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In the days when I went to grade school and secondary school, it was all a different world from today. One thing that would seem shocking today – there was obligatory classes in Christianity. It was not teaching of different religions, but Christianity and nothing else. 1 Goodness, in grade school, we even read parts of the Bible, here and there, and we had to learn hymns by heart. How backward could they be, the people in the education department who made the decisions concerning our school curricula! Those were indeed the Middle Ages. Only in the last year of Gymnasium did we learn a bit about other religions, but not in any depth at all.
Every year from the first grade in elementary school and through to secondary school graduation, studentexamen, Christianity was a compulsory subject. Two hours a week, on par with history, geography, biology and such learning subjects. More important subjects, such as English and Swedish got three or even four hours a week. The one and only subject that was ever taught five hours a week was Latin in Gymnasium.
A school excursion in the summer; my close friends Britt (left) and Ragnhild (right). I am in the back and on the upper left is a girl whose name escapes me. We were around 14 (so I was 13), but where is my dearest friend, Aina?
Only during the last two years in Gymnasium did we get some flexibility in the subjects we studied. Of course we all chose a group with English. With most of my class mates, I chose English, German and philosophy, which was actually one year of psychology and only one year of philosophy. The only philosopher I really liked was Spinoza and I thought Kant was a terrible bore. When we got to Kierkegaard I was pretty much lost anyway – from lack of motivation. And he was just too too black and weird for my taste. I have always been an optimist.
A special honorary paragraph is due here to our incredible English teacher. Her name was
My much beloved and admired Stina Lindeberg with a nice smile on the right. On the left is Sture Silverberg, our math teacher throughout all the seven years, for the ones among us who added math on top of the other eight subjects. In between is our religion teacher, Maria Cronquist.
Stina Lindeberg and she was no doubt the most outstanding English teacher in all of Sweden,
certainly at the time. We had the same teacher from our first year of English until graduation, that is six years, simply because she insisted on staying with us. The way she motivated us was remarkable. I believe that most of us saw it as a given that English was our most important and also our favorite subject. In Gymnasium there was no more piecemeal homework. She lent us books of her own; we picked one we thought looked interesting and we read two-week assignments. At the end of an assignment, we gave a summary report to the class about the section of the book we were reading. English was the language that was spoken in the class.
In fact, I remember now that this started with 'Winnie the Pooh' in one of the lower grades. One at a time, we read a section of this wonderfully funny book, and one at a time we gave a report to the class. As a parenthesis, my close friends Aina and Harriet had to repeat a year because they flunked Latin. Laziness certainly since they were both bright. They decided to change schools and continue at Katedralskolan in Lund. Their English grades soared from medium/good to the top grades.
A winter excursion to Frostavallen in the middle of Skåne. Our great beauty, Karin is on the lower right. We are still good friends. Front left is Britt with me behind, wearing a head scarf I bought in Abisko, Lapland in 1947. Ragnhild is behind Karin. Harriet is straddling the tiny stream. Sorry her head was cut off. Age around 15.
Stina Lindeberg's classes were used not to check homework but for conversation and discussion. What a difference from our boring and incompetent German and French teachers. "Siv, Karin, Gertrud…, translate today's text!" Even if I hadn't prepared it, I think I usually managed fairly well to give a satisfactory performance. Except one time when I was caught not having even taken a look at my French homework.
We were reading Prosper Mérimée's Colomba (or at least I think it was Colomba), but today I don't remember in the least what the story is about. And still I had to teach it myself when I substituted for one term in Helsingborg lycée for girls in the fall of 1957, teaching French and Swedish. I had no say in the choice of book to read with the class, and so I was stuck with Colomba.
One day I was caught red-handed. "Siv Molin, translate today's text!" said our French teacher, Anders Billquist. (The male teachers addressed us by our full names back in those days. What a drag!) The text I had to translate was about Colomba who had spent the night sleeping in a cave. As he woke up he caught sight of a lion who had also been sleeping in a far corner of the same cave. Seeing the lion, "il se dressa sur son séant". I forged ahead and guessed "He got dressed sitting up…". General laughter and I'm not sure that our kindly French teacher Billquist even blamed me for not having done my homework. German was the same. Why this pedagogically ignorant preference for boring books that were uninteresting to us and often far too difficult? Why not try to motivate us by having us read books we would enjoy? I later chose to read La Symphonie pastorale by André Gide as my special project, and I loved it.
We referred to our teachers by their last names only, when talking about them, which John finds very odd. No Mrs or Miss or Mr. Until the day of the baccalauréat, or maybe rather at the big party we held for all our teachers in a restaurant outside of Malmö, with sketches, singing and good food, when we were officially told that now we should say 'du' to them, and use first names.
Getting back to our English classes and incomparable Stina Lindeberg – My great friend Britt recently told me a thing that had completely escaped me. As students joined our Gymnasium classes from other schools and were subjected to English classes where only English was spoken and where you were expected to speak it fluently, some of the newcomers literally broke down in tears. However, they were intelligent students and little by little they caught up.
There were only two branches in Gymnasium in my days, the Latin branch (Latinlinjen) and the math-and-science branch (Reallinjen). Very soon thereafter a new branch was created, called the general branch (Allmänna linjen) where neither Latin nor high-level math was taught.
There were also morning prayers both in grade school and in secondary school. Listening to a brief sermon and hymn singing. A girl in my class played the organ while we all sang from our hymn books. Oh my god! What a backward era this was. Did they do the same in Germany I wonder, a country that influenced our culture so very heavily? Or were Nazi morning assemblies exclusively dedicated to patriotic lecturing and singing? Very likely.
In secondary school, right at the end of the hymn and after the person who had given the little sermon had stepped off the podium, our elderly woman principal, Marianne Mörner, got up on the podium and firmly told us, in no Christian tone of voice, that she wasn't going to see 'her girls' chew gum in school. Ever. "The one difference between a gum-chewing girl and a cow was the intelligent look in the eyes of the cow." I bet she thought she had found a very clever way of expressing herself. There was usually some kind of strict rules of that kind being spelled out to us by tall and sinister-looking Marianne Mörner after the hymn singing had died down. But only just died down.
Mademoiselle Mörner had been the principal for our school ever since 1929, long before the major and modern addition to the old school was built. She was a linguist and a docent in Romance languages at Lund University. Now she became the 'mother' of all her secondary school girls and the number grew every year, as more and more girls went on to studentexamen. Here is just a little incident that shows Marianne Mörner's other side.
In the lower grades, junior high school or so, we had sewing for two years and cooking the third year. In second grade we were supposed to sew a dress. Aina and I thought it would be nicer to make a skirt and a blouse, so we were of course allowed to do that. I learned to sew on an electric machine, since at home we only had a treadle machine. We made red wide skirts and white blouses and they really looked quite nice. One day we were wearing them in school both of us, and as we walked down the hallway, here comes Marianne Mörner towards us. She was so delighted to see us, two nice-looking little girls in identical outfits, that she stopped and talked to us. She was over-brimming with delight when she learned that we had made them ourselves – in sewing class.
Mörner, as we usually referred to her, retired the year we entered gymnasium, 1948, and I believe the school that same year ceased to be a partly private school. It also soon thereafter ceased to be a girls' school only. Bertil Block became our new principal, and the school gradually lost its exceptionalism.
Marianne Mörner was a tall, skinny spinster whose only close friend on the faculty was Ingeborg Tegnér, my philosophy teacher and my sister Gun's English teacher. This dried-out little woman was the descendant of our 'great national' poet Esaias Tegnér, one of the most outstanding members of the Gothic League (det Götiska Förbundet), from the early 19th century – an association of poets and writers that had clear nationalistic and patriotic features, an attempt at reviving the Viking spirit. Oh well, worse things have happened in history. The two spinsters found great satisfaction in knowing that they were both members of the Swedish intellectual and social elite. Mörner's father was Olof Stellan von Mörner, Ryttmästare (officer in the cavalry) at a German regiment and owner of vast properties in Germany as well as in Sweden, inherited in his family from the 16th century.
In the late 1990s when I was working on a big project about the escape over to Sweden of Danish Jews, which began in October 1943, I managed to get an academic paper from a library in Uppsala about Nazism in Sweden during and before World War II. I read all sorts of books on the subject of Danish Jews and even rescue organizations for German Jews set up in Berlin, and I was fascinated by my project. In this paper about Nazism in Sweden – what did I find? Marianne Mörner was one of the pillars of the aristocratic Nazi party. I wasn't surprised. There were at least three different Nazi parties in Sweden at the time, and one was especially dedicated to Swedish aristocratic admirers of Hitler. The three parties, for workers, farmers or the upper classes, were all named National Socialist this or that.
This woman had a historic ancestor, Baron Carl Otto Mörner (Friherre Mörner), the man who took it upon himself in 1810 to go to Paris to ask le Maréchal Bernadotte to become the Swedish Crown Prince. Prince Karl August, son of the present king, Karl XIII, had suddenly died and various political groups were vying for a candidate who would suit their own ambitions. Baron Mörner cheated them all by hurrying to Paris and persuading Jean Baptiste Bernadotte to become the Swedish Crown Prince.
Bernadotte was one of Napoleon's marshals and he had even taken over Napoleon's own betrothed, Désirée Clary, and married her in 1798. When he now made his entrance into Sweden he was accompanied by his wife Désirée, who became the Queen of Sweden a few years later, one big step better than being Napoleon's betrothed, especially since Napoleon in the meantime had married Joséphine de Beauharnais. As King of Sweden from 1818 Jean Baptiste became Karl XIV Johan. He governed Sweden for 34 years, but he never learned to speak Swedish.
My research about Swedish Nazism took me much further, however, than to the upper class National Socialist party of which Marianne Mörner was apparently a member. As I already knew, Sweden made a pact with Hitler, promising to deliver iron and arms, (cannons from Bofors, Boforskanoner), allowing German trains to carry soldiers and arms through Sweden to Narvik in the north of Norway, and to generally look the other way when the Germans needed their silent collaboration. So much for Swedish so-called neutrality. I now learned much more about the stand-off position of Swedish bourgeois Jews, who kept a low profile during the war years – be they Bonnier or Josephson – and the general fear of the Swedish war-time coalition government to interfere with Nazi plans.
Denmark was an occupied country very different from Norway or the continental German-occupied countries. Norway was of course ruled by the collaborator Vidkun Quisling, but the country was teaming with resistance fighters, many of whom made their way into Sweden across the border in the middle and the north that was not very distinguishable.
Germany was very much dependent on Denmark for food for the military, and so this fertile country got special treatment as a 'good' occupied nation. Until October 1943, their Jewish population was left undisturbed, so as to show the world what a benign occupier Germany really was. It got to be a model, and the Germans pointed to Denmark as proof of their harmless intentions as an occupying power. Sweden went along, and it even happened that German Jews, via Denmark, who had managed to make it over to Sweden, were forced to return to Denmark. All this suddenly changed in October '43.
In September 1943, a drastic order was issued from Hitler personally. He suddenly decided that all Danish Jews were to be rounded up and deported to German concentration camps. This took place during the night of Rosh Hashanah, October 1. However, thanks to Georg Ferdinand Duckwitz, a German working in Copenhagen as a naval specialist for the German transportation department, the dreadful news was leaked to Hans Hedtoft, president of the Danish Social Democratic Party, who then told Rabbi Marcus Melchior about the urgent need to spread the news among the entire Jewish population, especially in Copenhagen. This took place just two days before the fatal night. Most Jews managed to go into hiding, some just with neighbors, others with families they vaguely knew in the countryside.
It was also of importance to the fate of the Jews that the SS forces in Denmark were far from sufficiently numerous to really guard the coastline to Sweden. Many of the German soldiers and most of the Danish police purposely looked the other way when refugees were sighted close to the coast or transported inside a truck along the east coast. The 95% of the Danish Jews who were saved went through enormous ordeals to finally make it over to Sweden, mostly in small fishing boats and even canoes. Quite a few refugees drowned on nights when the winds were rough in the Sound. But 7,000 Jews out of 7,500 made it safely across the Sound, which must be considered as something of a miracle. Most of the remaining Jews were sent to Theresienstadt concentration camp in former Czechoslovakia and actually came back unharmed at the end of the war, thanks to endless involvement by the Danish Red Cross who regularly checked on the living conditions at the camp. And finally there was the remarkable rescue of thousands of prisoners from that camp by Folke Bernadotte's White buses. 2
The Swedish government learned about the radical change in the German stand relating to the Danish Jews at the time when Germany was factually a broken nation on all fronts. Even so, Hitler went on giving orders at Stalingrad "Not one step back" — the formal end of that horrible monhs-long winter war was in February 1943, Easy to say for Hitler in Berlin, but impossible for the half-starved and half-frozen German soldiers during the onslaught of Russian soldiers, who were well prepared for the climate.
The German war in Africa was lost. Field Marshal Rommel, the leader of the Afrika Korps, was a sick man who went to Germany to beg Hitler on his knees to order a stop to the fighting. The battle for El Alamein was lost in 1942 and any sane man could see that the war in Africa was unwinnable. But Hitler was not a sane man. He ordered Rommel to immediately go back to Africa and to make sure he would get to Egypt. Even the naval war in the Atlantic was lost and everything pointed to a soon upcoming surrender by the Germans. That was the situation when the Swedish government finally broadcast a message over to Denmark in the last days of September 1943, announcing that all Danish Jews and members of the resistance were now welcome to Sweden .
The White Buses on their way to Sweden going through Schleswig-Holstein in northern Germany. (Wikipediaà
My eyes were teary and my heart beat with joy and emotion, when I read about the strong searchlights being focused on the Danish boats, but this time, at last, by friends.
The little boats arrived at night in Swedish territory, and the loudspeakers announced "You are now in Swedish waters and we wish you welcome to Sweden." All kinds of Swedes were ready at the landing with warm clothes, warm food and warm welcomes.
A fairly large number of Danish Jews were rounded up nevertheless, in spite of the wonderful cooperation and support by the Danish people, and were taken to the concentration camp Theresienstadt in occupied Czechoslovakia. Miraculously, almost all of
Folke Bernadotte (1895-1948) was the leader of the action that brought back around 25,000 prisoners from the German concentration camps (Wikipedia)
them survived, thanks to regular visits by the Danish Red Cross at the camp and the ultimate rescue of the survivors by Count Folke Bernadotte and his 'White Buses'.3 The Danish Red Cross also brought food and made sure that living conditions were bearable.
In the spring of 1945 innumerable Scandinavian prisoners were saved from German concentration camps through Bernadotte's intervention. He had personally negotiated their release with Heinrich Himmler
"Himmler, Reichsführer of the Schutzstaffell (Protection Squadron; SS),,was one of the most powerful men in Nazi Germany and among those most directly responsible for the Holocaust."
while the Danish Aid Corps arranged for cars and buses to transport the prisoners. For an example, the white buses carried away, in one day only, 423 liberated prisoners.
Continued: Chapter 6 (Part 2) – Studies at Lund University
The teaching of Christianity (kristendom) changed in the sixties and became 'knowledge of religions' (religionskunskap). Obligatory teaching of Christianity would have been impossible today for several reasons, since there are students who are not Christians at all. There are Muslims, atheists, and not all the so-called Christians are Lutherans either. ↩
More about Folke Bernadotte and his White buses below. ↩
The White Buses © USHMM ↩
3 thoughts on "Chapter 6 (Part 1) – Studies and an interlude about World War II"
sara agrup on 4 March 2015 at 14:54 said:
What a wonderful English teacher you had! Not even today are there many of such modern teachers.
It was also interesting to read about the destiny of Danish jews.
admin on 4 March 2015 at 18:05 said:
Oh wow, Sara, you read this chapter too. I just rewrote it and posted the final version yesterday. I added all the part about the teachers and WWII. Yes, it was indeed fascinating doing that research, reading all those books back in the late 90s. I hope you saw the pictures I just added this morning. My friend Britt in Stockholm actually mailed me the picture of the three teachers, including Lindeberg and Silverberg. It's from our big dinner and entertainment for all our teachers at Vellinge gästis, and I, for some reason, never bought that picture.


I am writing to you as part of the consultation on the Draft Defamation Bill. I welcome the direction of the bill, but feel it doesn't go far enough to allow full freedom of speech. I would like to see the whole idea of defamation taken out of the legal system. My view is that any possible good that can come of censorship is greatly outweighed by its harm. I am afraid that any defamation law will be abused by the rich and powerful to suppress criticism and dissent. Thus, I believe that anybody should be able to state any view, whether or not some think it false or malicious.
You only say that because you're a gay con artist who has joined the Hittler youth and privately enjoy X factor.


Implement DLC feature, users are now able to download and manage owned DLC Content in their Library page.
Add repair button in submenu to quick recover content file missed case issue.
Enhanced country setting selection process during first login.
Changed VIVE Logo to VIVEPORT Logo on navigation bar.
Enhanced login browser login collaboration and mechanism.
Enhanced VIVEPORT VR performance and reduced start-up time.
Improved filters of content types, languages and genres.


However, the feeling of an involvement with the character in the outcome of her thoughts, captured. Beside that she is using the experience that her job, as a crime investigator has given her.
A piece of news is not verified the way a piece of knowledge iswhereas knowledge can be verified empirically, news can be verified empirically only after the hearer has already heeded its call.
we have only just two without her great love for me I would surely die Even though I am completely useless with regards to personal joys of life, she loves. Roys delight at his return from Changi turned to despair when he discovered his wife of 21 years, Ellen, mind reading computers research papers was no longer living in their family home. Diagnosed with a nervous condition, and in the mid-1950s still suffering the after-effects of beri-beri and malnutrition, Robert was forced to wear surgical boots. Wives themselves were frequently baffled and distressed by this turn of events.
Ellen was brief and resolute: All your writing will not alter me as its definitely No Ill never return to you. Some men believed that their experience of captivity had been, quite literally, emasculating. The wife repeatedly stated in conversations with her husband, I just want to be free, and ultimately left to live with her sister. Returning POWs were not immune to a broader trend in Australia in the late 1940s when there was a sharp spike in the divorce rate.
(No Ratings Yet loading). In contrast, Herberts first marriage had broken down when he returned from captivity and experienced what he called a loss of virility. Many wrote a simple question mark in response to the cosmetology essay thesis query. The timberworkers wife, Betty, corresponded over many years with the trustees on behalf of her husband, Robert. Years of fantasising about reunion, the experience of being loved, eating home-cooked meals, and rolling over in bed to see a familiar face came to naught. He was not alone in being perplexed by such self-destructive behaviour. Complaints about psychiatric disturbance were common, the recourse to violence less so, but the incidence of family violence in the homes of former POWs is impossible to gauge on the evidence of fund applications alone. The events tend to develop incrementally: the person will show the own behavior more and more violently each time. The trustees could not have anticipated that their form would open a vein, and that out of it would pour returned POWs stories of dysfunctional marriages, family breakdown and impotence. Family violence describes violent behavior between the family members, and it can take any form, and may occur not only between partners, but also it can be aimed at children. Betty described Robert as a difficult man to live with, still nasty and cutting when he is worried but he does work hard. Clarence had developed a deeply ambivalent relationship to his wife and home, at once idealising and demonising them.


text:
Q: Assign value for each row How to assign t-sql expression for each tables row?
For example:
Foreach user in Users
{
user.Place = SELECT COUNT(*) AS [value]
FROM [Users] AS [t0]
WHERE (COALESCE([t0].[Votes],@p0)) > @p1;
}

where Users is table. It's pseudo-code but I want to do that in t-sql stored procedure.

A: So you want all user records, and with each record, a count..
You can use a subquery - but what are @p0 and @p1?  Check this
SELECT u.*, Place = (
    SELECT COUNT(*) AS [value]
    FROM [Users] AS [t0]
    WHERE COALESCE([t0].[Votes],0) > u.Votes)
FROM Users u

From SQL Server 2005+, you can use windowing functions
SELECT u.*, Place = Rank() over (order by u.Votes)
FROM Users u

A: Are you looking for something like
ROW_NUMBER (Transact-SQL), RANK (Transact-SQL) or DENSE_RANK (Transact-SQL)
Also have a look at this example

A: Is this what you mean:
UPDATE Users
SET Place = (
  SELECT COUNT(*)
  FROM Users
  WHERE COALESCE(Votes, @p0) > @p1
)

?

redpajama_set_name: RedPajamaStackExchange


text:
Five asteroids will skin past Earth TODAY –and one is twice as big as the Empire State Building
According to the latest reports by the astronomers, five asteroids are about to skim past Earth, and the size of one of them is enormous than the Empire State Building, which is one of the tallest building in the world.
Asteroid 1998 HL1 is humongous, and the astronomers say that it will be visible from the Earth. The live streaming of the flypast of the colossal asteroids will be telecasted over YouTube under the Virtual Telescope Project from 6 pm, BST.
Astronomers spot dead 'monster' galaxy lurking in deep space
New kind of northern lights called 'dunes' discovered by amateur scientists
Gian Masi, the astronomer, said that we would have a close but safe encounter with the potentially dangerous Asteroid (162082) 1998 HL1, which will pass the Earth at a speed of 6.2 millions of km. People can join to watch the asteroid's encounter with Earth from their home with the help of the Virtual Telescope Project.
The diameter of the Asteroid 1998 HL1 is about 3248 feet wide, which makes it even taller than the Burj Khalifa. The Asteroid is being expected to make its closest approach to Earth around 6:21 pm, BST. It is likely to shoot past our planet at a speed of 25,076 miles per hours.
Over the years, scientists studying space have spotted countless galaxies in all different phases of their lives. From newborn galaxies...
British carbon tax leads to 93% drop in coal-fired electricity
Scientists start making Air out of Lunar Dust as Part of Plan to Live on the Moon
Scientists discovered a new 'super-Earth,' and want to know if it could sustain life
Astronomers reveal interstellar thread of one of life's building blocks
Durham scientists help create telescope which can see into the past
Huge 280-mile-wide asteroid is so big it could be classed as a dwarf planet

redpajama_set_name: RedPajamaCommonCrawl


text:
NEWS-PHOTOS
MESSAGES OF THE RECTOR MAJOR
United States – Rector Major begins Northern California Visit
(ANS – Berkeley) – Intense, with meetings and appointments, the visit of the Rector Major, Fr. Ángel Fernández Artime, to the "St. Andrew" Province of the Western United States (SUO) continues. On Friday, November 18, the 10th Successor of Don Bosco visited the community of the Daughters of Mary Help of Christians at Bellflower, before beginning the second part of his visit to Northern California, with stops in Berkeley and Richmond.
The St. Dominic Savio community of Salesian Sisters in Bellflower, Los Angeles, welcomed Fr. Angel into their home for an intimate Mass. Also joining them were the Salesian Sisters from the Community of St. Thomas the Apostle and the Postulants. The Salesian Sisters and Postulants were delighted to be able to celebrate the Eucharist with Fr. Ángel, Fr. Hugo Orozco (Interamerican Regional), and Fr. Mike Gergen (Pastor). After Mass, they walked to St. Dominic Savio Church to greet the school community as well as the other groups present in the Los Angeles region.
The teachers and students presented the Rector Major with gifts, as well as songs prepared by the Marian Club, Green Team (recycling team), Choir, and Student Council. After the school presentation a representative from ADMA-Y (ADMA Youth) from the St. Dominic Savio Church and some catechists from St. Thomas the Apostle Church, shared stories and info about their mission ministries in the Los Angeles region.
Soon after, the Provincial Fr. Mel Trinidad accompanied Fr. Angel and Fr. Hugo on a flight from Los Angeles to Oakland, California, to begin the Rector Major's visit to northern California Salesian communities and works. From the Oakland Airport, they traveled by BART (public electric train) to the city of Berkeley.
During his short visit to the Salesian community in Berkeley, Fr. Joe Boenzi and Fr. Alejandro Rodriguez (professors and staff) presented to the Rector Major an ongoing formation program proposal for the benefit of the Inter-American region of the Congregation and the Rector Major affirmed his support for Don Bosco Hall in Berkeley.
The next stop for the evening was a visit to Salesian College Preparatory in Richmond, California. The visit began with a brief prayer service at the Salesian Cemetery where Fr. Raphael Piperni was buried. Fr. Piperni was the leader of the first group of Salesian missionaries to arrive in the United States 125 years ago. The small crowd gathered for a beautiful dinner thereafter and was treated to some incredible performances by the student Jazz band and, of course, the Rector Major himself. Student representatives also presented the Rector Major with some gifts.
After dinner, Fr. Ángel gathered with the Salesian brothers and priests for a private meeting in the school library to express his fraternal words of affirmation and encouragement, and some continuing updates on the Congregation.
Title Title
https://www.infoans.org/en/sections/news/item/16674-united-states-rector-major-begins-northern-california-visit#sigProIda427a31a5e
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Governor PALIN: We Will Rise Up!
WASHINGTON (AP) — Voters will "rise up" in opposition if Republican power brokers try to take the presidential nomination away from Donald Trump or Ted Cruz at the GOP convention this summer, Sarah Palin said Thursday in a wide-ranging interview.
Republican presidential candidate businessman Donald Trump gestures during the Republican Presidential Debate, hosted by CNN, at The Venetian Las Vegas on December 15, 2015 in Las Vegas, Nevada.
Palin, who has endorsed Trump, said she is confident he will win the GOP nomination, but said she can support Cruz if he emerges as the nominee.
She said she backs Trump because he is "so reasonable and so full of common sense and knows that for America to be great again we have to develop our natural resources" such as oil and natural gas.
While some GOP leaders worry that Trump's disparaging comments about women, minorities and others have him struggling in the polls with key voter blocs, Palin said Trump would be the GOP's strongest nominee. Trump has created the "big tent" that party leaders have long been seeking, she said, citing the billionaire businessman's appeal to independents and "blue dog Democrats" in the South and other rural areas.
Former Republican vice presidential candidate Sarah Palin speaks to a crowd as she introduces Republican presidential candidate Donald Trump at a rally in Tulsa, Okla., Wednesday, Jan 20, 2016.
While Palin said she could support Cruz, she said it was "unfortunate that he has people around him who are not truthful. I sure want to believe it's the people around him and not Cruz as a person who would flip-flop on so many issues," including trade and immigration.
"He was there at the border incentivizing illegal families coming on over the border with gift baskets of soccer balls and teddy bears and now he says he was never for amnesty. Yes, you were, dude, come on," Palin said.
The former Alaska governor was in Washington to promote a new documentary that seeks to debunk what it calls myths and hype about human-caused global warming. The movie, "Climate Hustle," questions whether there is a genuine scientific consensus about global warming and features more than 30 scientists who reject mainstream climate science.
Palin said she wants viewers "to feel empowered to ask questions about what is being fed them by the scientific community." The movie opens May 2.
On other topics, Palin said Ryan has failed as House speaker because the House has not approved a federal budget as promised. And while she said she does not regret staying out of the presidential race, she did not rule out a possible presidential bid in the future.
"If I can trust that the electorate is as fed up as I am with the shenanigans and the corruption and the crony capitalism … and can handle a true outsider, then you know that the time would be right" to run for president, she said.


Trigger warning: The ending of this winter horror story includes violence which might be disturbing for some readers.
Snow. That was what they called it, the people down in the village. I wasn't supposed to know that or know them, but I just… couldn't help myself. They were so fascinating. They lived in such large groups and never moved their dens, and while that was peculiar, it was not what brought me back time and time again. Rather, it was the village peoples' relationship with snow.
You must understand, I am not one of the village people. My family does not even interact with the village people and we do not stay in one place for long. We do not have a permanent den and snow has never been beautiful or serene or a sign of good tidings. For my family, snow is a bad omen. It is a warning that food will be scarce, days will be short, and nights will be brutally cold. It is the beginning of the harsh season and no one smiles or laughs or celebrates the harsh season, but the village people do.


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text:
\section{Introduction}

Coupled chain systems with ladder structure
have attracted much current interest in the study of
strongly correlated electron systems.
It has been proposed
that these systems could show a superconducting state
in the metallic phase.\cite{drs,grs,nws}
Such ladder systems have been realized experimentally in
${\rm Sr_{n-1}Cu_{n+1}O_{2n}}$ and ${\rm VO_2P_2O_7}$.\cite{sr,vo}
More recently it has been reported that
hole-doped ${\rm Sr_{n-1}Cu_{n+1}O_{2n}}$ systems
actually undergo the transition
to the superconductivity under pressure.\cite{aki}
As microscopic models for such systems, two coupled Hubbard chains
and two coupled $t$-$J$ chains have been extensively
studied.\cite{drs,grs,nws,fin,fab,ima,ttr,kv,fuji1,nao,shu,baf}
The ground state of these models at half-filling is the Mott
insulator with spin gap.
When holes are doped into these systems, a metallic state with spin
gap is realized. The resulting massless charge mode belongs to a
class of the Tomonaga-Luttinger (TL) liquid
(sometimes referred to as the Luther-Emery class),
and has the enhanced fluctuation toward superconductivity
This superconducting state may be related with
that found in ${\rm Sr_{n-1}Cu_{n+1}O_{2n}}$.

Such ladder systems are quasi-one-dimensional, so that they
may be quite sensitive to random potentials.
It is thus important to investigate the
effects of quenched disorder on
two coupled Hubbard chains.
The case away from half-filling was studied
by Orignac and Giamarchi before.\cite{og}
However the case at half-filling has not been considered so far.
In this case, we can systematically study
how the Mott transition competes with the Anderson localization
when the disorder is introduced to the system.
This issue was previously addressed
by the present authors, but for one-dimensional (1D)
interacting electron systems.\cite{fuji}(see also refs.18 and 19)
In contrast to such single chain models, the Mott
insulating state for
two coupled Hubbard chains at half-filling has the
spin gap as well as the
charge gap.  Since the randomness affects not only the charge gap
but also  the spin gap, it is expected that the presence
of the spin gap may bring about novel properties,
which have not been
observed in the single chain model,
for the competition between
the Mott transition and the Anderson localization

In this paper,
we systematically study the effects of disorder on the
two coupled Hubbard chains at half-filling.
By exploiting bosonization and renormalization group methods,
we discuss how the introduction of disorder drives the system to
the Anderson localized state.
In particular, we focus on the role played by the
spin gap for the competition between the
Mott insulator and the Anderson localization.

We deal with two types of quenched disorder:
random impurity potentials within each band, and
random hopping between two bands.
We first discuss the effects of the former type of disorder.
As was pointed out in ref. 17,
the backward scattering due to impurities generates random gauge
field coupled with electron currents.  We find that
the resulting random gauge fields play an essential role
for destroying a charge gap in the coupled-chain systems.
At half-filling, sufficiently strong random forward scattering
due to impurities and random gauge fields
destroy charge gaps as well as spin gaps, and the
resulting massless charge modes are localized
by impurity  backward scattering.
However, for intermediate strength of random forward scattering,
only one of the charge gaps is collapsed, and
other modes are still gapful.
In this case, we find a parameter region where
one charge degrees of freedom is in the Anderson localized state,
while the other one is still in the massive state.

In the case of random hopping between two bands,
the analysis is more difficult, because two charge modes are
mixed by random hopping.
However introducing a unitary transformation,
we can avoid this difficulty and find the same low-energy
fixed point as the case without random interband hopping.

The organization of the paper is as follows:
In Sec. II, we introduce the model for the coupled Hubbard
chains, and apply bosonization methods
to obtain low-energy effective theory. For later convenience,
we also summarize the results known for a clean
system without randomness.
In Sec. III, we derive the scaling equations for
a disordered system at half-filling.
Sec. IV and V  are devoted to the discussions
about the fixed point properties in the half-filling case.
In particular, we discuss in detail how the competition
between the Mott transition  and the Anderson transition
occurs when the random potentials are introduced.
 In Sec. VI, we further take into account
the effects of random hopping  between two bands.
Conclusion is given in Sec. VII.

\section{Model and bosonization method: summary of the results
for clean systems}

We first summarize the results for a clean system
without random potentials, and then investigate the effects of
disorder on the resulting fixed points.
Our model Hamiltonian in the clean case is given by,
\begin{eqnarray}
H&=&-t\sum_{i,s,\alpha}
c^{\dagger}_{i s\alpha}c_{i+1 s\alpha}+h.c.
+U\sum_{i, \alpha}n_{i\uparrow\alpha}n_{i\downarrow\alpha}
\nonumber \\
&&-t_{\perp}\sum_{i, s}[c^{\dagger}_{i s 1}c_{i s 2}+h.c.],
\label{model}
\end{eqnarray}
where $c_{i s\alpha}$($c^{\dagger}_{i s\alpha}$) is
an annihilation (creation) operator for electrons with spin
$s=\uparrow \downarrow$ and chain-index $\alpha=1,2$ at a site $i$, and
$n_{i s\alpha}=c^{\dagger}_{i s\alpha}c_{i s\alpha}$.
The last term is the hopping term between two chains.
The kinetic energy part is diagonalized in terms of a new basis,
\begin{eqnarray}
\psi_{s 1}(k)&=&\frac{c_{s 1}(k)-c_{s 2}(k)}{\sqrt{2}}, \\
\psi_{s 2}(k)&=&\frac{c_{s 1}(k)+c_{s 2}(k)}{\sqrt{2}}.
\end{eqnarray}
The dispersion relations in  this basis are given by
$\varepsilon_{1 k}=-2t\cos k + t_{\perp}$ and
$\varepsilon_{2 k}=-2t\cos k - t_{\perp}$, resulting in
the two decoupled bands. For $t_{\perp}>2t$, there exists a band
gap between these two bands.
In the following we restrict our
arguments to the interesting case of $t_{\perp}<2t$,
where the chemical potential may cross the two bands
both at and near half-filling.
We apply abelian bosonization methods to this model
to obtain the scaling equations.\cite{eme,hal}
In the presence of spin degrees of freedom, non-abelian
bosonization is generally more convenient for preserving
SU(2) symmetry.\cite{witten,aff1}
However, in our model the backward scattering due to impurities
is expressed in terms  of the operators which are not included in
the operator content of SU(2) Wess-Zumino-Witten model.
Therefore we exploit conventional abelian bosonization methods
by carefully dealing with SU(2) symmetry
of spin degrees of freedom.
Passing to the continuum limit,
we linearize the dispersions around the Fermi momenta
$k_{\rm F1,2}=\vert \cos^{-1}((\pm t_{\perp}-\mu)/2t)\vert$
($\mu$: chemical potential), and
divide the electron operators into left- and right-going parts,
$\psi_{s 1}\sim \exp(i\sqrt{4\pi}\phi_{s 1 L}(x)+ik_{\rm F1}x)
+\exp(-i\sqrt{4\pi}\phi_{s 1 R}(x)-ik_{\rm F1}x)$, etc.
Defining the new boson phase fields,
\begin{eqnarray}
\phi_{+\rho}(x)&=&\frac{\phi_{\uparrow 1}(x)+\phi_{\downarrow 1}(x)
+\phi_{\uparrow 2}(x)+\phi_{\downarrow 2}(x)}{2}, \\
\phi_{-\rho}(x)&=&\frac{\phi_{\uparrow 1}(x)+\phi_{\downarrow 1}(x)
-\phi_{\uparrow 2}(x)-\phi_{\downarrow 2}(x)}{2}, \\
\phi_{+\sigma}(x)&=&\frac{\phi_{\uparrow 1}(x)-\phi_{\downarrow 1}(x)
+\phi_{\uparrow 2}(x)-\phi_{\downarrow 2}(x)}{2}, \\
\phi_{-\sigma}(x)&=&\frac{\phi_{\uparrow 1}(x)-\phi_{\downarrow 1}(x)
-\phi_{\uparrow 2}(x)+\phi_{\downarrow 2}(x)}{2},
\end{eqnarray}
with $\phi_{s 1(2)}(x)\equiv\phi_{s 1(2) L}(x)+\phi_{s 1(2) R}(x)$
($s=\uparrow,\downarrow$),
and their canonical conjugate momentum fields,
$\Pi_{+\rho}(x)=\partial_x\theta_{+\rho}(x)
\equiv\partial_x(\phi_{+\rho L}-\phi_{+\rho R})$, etc.,
we write down the low-energy effective Hamiltonian,
\begin{eqnarray}
H&=&H_0+H_1+H_2, \\
H_0&=&\sum_{a=\pm,\nu=\rho,\sigma}
\int dx\bigl[\frac{v_{a\nu}}{2K_{a\nu}}
(\partial_x\phi_{a\nu}(x))^2 \nonumber \\
&& \qquad\qquad +\frac{v_{a\nu} K_{a\nu}}{2}
\Pi_{a\nu}^2(x)\bigr], \\
H_1&=&U\int\frac{dx}{\alpha}\cos(\sqrt{8\pi}\phi_{1\rho}+\delta_1 x)
\nonumber \\
&&+U\int\frac{dx}{\alpha}\cos(\sqrt{8\pi}\phi_{2\rho}+\delta_2 x), \\
H_2&=&U_1\int \frac{dx}{\alpha}\cos\sqrt{4\pi}\theta_{-\rho}
\cos\sqrt{4\pi}\phi_{-\sigma} \nonumber \\
&&+U_2\int \frac{dx}{\alpha}
\cos\sqrt{4\pi}\theta_{-\rho}\cos\sqrt{4\pi}\phi_{+\sigma}
\nonumber \\
&&+U_3\int \frac{dx}{\alpha}\cos\sqrt{4\pi}\theta_{-\rho}
\cos(\sqrt{4\pi}\phi_{+\rho}+\delta x)  \nonumber \\
&&+U_4\int \frac{dx}{\alpha}
\cos\sqrt{4\pi}\theta_{-\sigma}\cos\sqrt{4\pi}\phi_{+\sigma}
\nonumber \\
&&+U_5\int \frac{dx}{\alpha}\cos\sqrt{4\pi}\theta_{-\sigma}
\cos(\sqrt{4\pi}\phi_{+\rho}+\delta x) \nonumber \\
&&+U_6\int \frac{dx}{\alpha}
\cos(\sqrt{4\pi}\phi_{+\rho}+\delta x)
\cos\sqrt{4\pi}\phi_{-\sigma} \nonumber \\
&&+U_7\int \frac{dx}{\alpha}\cos\sqrt{4\pi}\phi_{-\sigma}
\cos\sqrt{4\pi}\phi_{+\sigma} \nonumber \\
&&+U_8\int \frac{dx}{\alpha}
\cos(\sqrt{4\pi}\phi_{+\rho}+\delta x)
\cos\sqrt{4\pi}\phi_{+\sigma} \nonumber \\
&&+U_9\int \frac{dx}{\alpha}\cos\sqrt{4\pi}\theta_{-\sigma}
\cos\sqrt{4\pi}\theta_{-\rho}, \label{int1}
\end{eqnarray}
where $\delta_{1,2}=4k_{{\rm F} 1,2}-2\pi$, and
$\delta=2(k_{{\rm F}1}+k_{{\rm F}2})-2\pi$.
$H_1$ and $H_2$ are, respectively, the umklapp scattering terms
within each band and the other scattering terms of
electron-electron interaction.
For on-site Coulomb interaction, initially $U_i=U$ for $i=1\sim7$
and $U_8=U_9=0$.
We have dropped the terms
which include an oscillating factor
$\exp(\pm 2i(k_{\rm F1}-k_{\rm F2})x)$,
because these terms are irrelevant in the long wave-length limit.
We have also omitted the forward scattering terms of
electron-electron
interaction, which just renormalize the TL parameters.

We first consider the case of half-filling
$\delta=0$. Then $H_1$-term is irrelevant and can be discarded
after the renormalization procedure
because of the oscillating factors.
In general, the above Hamiltonian produces
the mass gaps both for the $(\pm)$-charge sectors and
$(\pm)$-spin sectors. In particular, it is to be noted that
there are two possibilities
for the gap formation of the $(-)$-spin sector;
the spin gap is generated owing to locking
of the $\phi_{-\sigma}$ or $\theta_{-\sigma}$ field.
In this connection, we wish to mention that
there is a special case for which some peculiar
behavior in the mass generation is observed, i.e. the case
of $U_1=U_2=U_3=U_4=U_5=U_6=U_7=U_8=U_9$
and $K_{-\sigma}=1$, where the model has
special symmetry enhancement.\cite{fin,shu}
This situation realizes in the case with a long-range
electron-electron interaction between parallel
spins, $H'=V\sum_{i, s=\uparrow\downarrow}n_{i s}n_{i+1 s}$,
where the initial values of $U_8$ and $U_9$ are non-zero.
To see the special symmetry, we rewrite the interaction term $H_2$ as,
\begin{eqnarray}
H_2&=&U_1\int \frac{dx}{\alpha}[(\cos\sqrt{4\pi}\phi_{+\rho}
+\cos\sqrt{4\pi}\phi_{+\sigma} \nonumber \\
&+&\cos\sqrt{4\pi}\phi_{-\rho})
(\cos\sqrt{4\pi}\phi_{-\sigma}+\cos\sqrt{4\pi}\theta_{-\sigma})
\nonumber \\
&+&\cos\sqrt{4\pi}\theta_{-\rho}\cos\sqrt{4\pi}\phi_{+\sigma}
\nonumber \\
&+&\cos\sqrt{4\pi}\theta_{-\rho}\cos\sqrt{4\pi}\phi_{+\rho}
\nonumber \\
&+&\cos\sqrt{4\pi}\phi_{+\sigma}\cos\sqrt{4\pi}\phi_{+\rho}].
\label{int2}
\end{eqnarray}
Then the Hamiltonian is invariant under the dual transformation
$\phi_{-\sigma}\leftrightarrow\theta_{-\sigma}$.
This is nothing but the Kramers-Wannier symmetry of the Ising model.
Thus only the half part of the boson degrees of freedom
in the $(-)$-spin sector is massive, and the other
half-part is decoupled and forms a massless
mode of the Ising class  with the central charge
 $c=1/2$.\cite{fin,shu}
The microscopic model given by eq.(\ref{model}) does not have
this special symmetry, so that
we can proceed with the following analysis
by assuming that  all the low-energy excitations are gapful.

We investigate the effects of disorder on these fixed points
in the following sections.  We note that the
effects of quenched disorder on 1D interacting electron systems
have been extensively studied before.\cite{chui,appel,suzu,gia}
Our approach is
a generalization of these previous studies.
We introduce weak disorder potentials into the model (\ref{model}),
\begin{eqnarray}
H_{dis}&=&H_f+H_b, \label{dish} \\
H_f&=&\int dx \sum_{s=\uparrow\downarrow}
\eta_1(x)[\psi^{\dagger}_{s1 L}(x)\psi_{s1 L}(x)
+\psi^{\dagger}_{s1 R}(x)\psi_{s1 R}(x)] \nonumber \\
&+&\int dx \sum_{s=\uparrow\downarrow}
\eta_2(x)[\psi^{\dagger}_{s2 L}(x)\psi_{s2 L}(x)
+\psi^{\dagger}_{s2 R}(x)\psi_{s2 R}(x)] \nonumber \\
&+&\int dx \sum_{s=\uparrow\downarrow}
\eta_3(x)[e^{i(k_{\rm F1}-k_{\rm F2})x}
(\psi^{\dagger}_{s1 L}(x)\psi_{s2 L}(x) \nonumber \\
&& +\psi^{\dagger}_{s2 R}(x)\psi_{s1 R}(x))+h.c.],
\label{for}\\
H_b&=&\int dx \sum_{s=\uparrow\downarrow}
[\xi_1(x)e^{2ik_{\rm F1}}\psi^{\dagger}_{s1 L}(x)\psi_{s1 R}(x)
+h.c.] \nonumber \\
&&+\int dx \sum_{s=\uparrow\downarrow}
[\xi_2(x)e^{2ik_{\rm F2}}\psi^{\dagger}_{s2 L}(x)\psi_{s2 R}(x)
+h.c.] \nonumber \\
&&+\int dx \sum_{s=\uparrow\downarrow}
[\xi_3(x)e^{i(k_{\rm F1}+k_{\rm F2})x}
(\psi^{\dagger}_{s1 L}(x)\psi_{s2 R}(x) \nonumber \\
&&+\psi^{\dagger}_{s2 L}(x)\psi_{s1 R}(x))+h.c.],\label{bac}
\end{eqnarray}
where $H_f$ is the forward scattering part, and
$H_b$ is the backward scattering part.
$\eta_i(x)$ and  $\xi_i(x)$ ($i=1,2$)  are the random potential
fields within each band for forward and backward
scatterings, respectively.
The $\eta_3(x)$ and $\xi_3(x)$ terms represent the random hopping
between two splitted bands.
We assume that these random fields obey
the Gaussian distribution law,
$\langle\eta_i(x)\eta_j(x')\rangle
=D_{\eta_i}\delta_{ij}\delta(x-x')$,
$\langle\xi_i(x)\xi_j^{*}(x')\rangle
=D_{\xi_i}\delta_{ij}\delta(x-x')$,
and $D_{\eta (\xi) 1}=D_{\eta (\xi) 2}$.
We first consider the case of $D_{\eta 3}=D_{\xi 3}=0$
in Sec. III, IV, and V,
and take into account these random hopping terms in Sec. VI.

\section{Scaling equations for disordered systems
in the case without random interband hopping}

In the absence of the $\eta_3$ and $\xi_3$ terms,
we bosonize the random potential terms (\ref{for}) and (\ref{bac}),
and average over the random fields using the replica trick as in ref. 27.
Then the random potential terms in the action are given by,
\end{multicols}
\begin{eqnarray}
S_{dis}&=&-\frac{2D_{\eta +}}{\pi}\int dx\int d\tau\int d \tau'
\sum_{i,j}^n\partial_x\phi_{+\rho}^{i}(x,\tau)
\partial_x\phi_{+\rho}^{j}(x,\tau')
-\frac{2D_{\eta -}}{\pi}\int dx\int d\tau\int d \tau'
\sum_{i,j}^n\partial_x\phi_{-\rho}^{i}(x,\tau)
\partial_x\phi_{-\rho}^{j}(x,\tau') \nonumber \\
&&-\frac{2D_{A +}}{\pi}\int dx\int d\tau\int d \tau'
\sum_{i,j}^n\frac{1}{v_{+\rho}^2}
\partial_{\tau}\phi_{+\rho}^{i}(x,\tau)
\partial_{\tau}\phi_{+\rho}^{j}(x,\tau') \nonumber \\
&&-\frac{2D_{A -}}{\pi}\int dx\int d\tau\int d \tau'
\sum_{i,j}^n\frac{1}{v_{-\rho}^2}
\partial_{\tau}\phi_{-\rho}^{i}(x,\tau)
\partial_{\tau}\phi_{-\rho}^{j}(x,\tau') \nonumber \\
&&-\frac{D_{\xi 1}}{\alpha^2}\int dx\int d\tau\int d \tau'
\sum_{i,j}^n\cos\sqrt{\pi}(\phi^i_{+\sigma}(x,\tau)
+\phi^i_{-\sigma}(x,\tau))
\cos\sqrt{\pi}(\phi^j_{+\sigma}(x,\tau)
+\phi^j_{-\sigma}(x,\tau)) \nonumber \\
&&\times
\cos\sqrt{\pi}(\phi^i_{+\rho}(x,\tau)+\phi^i_{-\rho}(x,\tau)
-\phi^j_{+\rho}(x,\tau')-\phi^j_{-\rho}(x,\tau')) \nonumber \\
&&-\frac{D_{\xi 2}}{\alpha^2}\int dx\int d\tau\int d \tau'
\sum_{i,j}^n\cos\sqrt{\pi}(\phi^i_{+\sigma}(x,\tau)
-\phi^i_{-\sigma}(x,\tau))
\cos\sqrt{\pi}(\phi^j_{+\sigma}(x,\tau)
-\phi^j_{-\sigma}(x,\tau)) \nonumber \\
&&\times\cos\sqrt{\pi}(\phi^i_{+\rho}(x,\tau)-\phi^i_{-\rho}(x,\tau)
-\phi^j_{+\rho}(x,\tau')+\phi^j_{-\rho}(x,\tau')),
\end{eqnarray}
\begin{multicols}{2}
\noindent
where $i,j$ are replica indices, and initially
$D_{\eta \pm}=(D_{\eta 1}+ D_{\eta 2})/2$.
The $D_{A\pm}$ terms are generated
in the process of renormalization,
 though initially $D_{A\pm}=0$.\cite{fuji}
The $D_{A\pm}$ term is equivalent to the interaction
with random gauge fields,
\begin{equation}
H_{rg\pm}=-\sqrt{\frac{2}{\pi}}\int dx
A_{\pm}(x)\partial_x\theta_{\pm\rho},
\end{equation}
where the random gauge fields $A_{\pm}(x)$ obey the Gaussian
distribution law,
$\langle A_{\alpha}(x)A_{\beta}(x')\rangle
=D_{A\alpha}\delta_{\alpha\beta}
\delta(x-x')$ ($\alpha,\beta=\pm$).
The random fields $\eta_{\pm}(x)$ and $A_{\pm}(x)$
can be incorporated
into the shift of the phase fields.\cite{gia,fuji}
Defining the new fields,
\begin{eqnarray}
\tilde{\phi}_{\pm\rho}(x)&\equiv& \phi_{\pm\rho}(x)
+\tilde{\eta}_{\pm}(x),  \\
\tilde{\eta}_{\pm}(x)&\equiv& \frac{K_{\pm\rho}}{v_{\pm\rho}}
\sqrt{\frac{2}{\pi}}
\int^x dx' \eta_{\pm}(x'), \\
\tilde{\theta}_{\pm\rho}(x)&\equiv&
\theta_{\pm\rho}(x)+\tilde{A}_{\pm}(x),
 \\
\tilde{A}_{\pm}(x)&\equiv&
\frac{1}{K_{\pm\rho}v_{\pm\rho}}\sqrt{\frac{2}{\pi}}
\int^x dx' A_{\pm}(x'),
\end{eqnarray}
where $\langle\eta_{\alpha}(x)\eta_{\beta}(x')\rangle
=D_{\eta\alpha}
\delta_{\alpha\beta}\delta(x-x')$ ($\alpha,\beta=\pm$),
we can absorb the $\eta_{\pm}(x)$ and $A_{\pm}(x)$ terms into
the Gaussian part of the Hamiltonian.
Thus these terms do not affect the massless modes
apart from a minor change in correlation functions.\cite{gia,fuji}
However, as we will see momentarily, these terms are important for
suppressing the charge and spin gaps in the massive sectors.

We now derive the scaling equations
for the above effective Hamiltonian.
Since we have so many coupling constants, it is pretty difficult
to analyze all scaling equations even at one-loop level.
We here ignore the renormalization of velocities for simplicity.
In some previous studies\cite{kv,baf}, it was shown that
even with this approximation, the mass generation can be described
rather well qualitatively for clean systems. This is partly because
in the clean case the mass generation is mainly determined by the
scaling dimensions of the relevant operators which are not affected
by the renormalization of velocities in the lowest order. Thus this
approximation gives a qualitatively correct answer for the mass
generation in the clean case. We expect that this may be also the
case so far as we are concerned with weak disorder.
Using the operator product expansion of the U(1) Gaussian model,
and taking the replica limit $n\rightarrow 0$,
we have the scaling equations for dimensionless couplings defined by
$\tilde{U}_i=U_i/v_{+\rho}$, $\tilde{D}_{\eta,A\pm}=
D_{\eta,A\pm}/v_{\pm\rho}^2$, and
$\tilde{D}_{\xi i}=D_{\xi i}/v_{+\rho}^2$,
\begin{eqnarray}
\frac{d \tilde{U}_1}{d l}&=&(2-\frac{1}{K_{-\rho}}
-K_{-\sigma})\tilde{U}_1
\nonumber \\
&&-\frac{\tilde{U}_3\tilde{U}_6}{2}
-\frac{\tilde{U}_2\tilde{U}_7}{2}
-\frac{2\tilde{D}_{A-}\tilde{U}_1}{\pi^2K_{-\rho}^2},
\label{sc1} \\
\frac{d \tilde{U}_2}{d l}&=&(2-\frac{1}{K_{-\rho}}
-K_{+\sigma})\tilde{U}_2
\nonumber \\
&&-\frac{\tilde{U}_3\tilde{U}_8}{2}
-\frac{\tilde{U}_1\tilde{U}_7}{2}
-\frac{2\tilde{D}_{A-}\tilde{U}_2}{\pi^2K_{-\rho}^2},
\label{sc2} \\
\frac{d \tilde{U}_3}{d l}&=&(2-\frac{1}{K_{-\rho}}
-K_{+\rho})\tilde{U}_3
-\frac{\tilde{U}_1\tilde{U}_6}{2}
-\frac{\tilde{U}_5\tilde{U}_9}{2} \nonumber \\
&&-\frac{2\tilde{D}_{A-}\tilde{U}_3}{\pi^2K_{-\rho}^2}
-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_3}{\pi^2},
\label{sc3} \\
\frac{d \tilde{U}_4}{d l}&=&(2-\frac{1}{K_{-\sigma}}
-K_{+\sigma})\tilde{U}_4
-\frac{\tilde{U}_2\tilde{U}_9}{2}
-\frac{\tilde{U}_5\tilde{U}_8}{2},
\label{sc4} \\
\frac{d \tilde{U}_5}{d l}&=&(2-\frac{1}{K_{-\sigma}}
-K_{+\rho})\tilde{U}_5
-\frac{\tilde{U}_4\tilde{U}_8}{2}
-\frac{\tilde{U}_3\tilde{U}_9}{2} \nonumber \\
&&-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_5}{\pi^2},
\label{sc5} \\
\frac{d \tilde{U}_6}{d l}&=&(2-K_{-\sigma}
-K_{+\rho})\tilde{U}_6
-\frac{\tilde{U}_1\tilde{U}_3}{2}
-\frac{\tilde{U}_7\tilde{U}_8}{2} \nonumber \\
&&-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_6}{\pi^2},
\label{sc6} \\
\frac{d \tilde{U}_7}{d l}&=&(2-K_{-\sigma}
-K_{+\sigma})\tilde{U}_7
-\frac{\tilde{U}_1\tilde{U}_2}{2}
-\frac{\tilde{U}_6\tilde{U}_8}{2} \nonumber \\
&&-\tilde{D}_{\xi 1}-\tilde{D}_{\xi 2}, \label{sc7} \\
\frac{d \tilde{U}_8}{d l}&=&(2-K_{+\sigma}
-K_{+\rho})\tilde{U}_8
-\frac{\tilde{U}_4\tilde{U}_5}{2}
-\frac{\tilde{U}_6\tilde{U}_7}{2} \nonumber \\
&&-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_8}{\pi^2},
\label{sc8} \\
\frac{d \tilde{U}_9}{d l}&=&(2-\frac{1}{K_{-\rho}}
-\frac{1}{K_{-\sigma}})
\tilde{U}_9
-\frac{\tilde{U}_2\tilde{U}_4}{2}
-\frac{\tilde{U}_3\tilde{U}_5}{2} \nonumber \\
&&-\frac{2\tilde{D}_{A-}\tilde{U}_9}{\pi^2K_{-\rho}^2},
\label{sc9} \\
\frac{d K_{+\rho}}{d l}&=&-\frac{\pi K_{+\rho}^2}{2}
[\tilde{U}_3^2+\tilde{U}_5^2+\tilde{U}_6^2
+\tilde{U}_8^2] \nonumber \\
&&-\frac{\pi K_{+\rho}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{sc10} \\
\frac{d K_{-\rho}}{d l}&=&\frac{\pi}{2}
[\tilde{U}_1^2+\tilde{U}_2^2+\tilde{U}_3^2+\tilde{U}_9^2]
\nonumber \\
&&-\frac{\pi K_{-\rho}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{sc11} \\
\frac{d K_{+\sigma}}{d l}&=&-\frac{\pi K_{+\sigma}^2}{2}
[\tilde{U}_2^2+\tilde{U}_4^2+\tilde{U}_7^2
+U_8^2] \nonumber \\
&&-\frac{\pi K_{+\sigma}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{sc12} \\
\frac{d K_{-\sigma}}{d l}&=&\frac{\pi}{2}
[\tilde{U}_4^2+\tilde{U}_5^2+\tilde{U}_9^2]
-\frac{\pi K_{-\sigma}^2}{2}
[\tilde{U}_1^2+\tilde{U}_6^2+\tilde{U}_7^2] \nonumber \\
&-&\frac{\pi K_{-\sigma}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{sc13} \\
\frac{d \tilde{D}_{\eta\pm}}{d l}&=&\tilde{D}_{\eta \pm}
+\pi^2(\tilde{D}_{\xi 1}^2+\tilde{D}_{\xi 2}^2),
\label{sc16} \\
\frac{d \tilde{D}_{A\pm}}{d l}&=&\tilde{D}_{A \pm}
+\pi^2(\tilde{D}_{\xi 1}^2+\tilde{D}_{\xi 2}^2),
\label{sc17} \\
\frac{d \tilde{D}_{\xi i}}{d l}&=&
(3-\frac{K_{+\sigma}}{2}-\frac{K_{-\sigma}}{2}
-\frac{K_{+\rho}}{2}-\frac{K_{-\rho}}{2})\tilde{D}_{\xi i}
\nonumber \\
&&-\tilde{U}_7\tilde{D}_{\xi i}, \qquad i=1,2. \label{sc18}
\end{eqnarray}
In the absence of disorder, $D_{\eta\pm}=D_{A\pm}=D_{\xi i}=0$,
the excitation gaps open
above the ground state energy in the $(\pm)$-charge and
($\pm$)-spin sectors.
We have two possibilities in the massive
phase of the $(-)$-spin sector
as mentioned in Sec. II.
If $\tilde{U}_1$, $\tilde{U}_6$,
and $\tilde{U}_7$ scale to strong-coupling
regime, and the $\tilde{U}_4$, $\tilde{U}_5$, and $\tilde{U}_9$ terms
are irrelevant, the phase field $\phi_{-\sigma}$ is locked,
and the correlations
for the operator $\exp(ia\theta_{-\sigma})$ shows exponential decay.
We refer to this case as {\it case A}.
To the contrary, if the $\tilde{U}_4$, $\tilde{U}_5$,
and $\tilde{U}_9$ terms are relevant,
and the $\tilde{U}_1$, $\tilde{U}_6$, and $\tilde{U}_7$ terms are
irrelevant, $\theta_{-\sigma}$ is locked.
We refer to this case as {\it case B}.
In the subsequent sections, we discuss the effects of disorder
on these two kinds of the fixed points.

\section{Disordered fixed point:  {\it case A}}

Here we consider the effects of quenched disorder
on the fixed point of {\it case A},  where
the phase field $\phi_{-\sigma}$ is locked,
and  $\exp(ia\theta_{-\sigma})$ is the disorder
parameter.
In this case, the $\tilde{U}_1$, $\tilde{U}_2$, $\tilde{U}_3$,
$\tilde{U}_6$, $\tilde{U}_7$, and $\tilde{U}_8$
terms are relevant, and the $\tilde{U}_4$, $\tilde{U}_5$,
$\tilde{U}_9$ terms are irrelevant in the clean system.
The sufficiently strong forward scattering due to impurities
$\tilde{D}_{\eta +}$
may suppress the mass-generating interactions,
$\tilde{U}_3$, $\tilde{U}_6$,
and $\tilde{U}_8$, as seen from
Eqs.(\ref{sc3}), (\ref{sc6}), and (\ref{sc8}).
It turns out from Eqs.(\ref{sc1}) and (\ref{sc2})
that this is not the case for
the $\tilde{U}_1$ and $\tilde{U}_2$ terms which
bear the charge gap in the $(-)$-sector, because they
are not coupled with the random forward scattering.
We note, however, that these terms should be
 suppressed in the presence of  sufficiently strong random
gauge fields $A_{-}$ generated
by random backward scattering due to impurities.
Therefore in order to close the charge gap in the $(-)$-sector,
one needs the random backward scattering due to impurities.
This point makes a clear contrast to the case of a single chain model
where the Mott-Hubbard gap is collapsed only by the random forward
scattering due to impurities.\cite{fuji}

Note that the $\tilde{U}_7$ term which generates the spin gap
in $\phi_{\pm\sigma}$ fields
is not affected by the random forward scattering, because
the random forward scattering couples only with
 the charge degrees of freedom.
On the other hand,
it is seen from Eq.(\ref{sc7}) that
the backward scattering due to impurities $\tilde{D}_{\xi i}$
may drive $\tilde{U}_7$ to
a large negative value.
However, if the initial values of $\tilde{D}_{\xi i}$
are small enough
compared to that of $\tilde{U}_7$, the spin
gaps of $\phi_{\pm\sigma}$ fields
generated by the $\tilde{U}_7$ term may survive.
Based on this observation, we start with the limiting case
$\tilde{U}_7\rightarrow +\infty$ for simplicity.
Then after the spin gap formation, we can eliminate
the spin degrees of freedom from the scaling equations.
Suppressing irrelevant terms,
we have the reduced scaling equations,
\begin{eqnarray}
\frac{d \tilde{U}_1}{d l}&=&(2-\frac{1}{K_{-\rho}})\tilde{U}_1
-\frac{\tilde{U}_3\tilde{U}_6}{2}
-\frac{2\tilde{D}_{A-}\tilde{U}_1}{\pi^2K_{-\rho}^2},
\label{sca1} \\
\frac{d \tilde{U}_2}{d l}&=&(2-\frac{1}{K_{-\rho}})\tilde{U}_2
-\frac{\tilde{U}_3\tilde{U}_8}{2}
-\frac{2\tilde{D}_{A-}\tilde{U}_2}{\pi^2K_{-\rho}^2},
\label{sca2} \\
\frac{d \tilde{U}_3}{d l}&=&
(2-\frac{1}{K_{-\rho}}-K_{+\rho})\tilde{U}_3
-\frac{\tilde{U}_1\tilde{U}_6}{2}-\frac{\tilde{U}_5\tilde{U}_9}{2}
\nonumber \\
&&-\frac{2\tilde{D}_{A-}\tilde{U}_3}{\pi^2K_{-\rho}^2}
-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_3}{\pi^2},
\label{sca3} \\
\frac{d \tilde{U}_6}{d l}&=&(2-K_{+\rho})\tilde{U}_6
-\frac{\tilde{U}_1\tilde{U}_3}{2}
-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_6}{\pi^2},
\label{sca6} \\
\frac{d \tilde{U}_8}{d l}&=&(2-K_{+\rho})\tilde{U}_8
-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_8}{\pi^2},
\label{sca8} \\
\frac{d K_{+\rho}}{d l}&=&-\frac{\pi K_{+\rho}^2}{2}
[\tilde{U}_3^2+\tilde{U}_6^2
+\tilde{U}_8^2] \nonumber \\
&&-\frac{\pi K_{+\rho}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{sca10} \\
\frac{d K_{-\rho}}{d l}&=&\frac{\pi}{2}
[\tilde{U}_1^2+\tilde{U}_2^2+\tilde{U}_3^2]
-\frac{\pi K_{-\rho}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{sca11} \\
\frac{d \tilde{D}_{\xi i}}{d l}&=&
(3-\frac{K_{+\rho}}{2}-\frac{K_{-\rho}}{2})\tilde{D}_{\xi i},
\qquad i=1,2. \label{sca18}
\end{eqnarray}
Since initially $\tilde{D}_{\eta\pm}>\tilde{D}_{A\pm}=0$,
and they
obey the same scaling equations (\ref{sc16}) and (\ref{sc17}),
$\tilde{D}_{\eta\pm}$ flows into strong-coupling regime
faster than $\tilde{D}_{A\pm}$.
Thus there exists a parameter region where
the $\tilde{U}_3$, $\tilde{U}_6$, and $\tilde{U}_8$ terms
are suppressed for sufficiently
large $\tilde{D}_{\eta +}$, though the $\tilde{U}_1$
and $\tilde{U}_2$ terms are still
relevant.
Then the charge gap in $(+)$-charge mode is collapsed.
Since the $\theta_{-\rho}$ field is locked due to the $\tilde{U}_1$
and $\tilde{U}_2$
terms, $K_{-\rho}$ scales to a large value, and
the $\tilde{D}_{\xi i}$ terms are suppressed,
as seen from Eq.(\ref{sca18}).
However, these terms are proved to be relevant,
after one integrates out the $\phi_{-\rho}$ field,
following the argument by Orignac and Giamarchi\cite{og}.
as in the case away from half-flling.\cite{og}
Then the $(+)$-charge mode is in the Anderson localized state,
while the $(-)$-charge mode is still in the Mott insulating state.

As the value of $\tilde{D}_{A-}$ becomes larger,
the $\tilde{U}_1$, $\tilde{U}_2$
terms are suppressed, and all excitation gaps are collapsed.
Then, the backward scattering due to impurities, $\tilde{D}_{\xi i}$
is relevant, and consequently the Anderson localization occurs.
$\tilde{U}_7$ flows to a large negative value
because of the $\tilde{D}_{\xi i}$ term,
as seen from Eq.(\ref{sc7}).
Thus the spin degrees of freedom are frozen.
The fixed point is identified with the pinned CDW state
which is similar to that found
in the disordered single chain model.\cite{gia}

In order to confirm the above arguments,
we solve the scaling equations
(\ref{sc1})-(\ref{sc18}) numerically for several initial values
of coupling constants.
The numerical results for the flow of
coupling constants are shown in FIG.1.
In order to realize the massive spin state due to the locking of
$\phi_{-\sigma}$ in the clean system,
we have chosen the initial value of the TL liquid
parameter in the $(-)$-spin sector as $K_{-\sigma}(0)=0.86$.
The other TL parameters are set to be
$K_{\pm\rho}(0)=0.98$ and $K_{+\sigma}(0)=1.0$.
For these initial values, the system definitely flows toward
the low-energy fixed point of {\it case A}
in the absence of disorder.
In FIG.1(a) the results in the case
without quenched disorder are shown.
$\tilde{U}_1$, $\tilde{U}_2$, $\tilde{U}_3$, $\tilde{U}_6$,
$\tilde{U}_7$, and $\tilde{U}_8$ flow into a
strong-coupling regime, while
$\tilde{U}_4$, $\tilde{U}_5$, and $\tilde{U}_9$ scale to $0$.
As a consequence, the phase fields
$\phi_{+\rho}$, $\theta_{-\rho}$. $\phi_{\pm\sigma}$
are locked. This fixed point
is stable against the disorder
weaker than mass-generating interactions, because of
the presence of excitation gaps in all modes.
As the quenched disorder becomes stronger,
some different fixed points
manifest themselves.
In FIG.1(b), we show the results
for the initial values
$\tilde{D}_{\eta\pm}(0)=0.9$ and $\tilde{D}_{\xi i}(0)=0.005$.
Although  $\tilde{U}_1$, $\tilde{U}_2$ and $\tilde{U}_7$
still scale to a strong-coupling regime,
$\tilde{U}_3$, $\tilde{U}_6$, and $\tilde{U}_8$ scale to $0$.
As a result, the charge gap
in the $(+)$-mode is collapsed.
Since $\tilde{D}_{\xi i}$ is relevant,
the charge mode in the $(+)$-sector is localized, though
the charge mode in the $(-)$-sector is still massive.
Thus the Anderson localized state and the Mott
insulating state coexist in this phase.
Since $\tilde{U}_7\rightarrow -\infty$,
the spin degrees of freedom are frozen, and may be in
the random spin singlet state.
In FIG.1(c),
the results for $D_{\eta\pm}(0)=0.9$
and $D_{\xi i}(0)=0.01$ are shown.
In this case, all mass-generating interactions except
$\tilde{U}_7$ in Eq.(\ref{int1})
flow to $0$. The charge gaps are thus collapsed in this phase.
Since $\tilde{D}_{\xi i}$ scales to a strong-coupling regime,
the Anderson localized state realizes.
The relevance of the $\tilde{U}_7$ term implies
freezing of spin degrees of freedom
in the $(\pm)$-sectors.

In summary, we have found three kinds of fixed points
in the presence of quenched disorder:
(i) the clean fixed point where the disorder effect
is entirely irrelevant,
(ii) Anderson-insulator (I) where
the $(+)$-charge mode is in the Anderson localized state,
the $(-)$-charge mode is in the Mott insulating state,
and the spin degrees of freedom are massive or frozen,
(iii) Anderson-insulator (II) where
all charge modes are in the Anderson localized state, and
spin degrees of freedom are frozen, {\it i.e.} pinned CDW.
We thus end up with the schematic phase
diagram for charge degrees
of freedom as shown in FIG.2.

\section{Disordered fixed point: {\it case B}}

Here we consider the effects of disorder on
the fixed point of {\it case B}:
$\theta_{-\sigma}$ is locked, and
$\exp(ia\phi_{-\sigma})$ is the disorder parameter field
in the absence of quenched disorder.
In this case, the $\tilde{U}_2$, $\tilde{U}_3$, $\tilde{U}_4$,
$\tilde{U}_5$, $\tilde{U}_8$, and $\tilde{U}_9$ terms
are relevant in the clean system.
We see from Eq.(\ref{sc13}) that $K_{-\sigma}$ is decreased by
the backward scattering due to impurities $\tilde{D}_{\xi i}$.
Thus it suppresses the growth of $\tilde{U}_4$, $\tilde{U}_5$,
and $\tilde{U}_9$
and prevents the spin gap formation due to locking of
the $\theta_{-\sigma}$ field.
Moreover it follows from Eq.(\ref{sc7}) that
if the $\tilde{D}_{\xi i}$ term is
relevant, $\tilde{U}_7$ flows to a large negative  value,
resulting in freezing of the spin degrees of freedom,
and the suppression of the $\tilde{U}_4$ term as in Sec. IV.
However for sufficiently small $\tilde{D}_{\xi i}$,
the $\tilde{U}_4$ term
is relevant and the spin gap opens
owing to locking of $\theta_{-\sigma}$.
After the generation of the spin gap due to locking of
the $\theta_{-\sigma}$ and
$\phi_{+\sigma}$ fields, we can eliminate
the spin degrees of freedom,
and obtain the scaling equations for
the charge degrees of freedom,
\begin{eqnarray}
\frac{d \tilde{U}_2}{d l}&=&(2-\frac{1}{K_{-\rho}})\tilde{U}_2
-\frac{\tilde{U}_3\tilde{U}_8}{2}
-\frac{2\tilde{D}_{A-}\tilde{U}_2}{\pi^2K_{-\rho}^2},
\label{scb2} \\
\frac{d \tilde{U}_3}{d l}&=&(2-\frac{1}{K_{-\rho}}
-K_{+\rho})\tilde{U}_3
-\frac{\tilde{U}_5\tilde{U}_9}{2} \nonumber \\
&&-\frac{2\tilde{D}_{A-}\tilde{U}_3}{\pi^2K_{-\rho}^2}
-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_3}{\pi^2},
\label{scb3} \\
\frac{d \tilde{U}_5}{d l}&=&(2-K_{+\rho})\tilde{U}_5
-\frac{\tilde{U}_3\tilde{U}_9}{2} \nonumber \\
&&-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_5}{\pi^2},
\label{scb5} \\
\frac{d \tilde{U}_8}{d l}&=&(2-K_{+\rho})\tilde{U}_8
-\frac{2\tilde{D}_{\eta +}K_{+\rho}^2\tilde{U}_8}{\pi^2},
\label{scb8} \\
\frac{d \tilde{U}_9}{d l}&=&(2-\frac{1}{K_{-\rho}})\tilde{U}_9
-\frac{\tilde{U}_3\tilde{U}_5}{2}
-\frac{2\tilde{D}_{A-}\tilde{U}_9}{\pi^2K_{-\rho}^2},
\label{scb9} \\
\frac{d K_{+\rho}}{d l}&=&-\frac{\pi K_{+\rho}^2}{2}
[\tilde{U}_3^2+\tilde{U}_5^2
+\tilde{U}_8^2],
\nonumber \\
&&-\frac{\pi K_{+\rho}^2}{16}(\tilde{D}_{\xi 1}
+\tilde{D}_{\xi 2}), \label{scb10} \\
\frac{d K_{-\rho}}{d l}&=&\frac{\pi}{2}
[\tilde{U}_2^2+\tilde{U}_3^2+\tilde{U}_9^2].
\nonumber \\
&&-\frac{\pi K_{-\rho}^2}{16}(\tilde{D}_{\xi 1}+\tilde{D}_{\xi 2}),
\label{scb11} \\
\frac{d \tilde{D}_{\xi 1,2}}{d l}&=&(3-2K_{+\rho}
-2K_{-\rho})\tilde{D}_{\xi i}. \label{scb12}
\end{eqnarray}
As pointed out in Sec. IV, $\tilde{D}_{\eta\pm}$
grows faster than $\tilde{D}_{A\pm}$.
Thus there may exist a parameter region where
$\tilde{D}_{\eta +}$ suppresses $\tilde{U}_3$, $\tilde{U}_5$,
and $\tilde{U}_8$,
but the $\tilde{U}_2$ and $\tilde{U}_9$ terms are still relevant,
bearing the mass gap in the $(-)$-charge mode.
In this parameter region, only the charge gap in the
$(+)$-charge sector is collapsed.
The low-energy properties of this sector is determined by
the $\tilde{D}_{\xi i}$-terms.
After integrating out the massive $(-)$-charge mode\cite{og},
we find that the scaling equation for
the $\tilde{D}_{\xi 1,2}$-terms are changed to,
\begin{equation}
\frac{d \tilde{D}_{\xi 1,2}}{d l}=(3-8K_{+\rho})\tilde{D}_{\xi i}.
\end{equation}
Thus the impurity backward scattering is suppressed
in comparison with
single chain systems.
For $K_{+\rho}>3/8$, the $D_{\xi 1,2}$-terms are irrelevant.
However it does not directly mean the occurrence of delocalization,
since we should actually include the random hopping between two bands,
$D_{\eta 3}$, $D_{\xi 3}$.
In the case of $D_{\eta 3}\neq 0$ and $D_{\xi 3}\neq 0$,
the CDW order couples directly to the $D_{\xi 3}$ term,
and then the pinning of the CDW occurs for infinitesimally small
$D_{\xi 3}$ as will be discussed in Sec. VI.
Therefore the $(+)$-charge mode is in the Anderson localized state
with the vanishing Drude weight and the non-zero charge susceptibility.
For a sufficiently large value of $\tilde{D}_{A-}$, the charge gap
in the $(-)$-mode may be also collapsed.
All the charge modes are in the Anderson localized state.

In order to specify the low-energy fixed points correctly,
we have solved the full scaling equations
(\ref{sc1})-(\ref{sc18})
numerically for some initial values of couplings.
The numerical results are shown in FIG.3.
In FIG.3(a), we display the results for clean systems.
In order to realize the spin gapped state due to locking of
$\theta_{-\sigma}$,
we choose the initial values of the parameters
$K_{-\sigma}(0)=1.1$, $K_{+\sigma}(0)=1.0$, and
$K_{\pm\rho}(0)=0.98$.
For these parameters,
$\tilde{U}_2$, $\tilde{U}_3$, $\tilde{U}_4$,
$\tilde{U}_5$, $\tilde{U}_8$,
and $\tilde{U}_9$ scale to
a strong-coupling regime, and $\tilde{U}_1$, $\tilde{U}_6$, and
$\tilde{U}_7$ scale to $0$.
Then the phase fields $\phi_{+\rho}$,
$\theta_{-\rho}$, $\phi_{+\sigma}$,
and $\theta_{-\sigma}$ are locked.
This fixed point with the spin and charge gaps
is stable as far as
disorder is much weaker than mass-generating interactions.
In FIG.3(b), the results for $\tilde{D}_{\eta\pm}(0)=0.9$,
and $\tilde{D}_{\xi i}(0)=0.007$ are shown.
This flow ends up at the same fixed point as that shown in FIG.1(b),
where the backward scattering due to impurities is relevant,
and thus
the Anderson localization occurs in the $(+)$-charge mode,
though the $(-)$-charge mode is
in the Mott insulating state.
The spin degrees of freedom are also massive or frozen.
Note that the freezing of the spin degrees of freedom
in the $(-)$-sector
is not due to locking of $\theta_{-\sigma}$
but due to $\phi_{-\sigma}$.
Thus the fixed point properties of the $(-)$-spin sector are
different from those for the clean case where locking of
$\theta_{-\sigma}$ occurs.
In FIG.3(c), we show the results for $\tilde{D}_{\eta\pm}(0)=0.9$
and $\tilde{D}_{\xi i}(0)=0.01$.
In this case, the flows of coupling
constants are qualitatively the same
as those shown in FIG.1(c); {\it i.e.}
all excitation gaps are collapsed, and the Anderson localized state
with frozen spin degrees of freedom is realized.
Thus for sufficiently strong impurity scattering,
the disordered fixed point
of {\it case B} flows into the same class as that of {\it case A}.
In summary, we have found three kinds of fixed points as in
{\it case A}.
The schematic phase diagram for the charge degrees of freedom
is also given by FIG.2.

\section{Effects of random hopping between two bands}

We now take into account the effects of
random hopping between two bands,
$\eta_3$ and $\xi_3$.
First we consider only random forward scattering, and omit
random backward scattering for a while.
Going back to the fermion representation,
we write down the kinetic energy part and
the impurity forward scattering part of the Hamiltonian,
\begin{eqnarray}
H_0'&=&\int dx \sum_{s=\uparrow\downarrow}
[v_1(\psi^{\dagger}_{s1 L}\partial_x\psi_{s1 L}
-\psi^{\dagger}_{s1 R}\partial_x\psi_{s1 R}) \nonumber \\
&+&v_2(\psi^{\dagger}_{s2 L}\partial_x\psi_{s2 L}
-\psi^{\dagger}_{s2 R}\partial_x\psi_{s2 R})] \nonumber \\
&+&\int dx \sum_{s=\uparrow\downarrow}
\eta_1(x)[\psi^{\dagger}_{s1 L}(x)\psi_{s1 L}(x)
+\psi^{\dagger}_{s1 R}(x)\psi_{s1 R}(x)]
\nonumber \\
&+&\int dx \sum_{s=\uparrow\downarrow}
\eta_2(x)[\psi^{\dagger}_{s2 L}(x)\psi_{s2 L}(x)
+\psi^{\dagger}_{s2 R}(x)\psi_{s2 R}(x)]
\nonumber \\
&&+\int dx \sum_{s=\uparrow\downarrow}
\eta_3(x)[e^{i(k_{\rm F1}-k_{\rm F2})x}
(\psi^{\dagger}_{s1 L}(x)\psi_{s2 L}(x) \nonumber \\
&& +\psi^{\dagger}_{s2 R}(x)\psi_{s1 R}(x))+h.c.].
\label{h06}
\end{eqnarray}
In the case of half-filling, $v_1=v_2\equiv v$ holds.
Defining the spinor fields,
\begin{equation}
\psi_{s L(R)}=\left(
\begin{array}{c}
\psi_{s1 L(R)} \\ \psi_{s2 L(R)}
\end{array}\right),
\end{equation}
we cast the Hamiltonian Eq.(\ref{h06}) into
the following form,
\begin{eqnarray}
H_0'&=&\int dx\sum_{s=\uparrow\downarrow}
v(\psi^{\dagger}_{s L}\partial_x\psi_{s L}
-\psi^{\dagger}_{s R}\partial_x\psi_{s R}) \nonumber \\
&+&\int  dx\sum_{s=\uparrow\downarrow}
\psi^{\dagger}_{s L}
\Bigl(\frac{\eta_1+\eta_2}{2}{\bf 1}+\frac{\eta_1-\eta_2}{2}\tau^z
\nonumber \\
&+&\eta_3 e^{i(k_{\rm F1}-k_{\rm F2})x}\frac{\tau^{+}}{2}
+\eta_3^{*} e^{-i(k_{\rm F1}-k_{\rm F2})x}\frac{\tau^{-}}{2}\Bigr)
\psi_{s L} \nonumber \\
&+&\int  dx\sum_{s=\uparrow\downarrow}
\psi^{\dagger}_{s R}
\Bigl(\frac{\eta_1+\eta_2}{2}{\bf 1}+\frac{\eta_1-\eta_2}{2}\tau^z
\nonumber \\
&+&\eta_3^{*} e^{-i(k_{\rm F1}-k_{\rm F2})x}\frac{\tau^{+}}{2}
+\eta_3 e^{i(k_{\rm F1}-k_{\rm F2})x}\frac{\tau^{-}}{2}\Bigr)
\psi_{s R},
\end{eqnarray}
where {\bf 1} is a $2\times2$ unit matrix,
and $\tau^z$, $\tau^{\pm}$
are the Pauli matrices.
This Hamiltonian can be diagonalized by using the unitary
transformations defined by,
\begin{eqnarray}
\tilde{\psi}_{s L}&=&U_L\psi_{s L}, \\
U_L&\equiv&\exp\Bigl(-\frac{i}{v}\int^x_{-\infty}dx'
\Bigl[\frac{\eta_1(x')+\eta_2(x')}{2}
{\bf 1} \nonumber \\
&&+\frac{\eta_1(x')-\eta_2(x')}{2}\tau^z \nonumber \\
&&+\eta_3(x')e^{i(k_{\rm F1}-k_{\rm F2})x'}\frac{\tau^{+}}{2}
\nonumber \\
&&+\eta_3^{*}(x')e^{-i(k_{\rm F1}-k_{\rm F2})x'}
\frac{\tau^{-}}{2}\Bigr]
\Bigr), \label{uni}
\end{eqnarray}
and $U_R=U_L^{\dagger}$.
We thus end up with the diagonalized form of $H_0'$,
\begin{equation}
H_0'=\int dx \sum_sv(\tilde{\psi}^{\dagger}_{s L}
\partial_x\tilde{\psi}_{s L}
-\tilde{\psi}^{\dagger}_{s R}\partial_x\tilde{\psi}_{s R}).
\end{equation}
Therefore the forward scattering part of the random hopping
can be incorporated into the free part of the Hamiltonian.
Under the unitary transformation (\ref{uni}),
the forward scattering terms of electron-electron interaction are
invariant.
The backward and umklapp scattering terms of electron-electron
interaction (\ref{int1}) which may generate the mass gaps are
multiplied with
exponentially decaying factors by this transformation.
Thus for sufficiently strong impurity forward
scattering both of the charge and spin gaps
collapse as in the previous sections.

Let us now take into account random backward scattering.
The backward scattering term of random hopping is expressed as
\begin{equation}
H_{\xi3}=\int dx [\xi_3(x)e^{i(k_{\rm F1}+k_{\rm F2})x}
O_{CDW}(x)+h.c.].
\end{equation}
The scaling equation for $D_{\xi 3}$ is given by
\begin{eqnarray}
\frac{d \tilde{D}_{\xi 3}}{d l}&=&\Bigl(3-\frac{K_{+\rho}}{2}
-\frac{1}{2K_{-\rho}}
-\frac{K_{+\sigma}}{2}-\frac{1}{2K_{-\sigma}} \nonumber \\
&&-\tilde{U}_4\Bigr)
\tilde{D}_{\xi 3},\label{s3}
\end{eqnarray}
where $\tilde{D}_{\xi 3}=D_{\xi 3}/v_{+\rho}^2$.
It is seen that the $D_{\xi 3}$ term couples with the $U_4$ term.
Then the scaling equation (\ref{sc4}) is altered to
\begin{eqnarray}
\frac{d \tilde{U}_4}{d l}&=&
\Bigl(2-K_{+\sigma}-\frac{1}{K_{-\sigma}}\Bigr)
\tilde{U}_4-\tilde{U}_2\tilde{U}_9-\tilde{U}_5\tilde{U}_8
\nonumber \\
&&-\tilde{D}_{\xi 3}.
\end{eqnarray}
Therefore in the case that the mass gaps open due to locking of
the $\phi_{+\rho}$,
$\theta_{-\rho}$, $\phi_{+\sigma}$, and $\theta_{-\sigma}$ fields
in the clean system ({\it case B}),
$3-K_{+\rho}/2-1/2K_{-\rho}
-K_{+\sigma}/2-1/2K_{-\sigma}-\tilde{U}_4>0$
holds, and the $D_{\xi 3}$ term is relevant. Then,
the Anderson localization inevitably occurs in the $(+)$-charge
sector provided that the impurity forward scattering
suppresses the charge gap as mentioned in Sec. V.
This fixed point is the pinned CDW.
If the random backward scattering $\tilde{D}_{\xi 1,2}$
drives $\tilde{U}_7$ to $-\infty$,
the $\phi_{-\sigma}$ field is locked.
Then in the presence of the gap in the $(-)$-charge
sector, integrating out the $\theta_{-\rho}$ field
as in ref. 16, we find that the scaling equation (\ref{s3})
is changed to,
\begin{equation}
\frac{d \tilde{D}_{\xi 3}}{d l}=(3-2K_{+\rho})\tilde{D}_{\xi 3}.
\label{s32}
\end{equation}
Thus although the $\tilde{D}_{\xi 3}$-term is suppressed,
it is still relevant for the repulsive interaction $K_{+\rho}<1$,
and the Anderson localization occurs in the $(+)$-charge sector.

If the spin gap opens owing to locking of
the $\phi_{-\sigma}$ field in the clean system
({\it case A}), the scaling equation for $\tilde{D}_{\xi 3}$
is also given by Eq.(\ref{s32}).
Then the random hopping between two bands enhances
the Anderson localization obtained in Sec. IV.

\section{Conclusion}

We have  systematically studied the effects of quenched disorder
on the spin-gapped state of two chain coupled Hubbard ladder
model at half-filling.
We have shown that
a sufficiently strong forward scattering
due to impurities and the random gauge fields generated by
impurity backward scattering collapse
all the charge gaps as well as the spin gaps,
and consequently the Anderson localization
takes place in all the charge sectors
owing to the impurity backward scattering.
We have also found the phase that  one charge mode is
in the Anderson localized state, while
the other one is in the massive state.

This work was partly supported by a Grant-in-Aid from the Ministry
of Education, Science and Culture.

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