public class Auth extends AppCompatActivity {

    private final String TAG = "log_tag";

    private EditText etUser;
    private EditText etPassword;
    private Button btnLogin;
    private Button btnSignUp;

    private Realm realm;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.auth);

        etUser = (EditText) findViewById(R.id.editTextUserName);
        etPassword = (EditText) findViewById(R.id.editTextPassword);
        btnLogin = (Button) findViewById(R.id.btnLogin);
        btnSignUp = (Button) findViewById(R.id.btnSignUp);

        btnLogin.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View view) {
                authorization(view);
            }
        });

        btnSignUp.setOnClickListener(new View.OnClickListener() {
            @Override
            public void onClick(View view) {
                Intent intent = new Intent(getApplicationContext(), SignUp.class);
                startActivity(intent);
            }
        });
    }

    private void authorization(View view) {
        Realm.init(getApplicationContext());
        RealmConfiguration realmConfiguration = new RealmConfiguration
                .Builder()
                .deleteRealmIfMigrationNeeded()
                .build();
        realm = Realm.getInstance(realmConfiguration);

        RealmResults<User> user = realm.where(User.class).findAll();

        for (int i = 0; i < user.size(); i++) {
            if (user.get(i).getUserName().equals(etUser.getText().toString())&&
                    user.get(i).getPassword().equals(etPassword.getText().toString())) {
                Log.d(TAG, "authorization: user size = " + user.size());
                Log.d(TAG, "authorization: userName = " + user.get(i).getUserName());
                Log.d(TAG, "authorization: password = " + user.get(i).getPassword());
                Profile.newInstance(user.get(i).getUserName());
                Intent intent = new Intent(getApplicationContext(), MainActivity.class);
                startActivity(intent);
            } else
                Snackbar.make(view, "Error! Check your entries.", Snackbar.LENGTH_LONG).show();
        }
    }
}

public class Profile extends Fragment {

    private final String TAG = "log_tag";

    private CircleImageView imageView;

    private TextView tvName;
    private TextView tvEmail;
    private TextView tvPhone;

    private Realm realm;

    @Override
    public View onCreateView(LayoutInflater inflater, ViewGroup container, Bundle savedInstanceState) {
        View view = inflater.inflate(R.layout.profile, container, false);

        imageView = (CircleImageView) view.findViewById(R.id.imageViewAvatar);

        tvName = (TextView) view.findViewById(R.id.tvFirstAndLastName);
        tvEmail = (TextView) view.findViewById(R.id.tvEmail);
        tvPhone = (TextView) view.findViewById(R.id.tvPhone);

        writeProfile();

        return view;
    }

    public static Profile newInstance(final String userName) {
        final Profile profile = new Profile();
        final Bundle bundle = new Bundle();
        bundle.putString("userName", userName);
        profile.setArguments(bundle);
        return profile;
    }

    private void writeProfile() {
        Realm.init(getActivity());
        RealmConfiguration realmConfiguration = new RealmConfiguration
                .Builder()
                .deleteRealmIfMigrationNeeded()
                .build();
        realm = Realm.getInstance(realmConfiguration);

        String userName = this.getArguments().getString("userName");//здесь падает ошибка

        Log.d(TAG, "writeProfile: userName = " + userName);

        RealmResults<User> user = realm.where(User.class).findAll();

        User userData = user.where().equalTo("userName", userName).findFirst();

        tvName.setText(userData.getFirstName() + " " + userData.getLastName());
        tvEmail.setText(userData.getEmail());
        tvPhone.setText(userData.getPhone());
    }
}

E/AndroidRuntime: FATAL EXCEPTION: main
                  Process: com.example.nikolai.engadgetnews, PID: 8073
                  java.lang.NullPointerException: Attempt to invoke virtual method 'java.lang.String android.os.Bundle.getString(java.lang.String)' on a null object reference
                      at com.example.nikolai.engadgetnews.Profile.Profile.writeProfile(Profile.java:61)
                      at com.example.nikolai.engadgetnews.Profile.Profile.onCreateView(Profile.java:40)
                      at android.support.v4.app.Fragment.performCreateView(Fragment.java:2184)
                      at android.support.v4.app.FragmentManagerImpl.moveToState(FragmentManager.java:1298)
                      at android.support.v4.app.FragmentManagerImpl.moveFragmentsToInvisible(FragmentManager.java:2323)
                      at android.support.v4.app.FragmentManagerImpl.executeOpsTogether(FragmentManager.java:2136)
                      at android.support.v4.app.FragmentManagerImpl.optimizeAndExecuteOps(FragmentManager.java:2092)
                      at android.support.v4.app.FragmentManagerImpl.execPendingActions(FragmentManager.java:1998)
                      at android.support.v4.app.FragmentManagerImpl$1.run(FragmentManager.java:709)
                      at android.os.Handler.handleCallback(Handler.java:739)
                      at android.os.Handler.dispatchMessage(Handler.java:95)
                      at android.os.Looper.loop(Looper.java:148)
                      at android.app.ActivityThread.main(ActivityThread.java:5417)
                      at java.lang.reflect.Method.invoke(Native Method)
                      at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:726)
                      at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:616)

Profile.newInstance(user.get(i).getUserName());
            Intent intent = new Intent(getApplicationContext(), MainActivity.class);
            startActivity(intent);