2. Logo placeholder: You can replace the "Logo" text with your actual logo.
3. Navigation links: Easy to customize by modifying the navItems array.
4. Mobile menu: Toggles on smaller screens with a hamburger icon.
5. Dark mode toggle: Users can switch between light and dark themes.
6. User avatar: Includes a dropdown menu for user-related actions.

The navbar is built using React and Tailwind CSS, with shadcn/ui components for the dropdown menu and buttons. It's also set up to work with Next.js, including the App Router.

To use this navbar, make sure you have the necessary dependencies installed, including next-themes for theme management and lucide-react for icons. You may need to run:

Also, ensure that your tailwind.config.js is set up correctly to work with the shadcn/ui components and the theme colors.

You can now customize the navbar further by:

1. Replacing the "Logo" text with your actual logo image.
2. Modifying the navItems array to include your specific navigation links.
3. Customizing the user menu items in the UserMenu component.
4. Adjusting the colors and styles to match your brand using Tailwind CSS classes.

This navbar should provide a solid foundation for a modern, responsive website with dark mode support and user interaction features.