\section{Introduction}
Over the last decade, Type Ia Supernovae (SNe Ia) have become important cosmological probes as they can be used to measure distances to high redshift ($z\lesssim1.5$).
In recent years, numerous samples of SN Ia have been compiled, e.g., CSP \citep{2006PASP..118....2H}, SNLS \citep{2006A&A...447...31A}, ESSENCE \citep{2007ApJ...666..694W}, SDSS \citep{2008AJ....135..338F}, CfA \citep{2009ApJ...700.1097H}, and combined, we are approaching $\sim1000$ spectroscopically-confirmed SNe Ia available
for cosmological analysis \citep{2010arXiv1004.1711A}. With such large samples, it is becoming increasingly important to understand the systematic uncertainties (photometric calibration, SN color variations, etc.) associated
with using SNe Ia for cosmology, including any additional physical parameters that could reduce the intrinsic scatter of the population.

One such parameter could be related to the environment of the supernova. First and foremost, one would expect differences in the colors of SNe Ia based on the different dust content of
their hosts, i.e., potential variations in local circumstellar dust around the progenitor star \citep{wang2005, 2008ApJ...686L.103G} and/or differences in the global dust content of
different galaxy types\footnote{Dust in our own Galaxy is usually corrected for using the dust maps of \citet{1998ApJ...500..525S}}.  Despite these concerns, most analyses account for dust
during the fitting of the supernova light curves by assuming a single absorption law ($R_V$) for all SNe, which minimizes the scatter around the Hubble diagram \citep{1998A&A...331..815T}.
This process, however, has led to a dust law that is significantly different from the canonical value for our Galaxy ($R_V \approx 3.1$), e.g., \citet{2007ApJ...664L..13C} find $R_V \approx 1$
for nearby SNe Ia, while  \citet{2009ApJS..185...32K} obtained a best fit of $R_V = 2.18 \pm 0.14{\rm (stat)} \pm {\rm 0.48(sys)}$ for the first year sample from the Sloan Digital Sky Survey II (SDSS-II) Supernova Survey. In a recent near-infrared study
of nearby SNe,  \citet{2010AJ....139..120F} found $R_V \approx 1-2$ for their whole sample, but obtained $R_V \approx 3.2$ if they exclude their most reddened objects.
Alternatively, in a study of 80 nearby SNe Ia, \citet{2008A&A...487...19N} found
a value of $R_V=1.75\pm0.27$ for their whole sample and a lower value of $R_V\sim1$ if they restrict the sample to low reddening values.
These differences could suggest that the effects of dust may also be dependent on the particular line of sight \citep{2007ApJ...666..694W} or on the inclination of the host
galaxies \citep{2010arXiv1001.1744M}.

Secondly, the details of the supernova progenitor system could systematically vary between the different galaxy types. Our present theoretical understanding of SNe Ia suggests they are the thermonuclear explosion of a carbon-oxygen white dwarf which has reached the Chandrasekhar limit \citep{1973ApJ...186.1007W, 2000ARA&A..38..191H}. The mechanism for how the progenitor system accretes
mass could be different between galaxy types, either accretion from a nearby companion star (which could have different metallicities depending on the stellar populations in the host galaxy types) or the merger with another white dwarf \citep{2009ApJ...699.2026R}.

Therefore, there are clear reasons to search for correlations between the properties of Ia's and the properties of their host galaxies. For example, there is a well-established difference between the rates of Ia's in passive and star-forming galaxies, potentially indicating two different paths or timescales for Ia's
\citep{1979AJ.....84..985O, 1990PASP..102.1318V, 2005A&A...433..807M, 2006ApJ...648..868S}. There have also been indications that the host galaxy type correlates with the observed residuals on the SN Hubble diagram, even after standardizing each SN Ia
\citep{2003MNRAS.340.1057S, 2008ApJ...685..752G, 2009arXiv0912.0929K}. For example,
\citet{sullivan2010}
recently reported that SNe Ia in massive host galaxies are 0.08 magnitudes brighter than those in lower mass hosts after correction for the light curve shape and color (at a statistical significance of 4$\sigma$).
Such correlations would have important consequences for supernova surveys and could improve the use of SNe Ia as ``standard candles" \citep{2009ApJ...699L.139W, sullivan2010}.

We investigate the environmental dependencies of SNe Ia by studying the residuals on the Hubble diagram (around the best fit cosmology) as a function of host galaxy type. We further ask if there are differences in the assumed dust law depending on the type of the host galaxy. In Section 2, we outline the data used in this analysis, which is taken from the full SDSS-II Supernova Survey \citep{2008AJ....135..338F}. This sample of SNe Ia has several advantages for such environmental studies including high survey efficiency, multi-color ($ugriz$) photometry for all host galaxies and a significant cosmological volume, thus providing a fair sampling of the galaxy distribution. Also, the overall SN rate, as a function of galaxy type, has been measured using this data \citep{smithm}

In Section 2, and Appendix A, we outline the details of our analysis using two, well-known, public light curve fitting procedures; SALT2 \citep{2007A&A...466...11G} and MLCS2k2 \citep{2007ApJ...659..122J, 2009ApJS..185...32K}. We also describe our methodology for defining passive and star-forming host galaxies.  In Section 3, we present our main results, while in Section 4, we discuss these results in light of other work in the field. We conclude in Section 5.

\section{Observations and Data Analysis}

\begin{deluxetable}{lcc|cc}
\tabletypesize{\scriptsize}
\tablecaption{Number of SNe in our sample.
\label{tbl_sample}}
\tablewidth{0pt}
\tablehead{
\colhead{Selection} & \multicolumn{2}{c}{Spec Confirm} & \multicolumn{2}{|c}{Total$^a$}\\
\colhead{} & \colhead{SALT} & \colhead{MLCS} & \multicolumn{1}{|c}{SALT} & \colhead{MLCS}
}

\startdata
All SNe                 & \multicolumn{2}{c}{258} & \multicolumn{2}{|c}{361} \\
\hline
After LC cut     & 192 & 187 & 253 & 256 \\
After LC fitter limits   & 185 & 161 & 234 & 214 \\
Valid host galaxy type & 127 & 104 & 162 & 135 \\
\hline
Passive         & 27 & 24 & 40  & 35 \\
Star-forming  & 100 & 80 & 122  & 100 \\ \hline
 \enddata
 \tablenotetext{a}{Total SNe used in our analysis including spectroscopically--confirmed and photometrically--classified SNe Ia}
\end{deluxetable}

\begin{deluxetable}{llrrrrrrrr}
\tabletypesize{\scriptsize}
\tablecaption{
Host properties used in the main SALT2 analysis. Hosts with negligible star-formation rates are indicated with $N/A$ and
members of the restricted sample with r.
This is a sample of the full version of the table published in electronic format and can be found at the end of the preprint.}
\tablewidth{0pt}
\label{sn_sample}
\tablehead{

\multicolumn{2}{c}{designation} &
\multicolumn{2}{c}{host position} &
\multicolumn{1}{c}{stellar mass} &
\multicolumn{1}{c}{SFR}\\

\colhead{SN ID} &
\colhead{IAU} &
\colhead{$\alpha(J2000)$} &
\colhead{$\delta(J2000)$} &
\colhead{$[\log M_{\odot}]$} &
\colhead{$[\log M_{\odot}/yr]$} &
\colhead{SN$^a$} &
\colhead{Host$^b$} &
\colhead{Sample}
}
\startdata
1032  &  2005ez  & $03^h07^m11.016^s$ & $+01^{\circ}07^`11.96^{``}$ & $10.47^{+ 0.09}_{- 0.07}$ & $N/A$ &sp& p&  \\
1580  &  2005fb  & $03^h01^m17.544^s$ & $-00^{\circ}38^`38.63^{``}$ & $ 7.72^{+ 1.00}_{- 0.32}$ & $-0.98^{+ 1.02}_{- 0.36}$ &sp&sf& r  \\
15421  &  2006kw  & $02^h14^m57.912^s$ & $+00^{\circ}36^`09.80^{``}$ & $10.17^{+ 0.12}_{- 0.10}$ & $ 0.80^{+ 0.18}_{- 0.22}$ &sp&sf& r  \\
11172  & N/A & $21^h29^m39.120^s$ & $-00^{\circ}12^`07.88^{``}$ & $10.18^{+ 0.14}_{- 0.03}$ & $ 1.00^{+ 0.02}_{- 0.33}$ &lc&sf& r  \\
          &   & & & ... & & & &   \\
\enddata
\tablenotetext{a}{SN classification as a Ia based on spectra (sp) or light-curve shape (lc)}
\tablenotetext{b}{Our host classification either as passive (p) or star-forming (sf)}
\end{deluxetable}

\subsection{SDSS-II SN Sample}

In this analysis, we use the full dataset from the  SDSS-II Supernova Survey
\citep{2008AJ....135..338F}, which provides one of the largest samples of SNe Ia currently available. The SDSS-II SN Survey was a dedicated search for transient objects using the SDSS 2.5-m telescope and imaging camera
\citep{2000AJ....120.1579Y, 2006AJ....131.2332G}
to perform repeat imaging of  the ``Stripe 82" region of the SDSS survey for 3 months a year from 2005 to 2007. The SDSS-II transient database contains many thousands of potential SN candidates, out to $z\sim0.5$, of which $\simeq$500 were spectroscopically confirmed as SNe Ia during the survey period \citep{2008AJ....135..348S, 2008AJ....136.2306H}. The first year (2005) of the SDSS-II SN sample was recently used for detailed cosmological analyses \citep{2009ApJS..185...32K, 2009ApJ...703.1374S, 2010MNRAS.401.2331L}.

For the host galaxy analysis presented herein, we focus on the low--redshift part ($z<0.21$) of the SDSS-II SN sample, where  the SN light curves are measured to high
accuracy, with multiple, high signal-to-noise ratio data-points per light curve, and the k-corrections are empirically determined to be more reliable, i.e., minimizing the influence of the UV--part of the SN spectrum (see Foley et al. 2008; Kessler et al. 2009a). Furthermore, the efficiency of the SDSS-II SN survey remains above 50\% below this redshift limit as demonstrated in
\citet{2010arXiv1001.4995D}.

To ensure our SN sample is more complete, we also include photometric SNe Ia that have a light curve consistent with being a Type Ia, based on the Bayesian light curve fitting of  \citet{2008AJ....135..348S}, and a known host galaxy spectroscopic redshift. The likely non-Ia contamination within these additional photometrically--classified Ia's is only $\simeq 3\%$
\citep{2010arXiv1001.4995D}.

In total, this provides a sample of 361 supernovae (for $z<0.21$), of which 258 are spectroscopically confirmed. We provide a breakdown of these SN numbers in Table \ref{tbl_sample} where ``Spec Confirm" gives just the spectroscopically confirmed SNe.
In the following section, we describe the light curve fitting procedure and the host galaxy classification which lead to a further reduction in the available SN sample (see Table \ref{tbl_sample}).

\subsection{Fitting SN Light Curves}
\label{fitters}

Several algorithms are available for fitting the light curves of SNe, and determining cosmological distances and SN properties. The two most common, publicly--available, fitting methods are SALT2  \citep{2007A&A...466...11G} and MLCS2k2 \citep{2007ApJ...659..122J, 2009ApJS..185...32K}; we use both of these techniques to explore the dependencies of our results on the details of the light curve analysis.  For the main results of this paper, we use the public SALT2 light curve fitter and, in Appendix A, we provide a similar analysis using MLCS2k2. We find that the results and conclusions of this paper are consistent for both light curve fitting algorithms and different SN selection criteria.

For our main SALT2 light curve fitting analysis, we impose the following additional criteria to our SN sample (described in Section 2.1) to ensure robust measurements for the stretch and color of each SN (based on our experience with the first year SDSS-II SN cosmology analysis). First,  we require at least five epochs in all the SDSS $gri$ passbands, with at least one measurement before the light curve maximum. We also require that the reduced $\chi^2$ of the light curve fit to the data in each filter is less than three.  These cuts reject 108 SNe from our sample and are shown in Table \ref{tbl_sample}  labelled as  ``After LC cuts".

SALT2 reports for each individual
SN an apparent brightness ($m_B$) in the B-band, a stretch value ($x_1$) and a color (or $c$) term, which can then be used to calculate a distance modulus ($\mu$) using,

\begin{equation}
\mu = (m_B - M)+\alpha x_1 - \beta  c,
\label{eq1}
\end{equation}

\noindent where $M$ is the ``standardized" absolute SN Ia magnitude (at $x_1=c=0$), $\alpha$ describes the overall stretch law for the sample and $\beta$ is the color law for the whole sample\footnote{If the color term is interpreted as solely due to dust absorption, then
the relationship $\beta = R_B = R_V + 1$ should hold. However, such an interpretation
is probably too simplistic due to intrinsic variations of the SN color.}. We only report the {\it uncalibrated} values of $M$ from SALT2 which are degenerate with our assumed value of $H_0$:  We are only interested in relative differences in the absolute brightness of SNe, not the true brightness. In most SN cosmological analyses, it is assumed that these parameters are invariant to the type of host galaxy in the
sample and do not evolve with redshift, but there is no a-priori reason for such an assumption.

Based on the observed values of our SN stretch and color distributions, we remove a further 19 light curves by imposing the limits of $-4.5<x_1<2$ and $-0.3<c<0.6$, which are labelled in Table \ref{tbl_sample} as  ``After LC fitter limits". These limits were empirically determined to remove SN events where SALT2 reports extreme values either for $x_1$ or $c$
not resembling the majority of SN in our sample (see Fig. 2), leaving 234 SNe Ia for our main SALT2 analysis (as shown in Table 1).

Following the usual SALT2 prescription, we determine $M$, $\alpha$ and $\beta$ in Eqn. \ref{eq1} by minimizing the scatter about a fiducal redshift-distance relation. We adopt as the reference cosmological model a flat universe with an energy density of matter of $\Omega_M=0.272$,
taken from \citet{2009ApJS..180..330K}, and $H_0=65\,{\rm km\,s^{-1}\,Mpc^{-1}}$. We have confirmed that the main results of this paper do not depend on the details of this assumed cosmological model.

\begin{figure}[t]
\epsscale{.85}
\plotone{fig1.eps}
\caption{
The distribution of stellar mass and star-formation rate (SFR) for the complete sample of host galaxies used herein calculated using P\'EGASE2. The (blue) open squares are star-forming host galaxies while solid (red) circles are passive hosts. The open (grey) diamonds are host galaxies excluded because either because of  their large error bar (shown in grey), which makes their classification less clear, or the fit to the SN light curve does not pass the selection cuts discussed in the text. The dashed line shows the  $\log(\textrm{sSFR}) < -10.5$ limit for star--forming galaxies discussed in the text. Passive galaxies (with zero SFR) are artificially plotted at $-4.75$ to be displayed in this figure.
\label{fig:pegase}}
\end{figure}

\subsection{Host Galaxy Properties}
\label{host_galaxy_properties}

A key part of our analysis is the host galaxy properties for each SN, which were determined using the techniques outlined in detail in Smith (2009). We begin by matching SN positions, within a 0.25 arcminute search radius, with SDSS galaxies detected in the deep optical stacked images of ``Stripe 82" constructed from the SDSS-I/II photometry \citep{2009ApJS..182..543A}
and choose the closest match as the host galaxy.  We also require that all SN host galaxies have a measured SDSS model magnitude of $r\le23$. These two constraints remove 7\% of our SNe with either a missing or too faint host. We then visually confirm each host, via inspection of images with and without the SN present, to ensure that the correct host has been associated with each SN.
In six cases, where the host is extended, or de-blended into multiple objects by the SDSS automated software, we
adjust the host position to the center of the underlying galaxy.

Each of the SNe Ia in our sample has a spectroscopic redshift, either from the SN itself or its host galaxy. This redshift is combined with the five SDSS photometric measurements (model magnitudes in $ugriz$ passbands corrected for Galactic extinction) for the host galaxy to determine the star-formation rate (SFR) and stellar mass of each system, using the P\'EGASE2 spectral energy distributions \citep{1997A&A...326..950F, 1999astro.ph.12179F} and the Z-PEG software package \citep{2002A&A...386..446L}. In detail, we used the eight star-forming scenarios, as listed in Table 1 of Le Borgne \& Rocca-Volmerange (2002), and assume a Kroupa (2001) initial mass function. In these scenarios, the SFR is defined for most galaxies via $\textrm{SFR} = \nu \times M_{\textrm{gas}}$, where $\nu$ (in units of $\textrm{Gyr}^{-1}$) ranges from 0.07 to 3.33, while for irregular galaxies, the SFR is defined as $\textrm{SFR} = 0.065 {M}_{\textrm{gas}}^{1.5}$ ($M_\textrm{gas}$ is the density of gas in solar masses). We use the default modeling of internal dust as discussed in Le Borgne \& Rocca-Volmerange (2002)
where a King profile is used for the elliptical template, whilst a plane-parallel slab distribution is used for spiral and irregular galaxies.
Each scenario is then evaluated at 69 different time-steps in their evolution, thus resulting in 552 possible galaxy template spectra covering a wide range of possible evolutionary scenarios.

\begin{figure*}[t]
\epsscale{1.}
\plotone{fig2.eps}
\caption{
The observed distribution of the SALT2 $x_1$ and $c$ (color) values.
Red solid circles denote SNe in passive galaxies, whereas open blue squares indicate SNe in star-forming galaxies. The dot--dashed box shows the restricted subsample discussed in the text in Section \ref{testlaw}. The histograms in the top panel of the figure show the normalized distribution
in $c$ for the star-forming (blue open) and passive (red solid) host galaxies. The right--hand panel shows similar histograms but now for the $x_1$ distributions. \label{fig2}}
\end{figure*}

These templates were fit to the galaxy fluxes (converted from their model magnitudes after correcting them to the AB-system), with the redshift fixed to the redshift of the SN or host galaxy, to determine the best template for each host galaxy. The normalization is related to the total stellar mass of these templates, and is a free parameter which is determined as part of the fitting procedure to the host. We also use the best-fit template to estimate the recent star-formation rate of the host galaxy by integrating the best-fit template over the last half a gigayear of its evolution, e.g., if the best-fit template had an age of 8 Gyrs, then the recent SFR of the host was calculated over the period 8 to 7.5 Gyrs. Error bars on these estimates were determined by propagating the observed galaxy photometric errors. Our technique is similar to that used by Sullivan et al. (2006), and the Z-PEG software and the spectral energy distributions, have been used substantially in the literature (Glazebrook et al.2004; Grazian et al. 2006; Smith et al. 2010).

To test the validity of our P\'EGASE2--based methodology, we have compared the stellar masses determined using the photometric data on over 350,000 SDSS Main galaxies (taken from DR4; Adelman-McCarthy et al. 2006) to masses derived by Kauffmann et al. (2004) using a technique based on SDSS spectral features. We find no mean difference between the two mass estimates, with a variance of only 3\%.

In Figure \ref{fig:pegase}, we show the separation of host galaxies according to their stellar mass and SFR obtained from the P\'EGASE2 analysis above. We classify each host as either  passive (i.e., shows no sign of recent star-formation activity), or star-forming (i.e., having evidence for recent star-formation). To ensure a clean separation between these two galaxy classes, we require that the measured $1\sigma$ error on the estimated SFR for each galaxy is smaller than the separation in SFR for the two classes of galaxies, i.e., it is then unlikely that statistical errors on an individual SFR measurement can scatter a galaxy from one host galaxy type to the other. We also exclude star-forming host galaxies which have a specific SFR (i.e., sSFR; defined as the SFR per stellar mass) in the range $\log(\textrm{sSFR}) < -10.5$ as illustrated in Figure 1 as a dashed line. This limit excludes the locus of star--forming galaxies at the bottom of the blue cloud of points in Figure \ref{fig:pegase}, which are predominantly fit by the P\'EGASE2  lenticular galaxy scenario, thus leading to an unclear interpretation of their star--formation activity. These cuts and limits ensure we have two, well-separated, samples of host galaxies. We show in Table 1 the numbers of SNe available in these two host galaxy classes and note that many SNe have been excluded from further analysis because of the ambiguity of their host galaxy type.
In Table \ref{sn_sample}, we provide the SN designation, host galaxy coordinates, host galaxy stellar mass and star-formation rate (derived from our best fit P\'EGASE2 model). We also provide whether the SN was spectroscopically confirmed (sp) or classified by just its light curve (lc), and if we classify the host galaxy as passive (p) or star-forming (sf).
More sophisticated stellar population models could have been used to determine the star--formation histories of our host galaxies \citep{2009A&A...493..425M} but we find that our classification of galaxies into two broad classes of star-formation activity is relatively unaffected by the choice of templates (see Smith 2009 for more details). Also, our host galaxy analysis is in good agreement with a simple cut on the color of the galaxies, e.g., dividing the galaxies at $u-r=2.22$ \citep{2001AJ....122.1861S}.

\begin{figure*}[t]
\epsscale{1.5}
\plotone{fig3.eps}
\caption{The Hubble residuals as a function of redshift. The open blue squares denote SNe Ia in star-forming galaxies, while the solid red circles are SNe Ia in passive galaxies. The dashed line is the reference absolute magnitude fit to the whole sample regardless of host galaxy type. We find SNe Ia in passive host galaxies are $\simeq0.1$ magnitudes brighter than in star--forming hosts even after light curve fitting.
 \label{fig:hubbleresiduals}}
\end{figure*}

\section{Results}

\subsection{Testing the SALT2 Stretch and Color Relations}
\label{testlaw}

In Figure \ref{fig2}, we show the SALT2 output ($x_1$ and $c$ values from Eqn. 1) for our low redshift sample of SNe. The open (blue) squares represent SNe in host galaxies classified as star--forming as described in Section \ref{host_galaxy_properties}, whereas the solid (red) circles represent SNe in passive galaxies. In agreement with \citet{2006ApJ...648..868S}, we find a clear difference in the $x_1$ distributions between SNe Ia with a rapid decline rate (small $x_1$ values), which favor passive galaxies, and brighter, slower SNe (larger $x_1$ values) that favor star-forming galaxies. This result is clearly seen in the right-hand panel of Figure 2 and can be quantified using a Kolmogorov-Smirnov test (KS-test), where the probability for the two $x_1$ distributions being drawn from the same underlying parent distribution is only $10^{-7}$. This result has been known for some time \citep{{2000AJ....120.1479H}}, but it is reassuring that we can clearly recover this well-known difference in the $x_1$ distributions.

However, we note that in Figure \ref{fig2} there is no clear separation in the color term ($c$) of SNe with respect to host galaxy type, i.e., both populations span the same range in color. This agreement is demonstrated in the top--panel of Figure 2 and a K-S test of the two $c$ distributions has a probability of $0.24$ of being drawn from the same underlying parent distribution, i.e., no evidence that they are drawn from different underlying distributions. This observation seems counter-intuitive, as we might expect some differences in the  global dust properties of these two host galaxy types, and maybe even an inclination dependence for the disk (star-forming) galaxies as outlined recently by \citet{2010arXiv1001.1744M}. This similarity in the color distributions implies that the rest-frame colors of SNe are dominated either by local, circumstellar dust, with the same color distributions, and/or SNe Ia have the same intrinsic color variations in all galaxy types.

\begin{deluxetable}{lcccccc}
\tabletypesize{\scriptsize}
\tablecaption{Best fit values for $M,\alpha, \beta$ as a function of host galaxy type}
\tablewidth{0pt}
\tablehead{
\colhead{Host Galaxies} &
\colhead{Restricted$^a$} &
\colhead{$M$} &
\colhead{$\alpha$} &
\colhead{$\beta$} &
\colhead{$\chi^2$} &
\colhead{No. of SNe}
}
\startdata
passive             &  no  & $-30.19\pm 0.03$ & $   0.16\pm 0.02 $ & $  2.42\pm 0.16 $ &    34.46  & 40   \\
                           &   yes  & $-30.23\pm 0.05$ & $   0.18\pm 0.03 $ & $  2.50\pm 0.41 $ &    12.60  & 27  \\
\hline
star-forming      & no &   $-30.10 \pm0.01$  & $ 0.12 \pm0.01$ &  $3.09 \pm0.10$  & 143.63   & 122  \\
                           & yes &  $-30.11 \pm0.02$  & $ 0.16 \pm0.02$ &  $3.22 \pm0.20$  &   94.55   &   89  \\
\hline
\enddata
\tablenotetext{a}{Restricted range in allowed $c$ and $x_1$ as shown in Figure 2.}
\end{deluxetable}

As discussed above, there is a clear trend for the $x_1$ distribution with
host type, but no obvious trend for the color distribution. We explored if the constants in Eqn. 1 ($M$, $\alpha $, $\beta$) are dependent on host type by using a Markov chain Monte Carlo (MCMC) simulation
where we minimize the $\chi^2$ for the fit to the distance modulus versus redshift, as a function of $(M, \alpha , \beta)$ separately for passive and star--forming galaxies. Fits were obtained by running the MCMC chains with 50,000 accepted steps and adjusting the step size empirically to achieve a typical frequency for accepted steps of $\approx 20\%$.  One sigma errors on each parameter are provided by marginalizing over the remaining other parameters from the MCMC chains. We perform this analysis assuming an intrinsic dispersion of $\sigma_{int}=0.14$ mags, which is added in quadrature to the errors on the distance modulus to achieve a reduced $\chi^2$ close to one (i.e., $\chi^2/{\rm ndf} \approx 1$, see Lampeitl et al. 2010 for further discussion of this intrinsic dispersion).

To ensure our results are not driven by a few outliers, we also perform our analyses on a restricted subset of SNe with tighter allowed ranges of $c$ and $x_1$ values. This restricted sample is illustrated in Figure 2 as the inner dot-dashed box and reduces the sample from 162 SNe (see Table 1) to 116 SNe.

In Table 2, we summarize our results for fitting $M,\alpha , \beta$ for the full sample and the restricted subsample discussed above. We see a correlation between the host galaxy type and the absolute magnitude ($M$) of the supernovae, i.e., after the SNe have been standardized using the SALT2 light curve fitting algorithm, there is still a difference of $\simeq0.1$ magnitudes, with SNe Ia in passive galaxies being brighter (more negative absolute magnitudes). To illustrate this effect, we present in Figure \ref{fig:hubbleresiduals} the residuals to the Hubble diagram (after removing the fiducial redshift--distance relation) for the best-fit SALT2 parameters of $(M,\alpha,\beta)=(-30.11, 0.12,2.86)$, which were determined from fitting the whole SN sample regardless of host galaxy type. As can be seen, there is a visible offset between SNe in passive and star--forming galaxies.

The interpretation for the other SALT2 parameters ($\alpha $ and $\beta$) is less clear. First, we see no clear evidence for differences in $\alpha $ with host galaxy type given the statistical errors. Next, the fitted values of $\beta$ (the color law) for star-forming galaxies do appear to be larger than that found for passive galaxies, i.e., $\beta$ values for star-forming galaxies are above three, while for passive galaxies we find values below three. The significance of this difference in $\beta$ varies between the full and restricted samples,  which is not too surprising, as excluding the outliers in the color range will clearly increase the statistical error on the slope of the color law seen in Table 2. The mean slope ($\beta$) is similar for both the full and restricted sample.

In Figure \ref{fig3}, we show the corrected absolute magnitude for SNe in passive galaxies as a function of their fitted color and stretch values. The left-hand panels show the {\it color--corrected} absolute magnitude as a function of $x_1$, i.e., only the color part of Eqn. 1 ($\beta c$) has been applied to
$m_B$. The right--hand panels show the {\it stretch--corrected} absolute magnitude, as a function of $c$, when only the stretch component of Eqn. 1 ($\alpha x_1$) has been applied to the distance modulus. In the upper panels, we show the best fitting law (Eqn. 1) assuming the best fit values of $M, \alpha, \beta$ for passive galaxies in Table 2, i.e., $(M, \alpha , \beta)_{P}=($-30.19$, 0.16, 2.42)$ respectively. In the lower row of panels, we show the best fit law again but now assuming the best fit parameters for star-forming galaxies, namely $(M, \alpha , \beta)_{SF} =($-30.10$, 0.12, 3.09)$.

Comparing the top and bottom left--hand panels in Figure 4, it is clear we see that the amplitude ($M$) of the best-fitted relationship is different between the two and clearly wrong in the bottom panels (i.e., using the star-forming best-fit SALT2 parameters for SNe in passive galaxies). In Figure 5, we show the same analysis as in Figure 4, but this time the data plotted is for the star--forming SN sample. Again, we see that the amplitude of the fitted law (Eqn. 1) is different and inappropriate if used to describe the wrong type of galaxy.

\begin{figure*}[t]
\epsscale{.99}
\plotone{fig4.eps}
\caption{(Left panels) The color--corrected absolute magnitudes of our SNe versus their $x_1$ parameters for our SN sample in passive host galaxies (without the $c$ and $x_1$ restrictions discussed in the text). (Right panels) The $x_1$-corrected absolute magnitudes versus color ($c$) of the SNe. In the upper row, we have applied to the data the best fit SALT2 parameters derived for the passive SN sample, whereas in the lower row we use the best fit SALT2 parameter for the star-forming SN sample. The vertical dashed lines indicate the restricted region in $x_1$ and $color$ discussed in Section 3.1 and shown as the inner dashed box in Figure \ref{fig2}. The inclined dashed line in the left plots indicates the applied
stretch-correction ($\alpha  x_1$) and similarly the solid line in the right plots are the color--correction law ($\beta c$).
\label{fig3}}
\end{figure*}

\section{Discussion}

\subsection{Systematic Uncertainties}

Before we interpret these results, it is important to understand potential systematic uncertainties in our analysis. First, we have tested if our result depends on
the inclusion of a subset of photometrically confirmed SNe. We see negligible changes in the central values of $M_0$, $\alpha$ and $\beta$ (for SALT2) which are significantly
smaller than the errors on these parameters. We also verify the robustness of our results to reasonable changes in the fiducial cosmological model and find no significant effect as expected. Likewise, we have increased the redshift range of the sample used in our analysis, e.g., increasing the limit to $z<0.45$, which more than doubles the size of the sample but makes the sample more incomplete. We find that the observed differences with host galaxy type are still present,  but the significance is decreased. This decrease in significance is likely caused by the decrease in signal-to-noise ratio for both the SN light curves and the galaxy photometry, as well as increases in the sample incompleteness (both spectral confirmations and Malmquist bias effects). The uncertainties in the k--corrections are also increased as the UV--part of the SN spectrum becomes more important. We note that Sullivan et al. (2010) sees similar results but for the higher redshift SNLS sample, thus suggesting that any decrease in significance we witness is probably caused by observational issues rather than evolution in the SN population.  For these reasons, we have chosen to focus on the cleanest, most efficient, sample of SDSS-II SN at $z<0.21$.

In Appendix A, we provide a parallel analysis of our data using the MLCS2k2 light curve fitting technique and find similar results to those seen in the SALT2 analysis in Section 3; namely differences in the absolute magnitude of SNe Ia between passive and star--forming host galaxies, as well as differences in the best-fit color laws  (as represented by $R_V$). This confirms that our results are not sensitive to the details of how the light curves were analysed and suggests the trends we see are either inherent to the supernovae, especially as we still see a correlation for the restricted sample of SN. However, further analysis is required to conclusively determine the fundamental origin of the observed correlations including potential improvements in the light curve fitting methodologies, e.g., a better representation of local SNe in passive hosts within the MLCS2k2 training sample or a more sophisticated parameterization of the $x_1$ and $c$ distributions in SALT2 to better accommodate the fast declining SNe Ia in passive hosts.

\begin{figure*}[t]
\epsscale{.99}
\plotone{fig5.eps}
\caption{Similar in Figure 4, but for SNe in star-forming host galaxies. The dashed lines are for the restricted subset of SNe discussed in Section 3.1.
\label{fig4}}
\end{figure*}

\subsection{Intrinsic Dispersion}

Throughout this analysis, we have assumed an intrinsic dispersion for our SN sample of $\sigma_{int}=0.14$ mags, which is consistent with the value obtained for the first year SDSS-II data analysed in Lampeitl et al. (2010). In Table 2 (and Table \ref{tab:mlcs}), we present the separate $\chi^2$ values for SNe in both passive and star-forming galaxies (for both SALT2 and MLCS2k2). The star--forming subsample has a reduced $\chi^2$ above unity, but for passive hosts, the reduced $\chi^2$ value is now less than one. This observation suggests
that SNe in passive galaxies would favor a smaller intrinsic dispersion and are thus a more homogeneous population, or the observed errors are more representative of the scatter
about the Hubble diagram.

To investigate this matter further, we have re--fit both the passive and star--forming SN samples (with no restrictions in $c$ and $x_1$) and adjusting the intrinsic dispersion
$\sigma_{int}$
to a value that gives a reduced $\chi^2$ close to 1. In the case where we fit the passive sample with the parameters derived from just the passive sample we
find $\sigma_{int}=0.13$, but
these values give $\chi^2$ of 321 (for 122 SNe) for the star-forming sample. In the reverse, we fit the star-forming sample with the best-fit star-forming parameters and find
$\sigma_{int}=0.17$, where now the passive sample yields $\chi^2=55$ for 40 SNe.
This result suggests that by using the larger $\sigma_{int}$ for SNe in passive hosts, we are effectively down-weighting these SNe (by increasing their errors) because of the offset in $M$ between the SNe Ia in the two types of galaxies.

\begin{figure*}[t]
\epsscale{1.}
\plotone{fig6.eps}
\caption{The residuals around the best-fit Hubble diagram as a function of host galaxy stellar mass (calculated from P\'EGASE2). The (red) solid circles are for passive host galaxies, while the (blue) open squares are star-forming hosts. The inclined dot--dot--dashed line is the best--fit to these data, while the dot--dashed line is the fiducial 3--parameter SALT2 fitted model (Eqn. 1) without regard to the host galaxy stellar mass or type.
\label{fig9}}
\end{figure*}

\subsection{Mass Dependency}

\begin{deluxetable}{lcccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Best fit values $(M, \alpha, \beta, \gamma)$ \label{tab:gamma}}
\tablewidth{0pt}
\tablehead{
\colhead{Dataset} &
\colhead{M} &
\colhead{$\alpha$} &
\colhead{$\beta$} &
\colhead{$\gamma$} &
\colhead{$\chi^2$} &
\colhead{No. SNe}
}
\startdata
Full SN sample &$ -30.09\pm0.01$ & $0.13\pm0.01$ & $3.04\pm0.07$ & $0.072\pm0.018$ & $ 179.53$ & $ 162$  \\
Restricted & $ -30.11\pm0.01$ & $0.16\pm0.02$ & $3.22\pm0.30$ & $0.088\pm0.008$ & $  98.87$ & $ 116$  \\

\hline
\enddata
\end{deluxetable}

As discussed in Section 2.3, we have classified our SN host galaxies into two well--separated classes, namely passive and star-forming. We initially separated the galaxies in this way because previous studies of the properties, and rates, of SNe Ia have shown clear correlations with the star--formation activity of the host, e.g., Hamuy et al. (2000), Sullivan et al. (2006), Manucci et al. (2005), Dilday et al. (2010b). Gallagher et al. (2008) showed that the measured metallicity for local SN host galaxies was correlated to the residuals around the best fit distance-redshift relation, which has prompted some authors to look for correlation with the host galaxy stellar mass as a proxy for the metallicity (using the known mass-metallicity relationship; Tremonti et al. 2004). Both Kelly et al. (2009) and Sullivan et al. (2010) find such a correlation with stellar mass and Sullivan et al. (2010) exploit this correlation to improve the cosmological fits to the 3-year SNLS dataset.

We present here a first analysis of the SDSS-II Hubble residuals as a function of the host galaxy stellar mass. The advantage of such an analysis is that the host stellar mass is a continuous parameter thus potentially avoiding uncertainties associated with binning galaxies into two distinct samples. Unfortunately, estimating the stellar mass from broad-band photometry is challenging and therefore, the measurements of stellar mass can be noisy and potentially biased. For the analysis below, we therefore restrict ourselves to the clean sample defined in Section 2.3, i.e., we still do not include galaxies with an ambiguous classification between star-forming and passive.

 In Figure \ref{fig9}, we show the Hubble residuals as a function of stellar mass. As expected, the passive galaxies have preferentially higher stellar masses than the star-forming subsample. A linear fit to the Hubble residuals is shown as the inclined line (dot-dot-dashed) and has a slope of $0.069\pm0.014$ magnitudes per $\rm{log(M_{\sun})}$. We see the same result, at the same statistical significance, with our MLCS2k2 analysis in Appendix A. Therefore, these results imply that adding an additional parameter, dependent on the host stellar mass, to Eqn. \ref{eq1} would give a better fit to the SDSS-II SN Hubble diagram.

To quantify this statement, we have repeated our MCMC analysis in Section 3.1, but now minimizing over 4 parameters $(M, \alpha, \beta, \gamma)$ where calculating the distance modulus using,
\begin{equation}
\mu = (m_B - M)+\alpha x_1 - \beta  c+\gamma m_{st},
\label{eq2}
\end{equation}
where $m_{st}$ is defined as
\begin{equation}
m_{st} = \log(m_{host}) - 9.5,
\label{log:hostmass}
\end{equation}
and $m_{host}$ is the stellar mass of the host in units of solar mass derived from our P\'EGASE2 fits to the host galaxy colors (Section 2.3).

The results of this new analysis are presented in Table \ref{tab:gamma}. First, we see that adding an additional parameter has not significantly changed the fitted values of $M$, $\alpha$ and $\beta$; they are close to the values in Table 2 for star-forming hosts (which dominate the whole sample). Secondly, we see a significant non-zero value for $\gamma$ at $>4\sigma$. Finally, we can compare the three and four-parameter fits (Eqns. 1 and 2 respectively) to the full SN sample, regardless of host galaxy type, using the Bayesian Information Criteria (BIC; Liddle 2004). BIC is a penalized likelihood statistic that accounts for models with different numbers of parameters, and we find the BIC score for Eqn. 2 is 117 compared to 134 for Eqn. 1. The smaller BIC score demonstrates the additional parameter is justified.

\section{Conclusions}

We present an analysis of the host galaxy dependencies for the SDSS-II Supernova Survey.  We have used 361 SNe Ia (see Table \ref{tbl_sample}) taken from the full three years of this survey, and then applied several data cuts to ensure we have a clean, well-understood, sample of low redshift SNe ($z<0.21$). We have analysed these data using two well-known light--curve fitting routines (SALT2 and MLCS2k2) to demonstrate that our results are not dependent on the details of the light curve analysis. We summarise below the main conclusions of this work:

\begin{itemize}

\item We confirm, to high significance, the strong correlation between host galaxy type and the observed width of the light curve, i.e., quick decline--rate SNe (small $x_1$ values in SALT2), favor passive host galaxies, while bright, slower decline SNe Ia (larger $x_1$ values) favor star-forming galaxies. This has been seen before by several authors. However, we find no correlation between the color of individual SNe Ia and their host galaxy, as illustrated in Figure 2.

\item We find that SNe Ia are $\simeq0.1$ magnitudes brighter in passive host galaxies after light curve fitting. This effect is true for both SALT2 and MCLS2k2 analyses. The statistical significance of this difference is between $2$ and $3\sigma$ dependent upon the details of the fitting methodology and the inclusion of outliers in the color and $x_1$ distributions of these data.

\item We find evidence for differences in the SN color relationship between passive and star--forming host galaxies. For SALT2, we detect differences in $\beta$, with passive hosts showing $\beta\simeq2.5$ and star--forming hosts prefering $\beta>3$. For MLCS2k2, we see a similar trend for passive hosts preferring a dust law with $R_V\simeq1$ and star-forming hosts giving $R_V\sim2$. The significance of these trends depends on the color range considered, but is greater than $3\sigma$ for the full SN sample considered herein.

\item We find that the required intrinsic dispersion for passive galaxy hosts is smaller than that needed for the whole SN sample (and for star--forming hosts), e.g., only
$\sigma_{int}=0.13$ mags is required to obtain a reduced $\chi^2$ close to unity for passive hosts compared to 0.17 mags for the star-forming sample. This lower intrinsic
dispersion for passive hosts is true for both SALT2 and MLCS2k2 light curve fitters.

\item We demonstrate that the dependence on host galaxy type can be parameterized using the stellar mass of the host galaxies. We show that a 4--parameter fit to the distance modulus of SNe Ia ($M$, $\beta$, $\alpha$, $\gamma$) -- where $\gamma$ scales with stellar mass -- is better than the usual 3--parameter model given in Eqn. 1. For the data in Figure \ref{fig9}, we find $\gamma=0.069\pm0.014$, or a $4\sigma$ detection of this parameter.
\end{itemize}

These conclusions are in good agreement with other work, especially Kelly et al. (2009) and Sullivan et al. (2010). In particular, Sullivan et al. (2010) see the same trends in both $M$ and $\beta$ discussed herein with a similar level of statistical significance. This indicates that these trends are common to several SN surveys and appear to not change significantly with redshift. One possible cause for these correlations is a difference in the host galaxy metallicity (see Gallagher et al. 2008), which is correlated with the host stellar mass and host star-formation activity, and could affect the metallicity of the progenitor star thus leading to changes in the peak brightness of SNe Ia. However, the origin of these correlations requires further study, especially to ensure deficiencies in the light curve fitting techniques are not directly responsible.

The host galaxy dependencies presented in this paper will be important for future supernova cosmology surveys, which may wish to exploit these dependencies to minimize the scatter on the SN Hubble diagram. This could be achieved by including further parameters in the light curve fitting or distance modulus calculation (e.g., Eqn. 2). We will explore these issues in the future with the full SDSS-II SN Survey similar to the recent analysis of Sullivan et al. (2010) for the SNLS.

\acknowledgments

\section*{Acknowledgements}

The authors thank the referee for their careful reading of the paper. We also like to thank Mark Sullivan, Janine Pforr and Isobel Hook for helpful discussions during the course of this research. We thank Rick Kessler for his comments on an earlier draft of this paper. HL and RCH were supported by STFC, while MS is funded by an SKA fellowship.
MI acknowledges the support from the grant No. 2009-0063616 from MEST. SWJ acknowledges support from US Dept. of Energy grant DE-FG02-08ER41562.

Funding for the creation and distribution of the SDSS and SDSS-II
has been provided by the Alfred P. Sloan Foundation,
the Participating Institutions,
the National Science Foundation,
the U.S. Department of Energy,
the National Aeronautics and Space Administration,
the Japanese Monbukagakusho,
the Max Planck Society, and the Higher Education Funding Council for England.
The SDSS Web site \hbox{is {\tt http://www.sdss.org/}.}

The SDSS is managed by the Astrophysical Research Consortium
for the Participating Institutions.  The Participating Institutions are
the American Museum of Natural History,
Astrophysical Institute Potsdam,
University of Basel,
Cambridge University,
Case Western Reserve University,
University of Chicago,
Drexel University,
Fermilab,
the Institute for Advanced Study,
the Japan Participation Group,
Johns Hopkins University,
the Joint Institute for Nuclear Astrophysics,
the Kavli Institute for Particle Astrophysics and Cosmology,
the Korean Scientist Group,
the Chinese Academy of Sciences (LAMOST),
Los Alamos National Laboratory,
the Max-Planck-Institute for Astronomy (MPA),
the Max-Planck-Institute for Astrophysics (MPiA),
New Mexico State University,
Ohio State University,
University of Pittsburgh,
University of Portsmouth,
Princeton University,
the United States Naval Observatory,
and the University of Washington.

This work is based in part on observations made at the
following telescopes.
The Hobby-Eberly Telescope (HET) is a joint project of the University of Texas
at Austin,
the Pennsylvania State University,  Stanford University,
Ludwig-Maximillians-Universit\"at M\"unchen, and Georg-August-Universit\"at
G\"ottingen.  The HET is named in honor of its principal benefactors,
William P. Hobby and Robert E. Eberly.  The Marcario Low-Resolution
Spectrograph is named for Mike Marcario of High Lonesome Optics, who
fabricated several optical elements
for the instrument but died before its completion;
it is a joint project of the Hobby-Eberly Telescope partnership and the
Instituto de Astronom\'{\i}a de la Universidad Nacional Aut\'onoma de M\'exico.
The Apache
Point Observatory 3.5 m telescope is owned and operated by
the Astrophysical Research Consortium. We thank the observatory
director, Suzanne Hawley, and site manager, Bruce Gillespie, for
their support of this project.
The Subaru Telescope is operated by the National
Astronomical Observatory of Japan. The William Herschel
Telescope is operated by the
Isaac Newton Group on the island of La Palma
in the Spanish Observatorio del Roque
de los Muchachos of the Instituto de Astrofisica de
Canarias. The W. M. Keck Observatory is operated as a scientific partnership
among the California Institute of Technology, the University of
California, and the National Aeronautics and Space Administration; the
observatory was made possible by the generous financial support of the
W. M. Keck Foundation.


\section{Introduction}
\label{sec:int}
The conductance of one-dimensional quantum wires has been a focus of
research in condensed matter physics for over 20 years.  In agreement
with seminal theoretical works by
Landauer~\cite{landauer57ibm233,landauer70pm863,landauer87zpb} and
B\"uttiker~ \cite{buettiker86prl1761}, early experiments in the late
80's observed a quantization of conductance in units of
$2e^2/h$~\cite{wees-88prl848,wharam-88jpc209}.  This suggested that
there might be a universal explanation for these observations even in
the presence of interactions, which had been neglected in the original
treatment.  However, the precision of measurements of these
conductance plateaus left enough room for the possibility of
non-universal corrections.  Additionally, different theoretical
beliefs concerning the fundamental nature of the conductance did not
converge fast. For nearly one whole decade, the question whether the
quantized conductance is renormalized by interactions in the quantum
wire~\cite{apel-82prb7063,kane-92prl1220,ogata-94prl468} or not~
\cite{kawabata95jpsj30, safi-95prb17040,safi97,ponomarenko95prb8666,
  maslov-95prb5539, alekseev-98prl3503, ponomarenko-99prb16865,safi99ejp451}
dictated the research in this field, where the latter was suggested by
experiments already from the very beginning.
In recent years, the influence and nature of the coupling between the
 leads and the wire on the observed
 value
 of the
conductance has been further emphasized
(e.g. Ref. \onlinecite{imura-02prb035313}, which builds on the notion
of "charge fractionalization" discussed in Ref.
\onlinecite{pham-00prb16397} which is suggested to be observable in
the noise signature of Luttinger liquids~\cite{trauzettel04}).
Moreover, the field had a significant revival when experiments
observed an additional plateau structure generally referred to as 0.7
anomaly~\cite{danneau-08prl016403,thomas-96prl135,yacoby-96prl4612,thomas-00prb13365,kristensen-00prb10950,reilly-02prl246801,biercuk-05prl026801,cronenwett-02prl226805},
and subsequent theoretical studies concentrated on this point as well
as other processes that lead to corrections of the standard quantized
conductance~\cite{meir-02prl196802,matveev04prb245319,sushkov01prb155319,meden06}.
Recently, experimental progress in tunneling spectroscopy of
one-dimensional wire structures promises more detailed studies of the
influence of electron-electron interaction on the conductance and
other transport properties~\cite{chen-09prl036804,auslaender-02s825}.

In this paper, we intend to use an
approach to the problem which is as elementary as possible.  We show
that the quantized conductance is fully accounted for through the
harmonic equations of motion~\cite{haldane81prl1840} of an
inhomogeneous interacting Luttinger liquid (LL), together with proper
boundary conditions.  We note that the use of equations of motion has
played a role in other works as well.  As a contemporaneous
independent approach to a related work by Safi and
Schulz~\cite{safi-95prb17040}, Maslov and Stone~\cite{maslov-95prb5539}
(MS) have used equation of motion techniques in a rather unusual
geometry, where the Luttinger liquid wire must be assumed infinite as
a matter of principle, and a non-zero electric field is applied only
over a finite region of the infinite wire. This idealized setup is
intimately related to the success of a physically correct but
non-standard application of the Kubo formula, where $\omega$ is taken
to zero before $q$~\cite{maslov-95prb5539} (c.f.  also
Ref.~\onlinecite{safi-95prb17040}).
We argue that the subtleties in
the harmonic fluid approach of MS can be circumvented by imposing
proper boundary conditions at the ends of a finite interacting quantum
wire, which enforce chemical equilibrium between the left lead and the
right-traveling modes of the wire, and vice versa.  Similar boundary
conditions were also emphasized in Refs.~\onlinecite{safi99ejp451}
and~\onlinecite{imura-02prb035313}.
While our physical assumptions are equivalent to those made by Safi
and Schulz~\cite{safi-95prb17040} and MS, it is crucial that we study
such boundary conditions in the presence of general position-dependent
interactions, as will become clear below.  For similar reasons, Safi
and Schulz mainly focus on the case where the Luttinger parameter has
two isolated jumps and is otherwise constant, in a semi-infinite
geometry (this is also the case considered in
Ref. \onlinecite{lebedev-05prb075416}).  There are various
similarities between subsequent work by Safi\cite{safi97,
  safi99ejp451} and the approach developed in the following, and we
will comment on these as well as some important differences as we go
along.

In the following,  we will consider the conductance of
a Luttinger liquid wire with
interactions that may be arbitrary in strength
(only constrained by the stability of the LL) and profile in the bulk
of the wire.  However, we will assume that they are screened near the
contacts with the metallic leads, as argued in
Ref.~\onlinecite{maslov-95prb5539}.  This introduces a set of boundary
conditions, which we call ``proximity'' conditions.  These are imposed
in addition to the
boundary condition enforcing "chemical equilibrium" between leads and
chiral modes, to be specified below.
The paper is organized as follows. In Section \ref{sec:ill}, we define
the Luttinger-type Hamiltonian of our model and in particular discuss
our assumptions on proximity boundary conditions. In Section
\ref{sec:emo}, we derive the equations of motion for the current and
review the constraints arising from chemical equilibrium at the
boundary. In Section \ref{sec:dcc} we show that these assumptions
unambiguously imply the quantization of the conductance.  In Section
\ref{sec:acc} we generalize the approach to the case of a low but
finite frequency bias, deriving Drude-type corrections to the DC
result. An explicit formula for the non-universal $\tau$-parameter as
function of spatially dependent interactions is obtained.  In Section
\ref{sec:dis}, we discuss possible modifications of our results for
different experimental setups, and further clarify the relation of our
formalism to other approaches found in the literature.
\section{Inhomogeneous Luttinger Liquid}
\label{sec:ill}
We model the one-dimensional conductor by writing down a Luttinger or
"harmonic fluid" \cite{haldane81prl1840} type of Hamiltonian with
spatially dependent parameters:
\begin{equation}
\begin{split}
H = \frac{1}{\pi}\sum_{\nu}\int dx\, v_{\nu}(x)\{
(\frac{\pi}{2})^2K_{\nu}(x)\Pi_{\upsilon}^{2}(x)\\
+     \frac{1}{K_{\nu}(x)}(\partial_x\Phi_{\nu}(x))^2 \}. \label{LL}
\end{split}
\end{equation}
In the above, $\nu=\rho,\sigma$ refers to charge and spin degrees of
freedom. We will drop this index from now on, as we will only be
concerned with the charge sector of the model.  We follow the
conventions used in Ref.~\onlinecite{seidel-05prb045113}, where the
field $\Phi(x)$
is related to the electronic density via
$\rho(x)=-\frac{2}{\pi}\partial_x\Phi(x)$, and $\Pi(x)$ is its
canonical momentum, $[\Phi(x),\Pi(x')]=i\delta(x-x')$.  The charge
part of the operator that creates a local right (upper sign) or left
(lower sign) moving electron is then proportional to
$\exp\{i\Theta(x)\pm i\Phi(x)\}$, where $\Theta(x)=-\frac{\pi}{2}
\int^x dx'\Pi(x')$, and the spin part is similar with charge fields
replaced by spin fields.

The Luttinger parameter $K(x)$ describes the electronic interactions,
where $K=1$ corresponds to the non-interacting case, and $K<1$ ($K>1$)
corresponds to repulsive (attractive) interactions. In the bulk of the
quantum wire, $K(x)$ and the mode velocity $v(x)$ may have arbitrary
spatial dependence. We will pay special attention to the case where
$K(x)$ smoothly approaches the non-interacting value $K=1$ at the
right and left contact ($x=x_L$ and $x=x_R$, respectively):
\begin{subequations}\label{bc0}
\begin{align}
  K(x_L)&= K(x_R)=1\label{bc0a}\\
  \partial_x K(x_L)&=\partial_x K(x_R) =0\label{bc0b}
\end{align}
\end{subequations}
This situation is depicted in Fig.~\ref{fig:twofermi}.
\begin{figure}[t]
  \begin{minipage}[c]{0.99\linewidth}
    \psfrag{B}{$\text{FL}$}
    \psfrag{A}{$\text{FL}$}
    \psfrag{C}{$\text{LL}$}
    \psfrag{D}{$K(x)$}
    \psfrag{E}{$K_L$}
    \psfrag{F}{$K_R$}
    \psfrag{L}{$\text{L}$}
    \psfrag{R}{$\text{R}$}
    \includegraphics[width=\linewidth]{twofermiv2}
  \end{minipage}
  \caption{(color online) Schematic plot of the Luttinger Liquid wire
    (LL) connected to two Fermi-liquid leads (FL). The value of $K$ is
    arbitrary inside the wire, but the left and right boundary values
    $K_L$ and $K_R$ are assumed equal to $1$ due to Fermi liquid
    screening effects. It is further assumed that these values are
    smoothly approached at the contacts (Eq.~\ref{bc0}).}
  \label{fig:twofermi}
\vspace{0pt}
\end{figure}
The physical assumption behind these conditions is that the
modes of the
reflectionless wire extend into the leads,
where interactions are irrelevant\cite{maslov-95prb5539},
due to Fermi liquid screening.
Detailed arguments explaining why this picture applies to typical
experiments are also given in Ref.~\onlinecite{maslov04}.  Identical
or similar points of view have been adopted by many other authors
\cite{alekseev-98prl3503,lebedev-05prb075416,lehur-08ap3037}.  The
approach discussed here will confirm that while interactions may be
arbitrarily strong in the bulk of the wire, Eq. \eqref{bc0} is a key
ingredient to the observation of the quantized conductance. In
addition, one needs to impose a certain local chemical equilibrium
between the wire and the leads in the sense of Landauer, to be
discussed below.  Note that, independent of $K(x)$, $v(x)$ may
approach different limits at the left and right contact.  We also
point out that while in the infinite wire geometry discussed by other
authors~\cite{maslov-95prb5539,safi-95prb17040}, \Eq{bc0b} is
essentially {\em implied} (explicitly so in Ref.
\onlinecite{safi-95prb17040}) by the requirement that $K(x)$ tends to
a constant at $\pm\infty$, we find that the finite wire geometry makes
it necessary to impose \Eq{bc0b} as a separate constraint.  This
imposes stronger requirements on the strength of the proximity effect
than just \Eq{bc0a} alone.

\section{Equations of motion}
\label{sec:emo}
We will now proceed by working out the equations of motion for the
fields in~\eqref{LL}. From the equation of motion of $\Phi(x)$, one
obtains the continuity equation
\begin{equation}\label{cont}
  \partial_t \rho(x)= -\partial_x j(x)\;,
\end{equation}
where $j(x) =v(x) K(x) \Pi(x)$.  Furthermore, from the equation of
motion of $\Pi(x)$ we obtain
\begin{equation}\label{j1}
  \partial_t j(x) = -v(x)K(x)\;\partial_x\left(\frac{v(x)}{K(x)}\,\rho(x)\right).
\end{equation}
We shall now consider the effect of switching on a static electric
field by letting $ H\rightarrow H+\tilde{H}$, where :
\begin{equation}\label{hartree}
 \tilde{H}
=e \int dx\, \varphi(x) \rho(x)
=-\frac{2e}{\pi}\int dx\, \varphi(x) \partial_x\Phi(x).
\end{equation}
The underlying assumptions in writing down the model
Eqs.\eqref{LL}+\eqref{hartree} are the following: The potential
$\varphi(x)$ includes both the externally applied potential, as well
as the Hartree potential due to the response in the electronic charge
distribution.  Furthermore, we assume that the {\em difference}
between the true electron-electron interaction and the Hartree
mean-field term included in \Eq{hartree} can be represented by a local
contact interaction as usual in the long wavelength limit of a 1D
conductor.  This determines the position dependent Luttinger parameter
$K(x)$ discussed above.
We now take into account the effect of the potential $\varphi(x)$ on
the expression for the current, Eq.  \eqref{j1}. This equation then
takes on a form which is manifestly that of London's equation of a
superconductor:
\begin{equation}\label{london}
  \partial_t j = -\Lambda(x)^{-1}\;\partial_x\left(\mu(x)+e\varphi(x)\right),
\end{equation}
where $\mu(x)=\frac{\pi}{2}v(x)K(x)^{-1} \rho(x)$ can be identified as
the the local chemical potential in the wire in the absence of an
electric field, and $\Lambda(x)=\pi/(2v(x)K(x))$.  Eq.  \eqref{london}
makes it clear that in a steady state situation, any variation of the
electric potential in the wire has to be counter-balanced by the
chemical potential, in other words we have
\begin{equation}\label{const}
  \mu(x)+e\varphi(x) = \text{const.}
\end{equation}
in the wire.
We emphasize that the quantity in Eq. \eqref{const}
is not directly related to the observed voltage
(which would then vanish).
Instead,
 $\mu(x)$
is just the average of the ``rightmoving'' and
``leftmoving'' chemical potential.
In our ballistic wire, we do not
require that this {\em average} approaches the chemical potential of
the leads. Rather, we require that
\begin{equation}\label{bc}
  \mu_L=\mu^+(x_L)\;,\qquad \mu_R=\mu^-(x_R),
\end{equation}
where $\mu_L$ and $\mu_R$ are the chemical potentials in the left and
in the right lead, respectively, and $\mu^+$ and $\mu^-$ are the
``chemical potentials'' of the right- and leftmoving populations,
which we still need to define (Eq. \eqref{mupm} below).
This situation is depicted in
Fig.~\ref{fig:fermielse}.
\begin{figure}[t]
  \begin{minipage}[c]{0.99\linewidth}
    \includegraphics[width=\linewidth]{landauer_bc}
  \end{minipage}
  \caption{(color online) Illustration of the Landauer-type boundary
    condition~\eqref{bc}. The right (left) moving eigenmodes are in
    chemical equilibrium with the electrons at the left (right) lead.}
  \label{fig:fermielse}
\vspace{0pt}
\end{figure}
The measured voltage is then the difference in the electro-chemical
potential between the leads:
\begin{equation}\label{V}
  V=\varphi_L+\mu_L/e-\varphi_R-\mu_R/e,
\end{equation}
where we have written $\varphi(x_{L,R})\equiv \varphi_{L,R}$.  This
quantity will be finite whenever a finite current is flowing.  Thus,
since no dissipation is assumed to occur within the wire, it is only
through a proper description of the contact between the wire and the
leads, where dissipation occurs, that the current-voltage relation can
be determined unambiguously
\cite{glazman-88jetp238,landauer87zpb,imry86}.  In order to proceed,
we now need to determine $\mu^+(x)$ and $\mu^-(x)$.  To this end, we
decouple the equations of motions, \emph{i.e.,\ } \eqref{cont} and~\eqref{j1}.
(In this part of the calculation, the electric potential is not
important, so we will set it to zero.) We define~\cite{safi-95prb17040}:
\begin{equation}\label{rhopm}
  \rho_{\pm}(x)=\frac 1 2 (\rho(x)\pm j(x)/v(x)),
\end{equation}
which by~\eqref{cont} and~\eqref{j1} yields
\begin{eqnarray}\label{rhopm2}
  \partial_t\rho_\pm &=&\mp \partial_x( v(x)\rho_\pm(x) ) \nonumber \\
 && \pm \frac 1 2 (\partial_x
K(x) ) \frac{v(x)}{K(x)}(\rho_+(x)+\rho_-(x)).
\end{eqnarray}
We see that a complete decoupling is achieved only in parts of the
wire where $K(x)\approx \text{const}$ holds. In this case, we have a
complete decoupling between the quantities $\rho_-$ and $\rho_+$,
which can be interpreted as rightmoving and leftmoving densities,
respectively.  However, whenever we have a non-constant $K(x)$,
rightmovers will be scattered into leftmovers at some finite rate and
vice versa.  As explained above, we assume that $K(x)$ is
approximately constant (and equal to $1$) at the ends of the wire, as
a consequence of the proximity to the metallic leads.  It is only in
this case that we have a truly ``ballistic'' situation of independent
rightmoving and leftmoving populations with separate chemical
potentials, and that the boundary conditions \eqref{bc} at the
contacts are well defined in an obvious way.
\section{DC conductance}
\label{sec:dcc}
Independent of the spatial dependence of $K(x)$, however, we can write
the Hamiltonian \eqref{LL} (or rather, the charge part thereof) as a
sum of a $\rho_+$--part and a $\rho_-$--part:
\begin{equation}\label{Hpm}
  H=\frac \pi 2 \int dx\, \frac{v(x)}{K(x)} \left(\rho_+(x)^2+\rho_-(x)^2\right).
\end{equation}
Note that with the definition \eqref{rhopm}, Eq. \eqref{Hpm} holds
even where $K(x)$ is {\em not} constant, i.e. in regions of the wire
where the decoupling into left- and right-movers does not hold at the
equation of motion level.  Eq. \eqref{Hpm} naturally leads to the
definition of a local rightmoving and leftmoving chemical potential:
\begin{equation}\label{mupm}
  \mu^\pm(x)=\frac{\delta H}{\delta \rho_{\pm}(x)}=\pi
\frac{v(x)}{K(x)}\,\rho_{\pm}(x).
\end{equation}
As anticipated, we have $\mu(x)=\frac 1 2 (\mu^+(x)+\mu^-(x))$. We are
now in a position to address the main question: Are the boundary
conditions \eqref{bc} sufficient to uniquely specify the
voltage--current relationship? We start with \eqref{const} in the
following form:
\begin{equation}
  \mu(x_L)+e\varphi_L=\mu(x_R)+e\varphi_R.
  \label{acpot}
\end{equation}
Expressing $\mu$ through $\mu^+$ and $\mu^-$, we can bring this
equation into the form:
\begin{equation}
\begin{split}
&\mu^+(x_L)/e+\varphi_L-\mu^-(x_R)/e-\varphi_R=\\
&\frac 1 2 \left(\mu^+(x_L)-\mu^-(x_L)+\mu^+(x_R)-\mu^-(x_R)\right)/e.
\label{spplit}
\end{split}
\end{equation}
From~\eqref{bc} and~\eqref{V}, the left hand side is just the voltage
$V$ across the wire. As for the right hand side, we note that using
Eqs. \eqref{rhopm} and \eqref{mupm}, we have $j(x)=\pi^{-1}
K(x)(\mu^+(x)-\mu^-(x))$.  Given that $j(x)\equiv I/e$ is constant in
a steady situation, where $I$ is the electric current, the last
equation becomes
\begin{equation}\label{V2}
  V=\frac \pi {2e^2} (\frac{1}{K_L}+\frac{1}{K_R})\,I.
\end{equation}
Here, $K_{L,R}$ are the values of $K(x)$ at the contacts. From
different approaches, this expression has been obtained before in the
literature. As such, our framework also allows to describe generalized
reservoirs in~\eqref{V2} that are either of Fermi liquid or Luttinger
liquid type (encoded in $K_L$ and $K_R$), given that the proximity
boundary conditions are still justified. If we put back our original
assumption \Eq{bc0} that $K(x)$ approaches its ``Fermi liquid'' value
$1$ near the leads, i.e. $K_L=K_R=1$, we obtain the conductance
$G=I/V=\frac{e^2}{\pi}$. Or, putting back $\hbar$, which we had thus
far set equal to $1$:
\begin{equation}
\label{refgun}
  G=\frac{2 e^2}{2\pi\hbar}=2\frac{e^2}{h}.
\end{equation}
Hence, the conductance is quantized {\em independent} of the
interactions in the bulk of the wire, but {\em depending} on the
validity of the boundary conditions \eqref{bc0} and \eqref{bc} only.

An alternative derivation of this result using equations of motion has
been given by Safi in Ref. \onlinecite{safi97}, where the equations of
motion are argued to determine a particle transmission coefficient $T$.
In our case, the use of
the boundary conditions \eqref{bc} allows us to completely avoid the
concept of single particle transmission. The same boundary conditions
are also used later by Safi in Ref.  \onlinecite{safi99ejp451}. Unlike
there, however, a complete decoupling of the dynamics of left- and
rightmoving degrees of freedom is not possible in the present context,
as evidenced by \Eq{rhopm2}.  Even earlier, \Eq{bc} has been employed
in Ref. \onlinecite{oreg-96prb14265} in an argument that is identical
to ours in the non-interacting case, but not otherwise.  We will
further comment on the relation of our work to
Ref. \onlinecite{oreg-96prb14265} below.

\section{AC conductance}
\label{sec:acc}

Aspects of the
dynamical response of Luttinger liquids and one-dimensional conductors in general have been frequently addressed
before~\cite{pretre}. In the time-reversal invariant case, the AC-conductance for
special interaction profiles has been studied by
Safi and Schulz\cite{safi-95prb17040}, and Safi\cite{safi97annal}.
Quantum
Hall
edges have been considered by Oreg and Finkel'stein\cite{yuval95}.
Here we derive a general result for the  time-reversal invariant case with arbitrary
interaction profile.
To this end, we will restrict
ourselves to frequencies much less than $v/L$, where $v$ is the
typical mode velocity.  Using Eq.~\ref{london}, in the presence of a
time-dependent current, Eq.~\ref{acpot} is modified according to
\begin{equation}
\mu(x_L) + e \varphi(x_L) - \mu(x_R) + e\varphi(x_R)= \int_{-L/2}^{L/2} dx\, \partial j_t(x) \Lambda(x).
\label{acmu}
\end{equation}
We assume the general mode expansion of the particle current
to be of the form
\begin{equation}
j_\omega(x,t) = e^{-i \omega t} \sum_k j_k(\omega) e^{ikx},
\label{mode}
\end{equation}
where the mode coefficients $j_k(\omega)$ are assumed to be analytic
functions of $\omega$. We know that in the DC limit, the current does
not depend on $x$, i.e.  $j_{k}(\omega)\vert_{\omega = 0} \propto
\delta_{k,0}$. This implies
\begin{equation}
 j_{k}(\omega) = \mathcal{O} (\omega)\;\;\; \forall k \neq 0.
 \label{finiteo}
 \end{equation}
 With these assumptions, we plug Eq.~\ref{mode} into
 Eq.~\ref{acmu}. We keep only the leading term in $\omega$, which only
 depends on $j_{k=0} (\omega) \equiv \bar{I}(\omega)/e$. With this,
 Eq.~\ref{acmu} yields an AC generalization of Eq.\ref{spplit}:
\begin{eqnarray}\label{ACbalance}
&& \mu^+(x_L)/e + \varphi_L-\mu^-(x_R)/e -\varphi_R = -\frac{i \omega \pi \tau}{ e^2}\bar{I}(\omega) e^{-i\omega t}\nonumber \\
&&+\frac{1}{2 e}\left(   \mu^+(x_L) - \mu^-(x_L) + \mu^+(x_R) - \mu^-(x_R)\right), \nonumber \\
\end{eqnarray}
where the parameter $\tau$ is given by
\begin{equation}
\tau=1/\pi \int_{-L/2}^{L/2} dx \Lambda(x),
\label{tau}
\end{equation}
with $\Lambda(x)$ as defined in~\eqref{london}.  As before, the
expression in the second line of Eq. \eqref{ACbalance} equals
$(I_R+I_L)\pi/2e^2$, where $I_R$ and $I_L$ denote the current in the
left and right lead, respectively. Due to possible capacitive effects,
we no longer assume $I_R = I_L$, but \emph{define}
$I(\omega)e^{-i\omega t} = (I_R(t)+ I_L(t))/2$.  From
Eq.~\ref{finiteo}, it is clear that $\bar{I}(\omega) = I(\omega) +
\mathcal{O}(\omega)$.  Identifying again the LHS of Eq.
\eqref{ACbalance} with $Ve^{-i\omega t}$, we finally find the
following relation between $V$ and $I(\omega)$:

\begin{equation}
I(\omega) = V \frac{2e^2 /h}{1-i\omega \tau}.
\label{AC}
\end{equation}
Strictly speaking, this relation is obtained only for the small
$\omega$ limit. However, we prefer to keep it in the above form which
is manifestly Drude's law. An interesting property of this low
frequency limit is that while non-universal corrections enter through
the parameter $\tau$ given by Eq. \eqref{tau}, details such as the
functional relation between the potential $\varphi(x)$ and the charge
distribution on the wire do not enter. These details would be
sensitive to aspects such as wire geometry, and could in principle
affect higher order corrections. In contrast, $\tau$ is solely
determined by the value of the Luttinger parameter within the wire.
This would allow one to compare $\tau$ as determined by AC
measurements to other means of extracting the Luttinger parameter.

\section{Discussion}
\label{sec:dis}
The derivation given here emphasizes that two ingredients are
sufficient to observe the quantized conductance in interacting quantum
wires. The ``proximity'' boundary condition \Eq{bc0} implies that the
interactions in the quantum wire become gradually weaker as the Fermi
liquid leads are approached, as originally emphasized in
Refs.~\onlinecite{maslov-95prb5539, safi-95prb17040}.  The
``chemical'' boundary conditions \Eq{bc} imply that the rightmoving
(leftmoving) electrons of the wire are in chemical equilibrium with
electrons of the left (right) reflectionless contact.  We note that if
we relax condition \eqref{bc0} by allowing $K(x_R)$ and $K(x_L)$ to
take on a common value $K\neq 1$ (and in particular if
$K(x)=K=\text{const.}$), \Eq{V2} yields the Kane-Fisher
result~\cite{kane-92prl1220}, according to which the conductance is
renormalized via $G=2Ke^2/h$.

We note that the physical assumptions leading to the quantized DC
conductance in our framework are compatible with those made by a
number of other workers (e.g. Refs.~\onlinecite{safi-95prb17040,
  maslov-95prb5539,
  alekseev-98prl3503,lehur-08ap3037,lebedev-05prb075416}). On the
other hand, alternative views are also found in the literature.
Notably, in Ref.~\onlinecite{oreg-96prb14265}, a quantized conductance
of $G=e^2/h$ is found for interacting spinless 1D fermions, within an
approach which does not directly involve boundary conditions of the
type \Eq{bc0}.  Instead, following Ref.
\onlinecite{kawabata95jpsj30}, it is argued
that the quantized value is always obtained when electron-electron
interaction effects are self-consistently included into the
definition of the total electric field.
We will now clarify the relation of
this approach to ours (see Ref. \onlinecite{safi97} for a different discussion).
 We first observe that such interaction effects
also enter the definition of our total voltage, \Eq{V}.  Here, the
Hartree mean field part of the interactions is included in the
potential $\varphi(x)$, while correlation effects are included in the
definition of the right- and left-moving local chemical potential,
\Eq{mupm}. In addition, however, these chemical potentials do not
depend on interactions alone, but equally depend on the kinetic part
of the Hamiltonian, through their dependence on the local density.
This is in accordance with the general notion that in gapless 1D
systems, kinetic energy and interactions should be treated on equal
footings, as is manifest by their form in the Luttinger Hamiltonian.

The difference between the approach discussed here and those
emphasizing self-consistent fields over boundary conditions of the
form \Eq{bc0} can be understood as follows.  It is useful to discuss
both spinless and spinful fermions for a {\em homogeneous} wire with
$K(x)=K=const$.

In the spinful case, our result reduces to that of Kane and Fisher,
$G=2Ke^2/h$.  This value coincides with that for non-interacting
spinless fermions, $e^2/h$, if $K$ equals $1/2$.  This again is
entirely consistent with the well-known fact that for $K=1/2$, the
charge part of \Eq{LL} is identical to the Hamiltonian of
non-interacting spinless fermions. (This remains true in the presence
of an electromagnetic field.)  Thus, in the absence of boundary
conditions, at least naively nothing should distinguish the two cases.  If, on the
other hand, one concludes that the conductance is unrenormalized, even
in the homogeneous wire case with unspecified boundary conditions, one
would obtain $G=2e^2/h$ for spinful fermions even at $K=1/2$, whereas
$G=e^2/h$ for non-interacting spinless fermions.
This may  seem somewhat
at odds with the fact that the charge part of the Hamiltonian is
exactly the same in both cases.
The only difference in these two
cases is the decomposition of the Hamiltonian into distinct kinetic
and interacting parts.  In the present approach, where these parts
are treated on equal footing throughout, a difference in conductance
may only arise through different boundary conditions.
Indeed, with our conventions, these boundary conditions must read
$K_L=K_R=1/2$ if the system, including leads, consists of spinless
fermions.
On the other hand, a major difference between our treatment
and that in Refs. \onlinecite{kawabata95jpsj30, oreg-96prb14265}
lies in the definition of the measured voltage.
While we include a chemical potential drop between the leads,
which is related to a chemical potential difference between rightmovers
and leftmovers by \Eq{bc}, in Refs. \onlinecite{kawabata95jpsj30, oreg-96prb14265}
 the voltage
is just the integral over the total electric field.
This may be a more natural definition
in a truly homogeneous wire without boundaries.
The chemical potential difference included into our voltage
is always finite (for the homogeneous wire) in the presence of a finite current.
This explains why in the homogeneous wire case, our result for the conductance
(renormalized) will in general differ from that of Refs. \onlinecite{kawabata95jpsj30, oreg-96prb14265} (unrenormalized).
We do believe that in most experiments conducted so far, where a voltage is
measured between two leads and Landauers basic ideas apply,
the voltage definition used here is the appropriate one, and
the observed conductance is thus entirely determined
by boundary conditions of the kind used here.
We note, however, that experimental setups have been proposed~\cite{maslov04}
which circumvent the application of leads altogether by measuring
alternating currents induced in the wire by a resonator.
The approach of Refs. \onlinecite{kawabata95jpsj30, oreg-96prb14265}
may indeed be better suited to describe this case.
Such
experiments would hence be very useful to distinguish the picture developed
here and elsewhere from others.

We finally remark that although we did not develop a detailed scenario
for possible deviations from the quantized value of the conductance
\cite{matveev04prb245319,danneau-08prl016403,thomas-96prl135}, we are
confident that the explanation for such deviations may be addressed
similarly if proper boundary conditions describing these cases are
identified.
One interesting possibility is that a spin gap exists in the entire wire including the contacts.
In this case one may
conjecture that the spinless case effectively applies with an
effective charge $e^\ast=2e$ (Ref. \onlinecite{KYpriv}, c.f. also
Ref.~\onlinecite{seidel-05prb045113}).  However, the interactions can
then not be considered weak at the boundary, and we defer a detailed
analysis to further studies.
Moreover, we conjecture that our treatment of the AC
conductance largely carries over to these cases. Since deviations from
the $\omega\rightarrow 0$ limit are expressible solely through bulk
properties of the wire, the natural modification of Eq. \eqref{AC}
would be to simply replace the numerator by the respective
non-universal value of the DC conductance.  The Drude-type expression
\eqref{AC} we find in the AC case may also suggest that this
expression could be obtained in an "RPA-like" approximation. If so,
this may suggest that the validity of this expression can be extended
beyond the $\omega\ll v/L$ regime.  We note, however, that $v/L$
typically well exceeds the gigahertz range under relevant conditions,
and so the restriction $\omega\ll v/L$ should not pose a stringent
limitation for current experiments.

\section{Conclusion}
In this article we addressed the fundamental nature of quantized
conductance in one-dimensional quantum wires.
Working in a framework based on the analysis of the equations of
motion of an inhomogeneous interacting Luttinger liquid, we argued for
the significance of two types of boundary conditions as the underlying
cause for a quantized conductance.  The formalism presented here
simplifies some earlier treatments without loss of essential physics, and
should serve as a useful starting point also when discussing possible
deviations from the ideal quantized conductance.  We have further
studied AC deviations from the universal conductance, and found that
these are expressible through general Luttinger parameters in a simple
manner in the low frequency limit.

\begin{acknowledgments}
  We are indebted to D.-H. Lee and Y.-C. Kao for bringing this problem
  to our attention.

  RT acknowledges insightful discussions with D.~B.~Gutman, as well as
  I.~Affleck and C.-Y.~Hou at the LXXXIX Les Houches Summer School.
  AS would like to thank K.  Yang  for stimulating
  discussions, and M. Grayson for insightful comments and a careful reading of an earlier version of this
  manuscript. We also thank I. Safi for very helpful remarks.  RT is supported by a Feodor Lynen fellowship of the
  Humboldt Foundation and by the Sloan Foundation.  AS is supported by
  the National Science Foundation under NSF Grant No. DMR-0907793.
\end{acknowledgments}