How to implement logout in Nuxt.js?

I want to implement logout in my app so I saw logout() function in the documentation of Nuxt, I don't know how it exactly works since its not explained in the documentation but I tried to use it but its not working for me, when I click on logout link the user stays authenticated but just redirected to home page. in my server I don't have a logout function, I'm using laravel/passport, the response of a login request is a token and user data which then I save them in vuex and I use persistedstate to avoid emptying the state if I manually refresh the page.

Navbar.vue where I use logout

<template>
  <li v-if="$auth.loggedIn" class="nav-item">
    <nuxt-link class="nav-link" to="/" @click="logout">logout</nuxt-link>
  </li>
</template>

<script>
import { mapGetters } from 'vuex'
export default {
  computed: {
    ...mapGetters(['isAuthenticated', 'loggedInUser']),
  },
  methods: {
    async logout() {
      await this.$auth.logout()
    },
  },
}
</script>

store/index.js

<script>
export const state = () => ({
  user: {},
  token: '',
  id: '',
})

export const mutations = {
  saveUser(state, payload) {
    state.user = payload
  },
  saveId(state, id) {
    localStorage.setItem('id', id)
    state.id = id
  },

  saveToken(state, token) {
    localStorage.setItem('authToken', token)
    localStorage.getItem('authToken')
    state.token = token
  },
}
export const actions = {
  saveUserAction({ commit }, UserObject) {
    commit('saveUser')
  },
  logoutUser({ commit }) {
    commit('logout_user')
  },
}
export const getters = {
  getUser: (state) => {
    return state.user
  },
  isAuthenticated(state) {
    return state.auth.loggedIn
  },

  loggedInUser(state) {
    return state.user.user
  },
}
</script>

login.vue

<script>
export default {
  methods: {
    async login() {
      const succesfulLogin = await this.$auth.loginWith('local', {
        data: {
          email: this.email,
          password: this.password,
        },
      })
      this.$store.commit('saveUser', succesfulLogin.data.user)
      this.$store.commit('saveToken', succesfulLogin.data.token)
      this.$store.commit('saveId', succesfulLogin.data.user.id)

      if (succesfulLogin) {
        await this.$auth.setUser({
          email: this.email,
          password: this.password,
        })
        this.$router.push('/profile')
      }
    },
  },
}
</script>

thats how the store looks after I press on logout, loggedIn stays true

localstorege

Did you tried this?

await this.$auth.logout()

Also, try to bind it to a button rather than a link.
I'm not sure that the @click event happens properly when you have a nuxt-link.