Abstract
We present a renormalization group construction of a weakly interacting Bose gas at zero temperature in the two-dimensional continuum, both in the quantum critical regime and in the presence of a condensate fraction. The construction is performed within a rigorous renormalization group scheme, borrowed from the methods of constructive field theory, which allows us to derive explicit bounds on all the orders of renormalized perturbation theory. Our scheme allows us to construct the theory of the quantum critical point completely, both in the ultraviolet and in the infrared regimes, thus extending previous heuristic approaches to this phase. For the condensate phase, we solve completely the ultraviolet problem and we investigate in detail the infrared region, up to length scales of the order \((\lambda ^3\rho _0)^{-1/2}\) (here \(\lambda \) is the interaction strength and \(\rho _0\) the condensate density), which is the largest length scale at which the problem is perturbative in nature. We exhibit violations to the formal Ward Identities, due to the momentum cutoff used to regularize the theory, which suggest that previous proposals about the existence of a non-perturbative non-trivial fixed point for the infrared flow should be reconsidered.
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Notes
It would be very interesting to see whether the ideas of Balaban et al. [4] could be applied to the actual construction of this theory, which is much easier than the condensed one, and still very relevant for the physics of the Mott transition.
With this choice, the dimensions of the physical quantities speed (\(c\)), momentum (\({\vec {k}}\)), frequency (\(k_0\)) and energy (\(E\)) are respectively: \( [c] = [|{\vec {k}}|] = [L]^{-1}\) and \([E] =[k_0]= [L]^{-2}\).
If some of the time coordinates are equal to each other, each group of operators with the same times is reordered in such a way that the creation operators are all on the left of the annihilation operators, within that group.
For some of the bounds in the following, it is useful to assume that \(\chi \) is a function of \(t^2\) only, i.e., \(\chi (t)=\mathcal K(t^2)\) for some smooth \(\mathcal K\), and we shall do so from now on.
In order to prove (2.49), it is enough to rewrite each factor \(x_{2,0}-x_{1,0}\), etc., as a sum of differences \(x_{\ell -,0}-x_{\ell +,0}\) along the spanning tree \(\cup _{w>v}T_w\) (see the discussion preceding (2.31)), and to recognize that each term \(x_{\ell -,0}-x_{\ell +,0}\) goes together with the corresponding propagator \(G_{h_w}(\mathbf{x}_{\ell -}-\mathbf{x}_{\ell +})\); therefore, in the analogue of (2.31), the \(L_1\) norm of \(G_{h_w}(\mathbf{x}_{\ell -}-\mathbf{x}_{\ell +})\) is replaced by the one of \((x_{\ell -,0}-x_{\ell +,0})G_{h_w}(\mathbf{x}_{\ell -}-\mathbf{x}_{\ell +})\), which is \((\mathrm{const.})\gamma ^{-h_{w}}||G_{h_w}||_1\le (\mathrm{const.})\gamma ^{-h_{v}}||G_{h_w}||_1\). Similar considerations are valid with \(x_{2,0}-x_{1,0}\) replaced by \(x_{2,1}-x_{1,1}\), etc.
Actually, it could also happen that some of the fields in \(P_{v}\) are also in \(P_{v_0}\), in which case the corresponding propagators should be replaced by external fields. For simplicity, we exclude this possibility here.
A complete proof of (2.54) requires the discussion of a few other technical points, including the change of variables from the original to the interpolated variables, as well as the possible accumulation of the derivatives coming from the interpolation procedure on a given propagator (which may a priori worsen the combinatorial factors in (2.54)). For these and other closely related issues we refer the reader to previous literature, see in particular [10, Sect. 3].
The vanishing of the non-trivial diagrams with \(|P_v| = 2\) and \((n_{2,v},n_{3,v},\tilde{n}_{r,v})=(1,0,0)\) is valid if the 2-legged vertices are particle-conserving, i.e., if they are proportional to \(\int d\mathbf{x}|\psi _\mathbf{x}|^2x\), as in our case, see (3.11). However, we will see in a moment that non-particle-conserving 2-legged vertices, proportional to \(\int d\mathbf{x}(\psi ^+_\mathbf{x}\psi ^+_\mathbf{x}+h.c.)\), are generated by the iterative integration, as soon as we reach scales \({\bar{h}}\le h\le 0\). Therefore, a posteriori we will also have non-vanishing contributions with \(|P_v| = 2\) and \((n_{2,v},n_{3,v},\tilde{n}_{r,v})=(1,0,0)\), with the 2-legged vertex that is necessarily of type \(\int d\mathbf{x}(\psi ^+_\mathbf{x}\psi ^+_\mathbf{x}+h.c.)\).
A special but important case realizes when the first non-trivial truncation of the beta function displays a cancellation, which makes the beta function zero at that order: in such a case we need to go to higher orders in order to see whether the beta function is truly zero or not. Remarkable examples where this happens are models of spinless fermions in one dimension [11, 13, 26], for which it is possible to prove that the beta function is zero at all orders, by making use of remarkable, very subtle, cancellations at all orders in perturbation theory, following from the Schwinger–Dyson equation combined with local Ward Identities, see [11].
Here by super-renormalizability of the ultraviolet theory we mean that all the interactions are effectively irrelevant: even those that are superficially marginal have dimensional gains in the ultraviolet that make them exponentially insensitive to the ultraviolet cutoff as \(N\rightarrow \infty \).
In this section, for simplicity, we formally write all the involved expressions in the limit \(\beta ,L\rightarrow \infty \), as we already did in the bulk of the paper. It is implicit in the discussion that all the involved quantities have a finite temperature/volume counterparts, which can be easily written down, but are slightly more cumbersome than their formal \(\beta ,L\rightarrow \infty \) limits (this is the only reason why we prefer not to write them explicitly).
We will see in Appendix 2 below that the local parts of the kernels of \({\mathcal {W}}^{(h-1)}\) coincide with those of \({\mathcal {V}}^{(h-1)}+\sum _{k=h}^{{\bar{h}}}{\mathcal {L}}_Q{\mathcal {V}}^{(k)}\) up to a minor correction due to the “last integration step”, which is not visible at the level of the one-loop beta function, see (8.54) and the preceding discussion.
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Acknowledgments
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme ERC Starting Grant CoMBoS (Grant Agreement No. 239694). We thank Giuseppe Benfatto and Vieri Mastropietro for several discussions and illuminating comments. S.C. acknowledges the Hausdorff Center for Mathematics in Bonn for financial support.
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To the memory of Kenneth Wilson, for his inspiring and influential ideas.
Appendices
Appendix 1: Ultraviolet Flow in the Condensate Phase
In this section we prove Eqs.(3.29)–(3.30). We denote by \(\bar{z}_h\) and \(\delta \nu _h\) the quantities defined via the second and third lines of (3.27), both for \(h\ge 0\) and \({\bar{h}}\le h<0\). The graphs contributing to \(\delta \nu _h\) are all and only ladder diagrams as in the first line of Fig. 12, for all \(h\ge {\bar{h}}\); those contributing to \(\bar{z}_h\) are all and only ladder diagrams as in the second line of Fig. 12, if \(h\ge 0\), while they can be either as in the second line of Fig. 12 or as in Fig. 13, if \({\bar{h}}\le h<0\). For both couplings, we distinguish the regime \(h\ge 0\) from \({\bar{h}}\le h<0\).
Let us first consider \(\delta \bar{\nu }_h\). If \(h\ge 0\) we proceed as in the “ultraviolet regime” subsection of Sect. 2.4, thus obtaining a bound on the \(n\)-th order diagrams of the same form as the \(n\)-th order term in the right side of (2.61). Since \(n\ge 2\), we find that \(|\delta \bar{\nu }_h|\le C \lambda ^2\gamma ^{-h}\) for all \(h\ge 0\). For \(h<0\) we write the beta function equation for \(\delta \bar{\nu }_h\), analogous to (2.63), with initial datum \(\delta \bar{\nu }_0\) of the order \(\lambda ^2\): \(\delta \bar{\nu }_{h-1}=\delta \bar{\nu }_h+\beta _h^{\delta \bar{\nu }}\), with \(h\le 0\). Using the (renormalized analogoue of) (3.20), we see that \(|\beta _h^{\delta \bar{\nu }}|\le C\lambda ^2\), \(\forall {\bar{h}}\le h\le 0\), which implies that \(|\delta \bar{\nu }_h|\le C\lambda ^2|h|\), \(\forall {\bar{h}}\le h\le 0\), as desired.
We now apply the same strategy to the study of the flow of \(\bar{z}_h\). If \(h\ge 0\), we find that the diagrams of order \(n\) are bounded by \(\lambda (K\lambda )^{n-1}\gamma ^{{\bar{h}}-h}\), with \(n\ge 1\); here the factor \(\gamma ^{{\bar{h}}-h}\) is a dimensional estimate on the \(L_\infty \) norm of the off diagonal propagator \(r_h\). Therefore, \(|\bar{z}_h|\le (\mathrm{const.})\lambda \gamma ^{{\bar{h}}-h}\), for all \(h\ge 0\). In particular, \(\bar{z}_0\) is of order \(\lambda ^2\). From smaller scales, we study again the beta function equation \(\bar{z}_{h-1}=\bar{z}_h+\beta _h^{\bar{z}}\), where \(|\beta _h^{\bar{z}}|\le C\lambda ^2\), for all \({\bar{h}}\le h\le 0\) (see Fig. 16 for a graphical representation of the first non trivial contribution to \(\beta _h^{\bar{z}}\)). Therefore, as for \(\delta \bar{\nu }_h\), we find \(|\bar{z}_h|\le C\lambda ^2|h|\), \(\forall {\bar{h}}\le h\le 0\), which concludes the proof of (3.29)–(3.30).
Appendix 2: Bounds on the Propagators
In this section we prove the decay bound (3.39). We first focus on \({\tilde{g}}^{(1)}_{\bar{h}}\). Of course, it is enough that we restrict to its \(-+\) component, which we rewrite as the sum of two terms, whose definition is induced by the following rewriting of the cutoff function appearing under the integral sign in (3.42):
where
The corresponding decomposition for the \(-+\) component of \(\tilde{g}^{(1)}_{\bar{h}}\) is \([{\tilde{g}}^{(1)}_{{\bar{h}}}(\mathbf{x})]_{-+}= g_{\bar{h}}^{(1a)}(\mathbf{x}) - g_{\bar{h}}^{(1b)}(\mathbf{x})\). We now show that the two terms separately satisfy the same bound as (3.39). The easiest term to treat is \(g_{\bar{h}}^{(1b)}(\mathbf{x})\), which we write as
Using the compact support properties of \({\tilde{\chi }}_{\bar{h}}(\mathbf{k})\), it is immediate to check that
which implies (3.39) for \(n_0=|{\vec {n}}|=0\). In order to estimate the derivatives of \({\tilde{g}}_{{\bar{h}}}^{(1b)}(\mathbf{x})\), note that each derivative \(\partial _{x_0}\) (resp. \(\partial _{x_i}\) with \(i=1,2\)) acting on \({\mathfrak {h}}_{\bar{h}}(\mathbf{x})\) produces a factor proportional to \(\gamma ^{{\bar{h}}}\) (resp. \(\gamma ^{{\bar{h}}/2}\)), as desired. Moreover, each derivative \(\partial _{x_0}\) (resp. \(\partial _{x_i}\) with \(i=1,2\)) acting on the integral in the right side of (6.3) produces a factor \(-ik_0\) (resp. \(-ik_i\)) under the integral sign, which is bounded proportionaly to \(\gamma ^{{\bar{h}}}\) (resp. \(\gamma ^{{\bar{h}}/2}\)), thanks to the compact support properties of \(\tilde{\chi }_{\bar{h}}\). This concludes the proof that \(\tilde{g}^{(1b)}_{\bar{h}}\) satisfies a bound like (3.39).
Let us now focus on \(\tilde{g}_{\bar{h}}^{(1a)}(\mathbf{x})\). In order to bound it, we rewrite the cutoff function \((1-{\tilde{\chi }}_{\bar{h}}(\mathbf{k}))\) appearing in its definition as \(\sum _{h> {\bar{h}}} {\mathfrak {f}}_h(\mathbf{k})\), with \({\mathfrak {f}}_h(\mathbf{k})\) defined after (3.17) and note that, thanks to compact support properties of \({\mathfrak {f}}_h\),
Moreover, integrating by parts and using again the compact support properties of \({\mathfrak {f}}_h\), we find
and, similarly,
Combining the previous three equations we find
which leads to
Now, on the support of \((1-{\mathfrak {h}}_{\bar{h}}(\mathbf{x}))\), the combination \((|x_0|+|{\vec {x}}|^2)\) is larger than (const.)\(\gamma ^{{\bar{h}}}\) and, therefore, for all \(N_1,N_2\ge 0\) such that \(N_1+N_2=N\),
which is summable over \(h\) as soon as \(N_1>n_0+|{\vec {n}}|/2\). By picking such an \(N_1\) we obtain the desired bound on \(\tilde{g}^{(1a)}_{\bar{h}}\).
The proof of the desired decay bound for \(\tilde{g}^{(2)}_{\bar{h}}\) is completely analogous to the proof of (6.8) and, therefore, we do not give its details here. The same is true for the proof of (3.43), which we also leave to the reader.
Appendix 3: Lowest Order Computations
In this section we prove (3.81)–(3.83) and (3.89)–(3.92), i.e., we explicitly compute the beta functions for \(\lambda _h\), \(\mu _h\), \(\nu _h\), \(Z_h\), \(E_h\), \(B_h\) and \(A_h\) at the lowest non-trivial order (which turns out to coincide with the one-loop computation). Rather than presenting the computations in the same order as we presented the formulas after (3.81), we proceed in order of increasing difficulty, starting from the easiest computation, which is the beta function of \(Z_h\), then moving to the beta function of \(\nu _h\), etc.
In the following, in the computation of the one-loop contributions to the beta function, we systematically perform a number of approximations, which either induce corrections of higher order than the one we are computing, or are dimensionally irrelevant for \(h\le {\bar{h}}\). The approximations we do are the following: (1) we systematically replace \(\gamma ^{h+1}\lambda _{h+1}\) by \(\gamma ^{h}\lambda _h\), \(\gamma ^{(h+1)/2}\mu _{h+1}\) by \(\gamma ^{h/2}\mu _h\), \(\gamma ^{2(h+1)}\nu _{h+1}\) by \(\gamma ^{2h}\nu _h\), \(A_{h+1}\) by \(A_h\), \(B_{h+1}\) by \(B_h\), \(E_{h+1}\) by \(E_h\), and \(Z_{h+1}\) by \(Z_h\), since these replacements induce an error that is of higher order in \(\lambda \); (2) similarly, we replace the renormalization functions \(\tilde{A}_{h-1}(\mathbf{k})\), etc, appearing in the definition of the propagator on scale \(h\) by \(A_h\), etc; moreover, recalling the definition of \(\chi _h(\mathbf{k})\) in (3.51), we replace the support function \(\tilde{f}_h(\mathbf{k})=\chi _h(\mathbf{k})-\chi _{h-1}(\mathbf{k})\) by \({\mathfrak {f}}_h(|\mathbf{k}'|):=\chi (\gamma ^{-h}|\mathbf{k}'|)-\chi (\gamma ^{-h+1}|\mathbf{k}'|)\), where with some abuse of notation we let \(\mathbf{k}':=(k_0,\sqrt{A_h/C_h}\,{\vec {k}})\equiv (k_0,{\vec {k}}')\); note that also these replacements induce errors of higher order in \(\lambda \); (3) finally, we systematically neglect the terms coming from trees with at least one endpoint on scale \({\bar{h}}\), since these terms have relative size \(\gamma ^{\theta (h-{\bar{h}})}\) with \(0<\theta <1\) as compared to the main contributions to the beta function (see (3.75)).
1.1 Appendix 3.1: Lowest Order Beta Function for \(Z_h\)
The lowest order contribution to \(\beta _{h}^{Z}\) (see (3.80) and (3.63)) is of order \(\lambda _h\) (or, equivalently, of order \(\mu _h^2\)) and can be represented graphically by diagrams of the form in Fig. 17, computed at zero external momentum, with the two propagators labelled by two scale labels \(h_1,h_2\) such that \(\min \{h_1,h_2\}=h\). Note that by the compact support properties of the propagator \(|h_1-h_2|\le 1\). The sum of the values of these diagrams is
where
where \({\tilde{Z}}_{h-1}(\mathbf{k})\tilde{C}_{h-1}(\mathbf{k})={\tilde{Z}}_{h-1}(\mathbf{k})\tilde{B}_{h-1}(\mathbf{k})+\tilde{E}_{h-1}^2(\mathbf{k})\). Under the approximations explained above, this propagator can be replaced by
For future reference, let us write down the analogues of this equation, as far as the propagators with labels \(lt\) and \(ll\) are concerned:
Using these replacements, as well as the ones spelled above, we find that (7.1) is equal (up to higher order corrections) to
so that after rescaling and after the change of variables \(k_0=\rho \cos \vartheta \), \(|{\vec {k}}|=\rho \sin \vartheta \), with \(\rho \ge 0\) and \(\vartheta \in [0, \pi ]\), we get
which is the same as (3.89).
1.2 Appendix 3.2: Lowest Order Beta Function for \(\nu _h\)
The lowest order contributions to \(\beta _h^\nu \) (see (3.78) and (3.59)) are of the order \(\lambda _h\) and can be represented graphically as in Fig. 18. Once again, the propagators are associated with scale labels such that the minimum scale is equal to \(h\). We denote by \(\beta ^{\nu ,(1)}_h\) the contribution from the first diagram in Fig. 18, and by \(\beta ^{\nu ,(2)}_h\) those from the last two diagrams. Proceeding as in the previous section we find that, up to higher order corrections,
In order to calculate \(\beta ^{\nu ,(2)}_h\) we notice that the sum of the remaining two diagrams in Fig. 18 involves the following combinations of propagators (after the usual replacements):
where \(h.o.\) in the last equation means “higher orders”. Using these identities we find, up to higher order corrections:
Putting things together:
Using the definitions (3.84)–(3.88), this can be rewritten as
which implies (3.83).
1.3 Appendix 3.3: Lowest Order Beta Function for \(\mu _h\)
The lowest order contributions to \(\beta _h^\mu \) are of the order \(\mu _h\lambda _h\) and can be represented graphically as in Fig. 19. Once again, the propagators are associated with scale labels such that the minimum scale is equal to \(h\). We denote by \(\beta _h^{\mu ,(2) }\) (resp. \(\beta _h^{\mu ,(3)}\)) the contribution to \(\beta _h^{\mu }\) coming from the first diagram (resp. second \(+\) third \(+\) fourth diagrams) in Fig. 19. We find:
and, using the analogues of (7.8)–(7.9),
In the last expression, we can rewrite
where \(\beta _0^{(2,\chi )}\) is defined as in (3.87), so that, putting things together,
which implies (3.82)
1.4 Appendix 3.4: Lowest Order Beta Function for \(\lambda _h\)
The lowest order contributions to \(\beta _h^\lambda \) are of the order \(\lambda _h^2\) and can be represented graphically as in Fig. 20. We denote with \(\beta ^{\lambda ,(2)}_h\), \(\beta ^{\lambda , (3)}_h\) and \(\beta ^{\lambda , (4)}_h\) the contributions to \(\beta _h^\lambda \) coming from the diagrams in Fig. 20 with two, three and four end-points, respectively. Proceeding as in the previous sections, we find:
and
where
Using the definitions (3.85)–(3.88), it can be checked that
so that, putting things together,
which implies (3.81).
1.5 Appendix 3.5: Lowest Order Beta Function for \(E_h\)
The lowest order contribution to \(\beta ^E_h\) (see (3.80) and (3.62)) is of order \(\lambda _h\) and is represented graphically in Fig. 21. The value of the graph is (up to higher order corrections):
with
where \(T_2(\mathbf{k}):= {\mathfrak {f}}_0(|\mathbf{k}|) + {\mathfrak {f}}_1(|\mathbf{k}|)\), \(T_1(\mathbf{k}):={\mathfrak {f}}_1(|\mathbf{k}|)\). By integrating by parts:
so that
which gives (3.90).
1.6 Appendix 3.6: Lowest Order Beta Function for \(B_h\)
The lowest order contribution to \(\beta ^B_h\) (see (3.79) and (3.61)) is of the order \(\lambda _h\) and is represented graphically in Fig. 22. The sum of the values of the two graphs is (up to higher order corrections):
where, defining \(\alpha _h=\frac{E_h^2}{Z_h C_h}\), and letting \(T_1,T_2\) be the same as in the previous section,
By computing explicitly the derivative, and then integrating by parts, we find:
Let us now denote by \(\beta ^{B,(1)}_{i,h}\) the contribution associated with the first term in square brackets, and by \(\beta ^{B,(2)}_{i,h}\) the rest. We have:
which gives the first term in the r.h.s. of (3.91). Similarly, by passing to polar coordinates \(k_0=\rho \cos \vartheta \), \(|{\vec {k}}|=\rho \sin \vartheta \), with \(\vartheta \in [0,\pi ]\), and by explicitly computing the integral over \(\vartheta \), we find:
which gives the second term in the r.h.s. of (3.91).
1.7 Appendix 3.7: Lowest Order Beta Function for \(A_h\)
The lowest order contribution to \(\beta ^A_h\) is represented graphically in a way similar to Fig. 22, with \(\partial _{p_0}^2\) replaced by \(\partial _{p_1}^2\). Therefore, \(\beta ^A_h = \beta ^A_{h,2} - \beta ^A_{h,1}\), with
On the other hand, the integral in the r.h.s. is invariant under the exchange \(k_0 \longleftrightarrow k_1\) and, therefore, using the result of the previous section,
which proves (3.92).
Appendix 4: Ward Identities
1.1 Appendix 4.1: Derivation of the Global Ward Identities
In this section we derive the Ward Identities associated with a phase transformation \(\psi ^\pm _\mathbf{x}\rightarrow e^{\pm i\alpha (\mathbf{x})}\psi ^\pm _\mathbf{x}\), with \(\alpha (\mathbf{x})\equiv \alpha \). Next, in the following section, we will show how to modify the computation in order to deal with a local phase transformation in which \(\alpha (\mathbf{x})\) is a non-trivial function of \(\mathbf{x}\). We introduce a sequence of reference models, labelled by an integer \(h\), defined in terms of a functional integral similar to (3.18), but with an extra infrared cutoff at a pre-fixed scale \(h\) (that, for definiteness, we shall assume to be \(\le {\bar{h}}\)):
Here \(P_{\ge h}^B(d\psi )\) is the complex gaussian measure with propagator given by the analogue of (3.13), modulo an extra infrared cutoffFootnote 14:
and \(\hat{\chi }_{\ge h}(\mathbf{k})=1-\chi _{h-1}(\mathbf{k})\), where \(\chi _{h-1}(\mathbf{k})\) was defined in (3.51). Moreover, \(\bar{V}(\psi )\) is given by (3.11) (we dropped the label \(\Lambda \), for simplicity). The interest of the definition (8.1) is that the local parts of the kernels of \({\mathcal {W}}^{(h-1)}\), with \(h\le {\bar{h}}\), are essentiallyFootnote 15 the same as the local part of the kernels of \({\mathcal {V}}^{(h-1)}+\sum _{k=h}^{{\bar{h}}}{\mathcal {L}}_Q{\mathcal {V}}^{(k)}\), which we computed in Sect. 3.3.2 via a renormalized multiscale construction. Therefore, identities among the kernels of \({\mathcal {W}}^{(h-1)}\) induce identities between the running coupling constants at scales \(h\le {\bar{h}}\), which are the relations we are interested in. See Appendix 2 below for a detailed discussion about the connection between \({\mathcal {W}}^{(h-1)}\) and \({\mathcal {V}}^{(h-1)}\).
In order to obtain the desired identities among the kernels of \({\mathcal {W}}^{(h-1)}(\phi )\) it is convenient to preliminarily manipulate the r.h.s. of (8.1). We recall that the Bogoliubov’s reference gaussian measure \(P^B(d\psi )\) was obtained by first combining the negative exponential of the first two lines in (3.3) with the free gaussian measure \(P^0\), see the discussion after (3.3), and then by performing the c-number substitution \(\xi \rightarrow \sqrt{\rho _0}\) spelled out after (3.9). A similar connection is valid, of course, between the cut-offed measures \(P_{\ge h}^B(d\psi )\) and \(P_{\ge h}^0(d\psi )\), where \(P_{\ge h}^0(d\psi )\) is the free gaussian measure with propagator
Therefore, we can write:
where, in the limit \(\Lambda \nearrow \mathbb R^3\) (and using the fact that the term in the second line of (3.3) vanishes in this limit),
Plugging (8.4) into (8.1), we find:
We now first perform the change of variables \(\psi ^\pm _\mathbf{x}\rightarrow e^{\pm i\alpha }\psi ^{\pm }_\mathbf{x}\), and then we derive w.r.t. \(\alpha \), so that:
In order to compute the derivative explicitly, it is convenient to rewrite \(\bar{V}=\bar{W} (\psi )+\bar{Q}(\psi )\) with
and then note that
from which one gets (after an explicit computation of the contributions coming from \(\bar{Q}\)):
Note that the term in the last line is equal to \(-i\bar{\nu }\sqrt{\rho _0} \int d\mathbf{x}\big (\phi ^+_\mathbf{x}- \phi ^-_\mathbf{x}\big )\), simply because \(\psi \) has zero average, by construction. If we now use (8.12) into (8.7), after an explicit computation of the contribution coming from \(\bar{Q}^B_{\ge h}\) we find:
where \(w_{\ge h}(\mathbf{x})\) is the Fourier transform of \(\hat{v}({\vec {k}})[\hat{\chi }_{\ge h}(\mathbf{k})]^{-1}\). The Eq. (8.13) can be rewritten in terms of the fields \(\psi ^{l},\psi ^t\) and \(\phi ^{l},\phi ^t\) defined via (3.41), in terms of which we find:
Defining \(\langle [\ \cdot \ ]\rangle _h^\phi =e^{|\Lambda |{\mathcal {F}}_{h-1}+{\mathcal {W}}^{(h-1)}(\phi )}\int P_{\ge h}^B(d\psi ) e^{-\bar{V}(\psi +\phi )}[\ \cdot \ ]\), we can rewrite the last equation as
where in the second line \(\Delta _h(\mathbf{x})\) is the correction due to the presence of the cutoff:
Equation (8.15) can be also rewritten in a convenient form by introducing the auxiliary functional \(\tilde{\mathcal {W}}^{(h-1)}(\phi ,\Phi ^1,\Phi ^2,J^\Delta )\):
with
in terms of which (8.15) takes the form:
and \((\phi ,{\mathbf {0}})\) is a shorthand for \((\phi ,0,0,0)\). At this point, we can obtain infinitely many identities among the kernels of \({\mathcal {W}}^{(h-1)}\), known as global WIs, by further deriving (8.15) or (8.19) w.r.t. \(\phi ^{l},\phi ^t\), and then taking \(\phi \equiv 0\). The “formal” global WIs (discussed e.g. in [15, 39] in the framework of dimensional regularization) are those obtained by neglecting the effect of the cutoff, i.e., by neglecting \(\langle \Delta _h(\mathbf{x})\rangle _h^\phi \) in the second line of (8.15).
A few global WIs that we are interested in are those obtained by: (1) deriving w.r.t. \(\phi ^t_\mathbf{x},\phi ^l_\mathbf{y}\), then integrating w.r.t. \(\mathbf{y}\); (2) deriving w.r.t. \(\phi ^t_\mathbf{x},\phi ^t_\mathbf{y},\phi ^t_\mathbf{z}\) and then integrating w.r.t. \(\mathbf{y},\mathbf{z}\); (3) deriving w.r.t. \(\phi ^t_\mathbf{x}\) (of course in all these cases we put \(\phi \equiv 0\) after the derivation). Their explicit expression is (neglecting for simplicity the issue of the “last integration scale” mentioned in footnote 15, a couple of pages above):
The first two identities are clearly useful, because they relate the running coupling constants among each other. The last identity can be read as a renormalization condition, which fixes the chemical potential to the “right value”, and is known as the Hugenholtz-Pines identity.
It is easy to check that the correction terms in these identities are “trivial”, in the sense that they vanish in the limit of sharp cutoff function, in which case we can drop all these terms at once.
1.2 Appendix 4.2: Local Ward Identities
The local Ward identities are derived in a way completely analogous to the global ones, with the only difference that the phase factor \(\alpha (\mathbf{x})\) appearing in the change of variables \(\psi ^\pm _\mathbf{x}\rightarrow e^{\pm i \alpha _\mathbf{x}}\psi ^\pm _\mathbf{x}\) is a non-trivial function of \(\mathbf{x}\), and the derivatives w.r.t. \(\alpha \) performed in the previous section should be replaced by functional derivatives w.r.t. \(\alpha (\mathbf{x})\). When we derive (8.1) w.r.t. \(\alpha (\mathbf{x})\) we produce a number of terms that are essentially the same as those discussed in the previous section, except for the fact that there is an integration over \(\mathbf{x}\) missing; e.g., the analogue of (8.11) is
etc. In addition to these terms (i.e., the “local” analogues of those of the previous section), there is an extra contribution coming from the measure \(P^0_{\ge h}(d\psi )\), which is not invariant under a local gauge transformation (while it was invariant under a global one). The gaussian weight entering the definition of \(P^0_{\ge h}(d\psi )\) has the form
where the pseudo-differential operator \(D_{\ge h}\) is defined here for the first time. Taking the functional derivative of the expression in braces, we find, after some algebra:
where \(\partial _0=\partial _{x_0}\), \(\vec \partial =(\partial _{x_1},\partial _{x_2})\), and
with
The equations above can be rewritten in terms of the fields \(\psi ^{l},\psi ^t\), in terms of which we find
with
and
After having performed the functional derivative w.r.t. \(\alpha (\mathbf{x})\) and having re-expressed everything in terms of the \(\psi ^l,\psi ^t\) fields, we finally arrive at the analogue of (8.15), which reads
If desired, this equation can be put in a form similar to (8.19). It is enough to introduce the auxiliary potential
where: \({\mathbf {J}}=(J_0,J_1,J_2)=(J_0,\vec J)\), \(\vec \Phi =(\Phi ^1,\Phi ^2)\), and \(({\mathbf {J}},{\mathbf {J}})=\int d\mathbf{x}(J_{0,\mathbf{x}}j_{0,\mathbf{x}}+\vec J_\mathbf{x}\,\vec j_{\mathbf{x}})\), \((\vec \Phi , \vec {F})=\sum _{i=1}^2\int d\mathbf{x}\Phi ^i_\mathbf{x}F^i_\mathbf{x}\), etc. Using the auxiliary potential (8.33), Eq.(8.32) takes the form:
At this point, we can obtain infinitely many identities among the kernels of \({\mathcal {W}}^{(h-1)}\), known as local WIs, by further deriving this identity w.r.t. \(\phi ^{l,t}\), and then taking \(\phi \equiv 0\). The “formal” local WIs (discussed e.g. in [15, 39] in the framework of dimensional regularization) are those obtained by neglecting the effect of the cutoffs, i.e., by dropping the terms \(\langle \Delta _h(\mathbf{x})\rangle _h^\phi \) and \(\langle \delta T_h(\mathbf{x})\rangle _h^\phi \).
Two local WIs we are interested in are those obtained by deriving w.r.t. \(\phi ^l\), or w.r.t. \(\phi ^t\), and then taking the Fourier transform at \(\mathbf{p}=(p_0,\vec {0})\). Their explicit expression in momentum space is (neglecting for simplicity the issue of the “last integration scale”):
where \(W^{(h)}_{1,0;J_0}(\mathbf{x},\mathbf{y})\) is the kernel of \(J_{0,\mathbf{x}}\psi ^l_\mathbf{y}\) in \(\bar{\mathcal {W}}^{(h)}(\phi ,J_0,{\mathbf {0}})\), and \( \hat{W}^{(h)}_{1,0;J_0}(\mathbf{p})=\int d\mathbf{x}e^{i\mathbf{p}\mathbf{x}}\) \(W^{(h)}_{1,0;J_0}(\mathbf{x},{\mathbf {0}})\) (and similarly for \(\hat{W}^{(h)}_{1,0;J^\Delta }(\mathbf{p})\), etc). Eqs.(8.35)–(8.36) are the analogues of [39, (3.11)-(3.12)]. If we divide (8.35) by \(p_0\) and then take the limit \(p_0\rightarrow 0\) we find:
where \(Z_{h-1}^{J_0}:=\hat{W}^{(h)}_{1,0;J_0}({\mathbf {0}})\), and the name is justified by the fact that the Feynman diagram expansion for \(Z_h^{J_0}\) has the same structure as that for \(Z_h\). In particular, inspection of perturbation theory shows that
where the error terms in parentheses come from the irrelevant terms on scale \({\bar{h}}\). In a similar way, if we divide (8.36) by \(p_0^2\) and then take the limit \(p_0\rightarrow 0\) we find, defining \(E_{h-1}^{J_0}:= -\partial _{p_0}\hat{W}^{(h)}_{0,1;J_0}({\mathbf {0}})\):
where
Contrary to the correction terms of the global Ward Identities, the corrections in (8.37), (8.39) are not “trivial”, i.e., they do not vanish in the sharp cutoff limit. On the contrary, they give a finite (cutoff-dependent) contribution to the beta function, which shows up as an “anomaly” already at the level of the one-loop beta function, as discussed in Sect. 3.3.4.
1.3 Appendix 4.3: Comparison Between \({\mathcal {W}}^{(h-1)}\) and \({\mathcal {V}}^{(h-1)}\)
In order to establish the exact relation between \({\mathcal {W}}^{(h-1)}\) and \({\mathcal {V}}^{(h-1)}\), we compute the r.h.s. of (8.1) via a multiscale integration analogous to the one used in the bulk of the paper for the computation of \({\mathcal {V}}^{(h-1)}\). The integration of the ultraviolet fields on scales \(>{\bar{h}}\), as well as the integration of the field \(\tilde{\psi }^{({\bar{h}})}\) (see (3.40)), is identical to the one discussed in Sect. 3.2 and at the beginning of Sect. 3.3, after which we can rewrite (8.1) as
where \(P_{[h,{\bar{h}}]}(d\psi )\) is the same as the gaussian measure \(P_{\le {\bar{h}}}(d\psi )\) in (3.40), modulo the presence of an infrared cutoff on scale \(h\) in the corresponding propagator (i.e., while the cutoff function appearing in the propagator of \(P_{\le {\bar{h}}}(d\psi )\) is \(\chi _{{\bar{h}}}(\mathbf{k})\), the one in the propagator of \(P_{[h,{\bar{h}}]}(d\psi )\) is \(\chi _{{\bar{h}}}(\mathbf{k})-\chi _{h-1}(\mathbf{k})\)). Moreover, using the same convention of Sect. 3.3.2, \({\mathcal {V}}^{(\bar{h})}\) coincides with the function \(\bar{V}^{(\bar{h})}\) introduced after (3.40). At this point, we start dressing the gaussian measure in (8.41), in the same way as in Sect. 3.3.2. Let us describe the first integration step explicitly. Using the notations introduced in (3.64) and following equations, we rewrite the r.h.s. of (8.41) as
where \(\widehat{\mathcal {V}}^{({\bar{h}})}\) is a shorthand for \({\mathcal {L}}_C{\mathcal {V}}^{({\bar{h}})}+{\mathcal {R}}{\mathcal {V}}^{({\bar{h}})}\). We now combine \({\mathcal {L}}_Q{\mathcal {V}}^{({\bar{h}})}(\psi )\) with the gaussian measure, as in (3.66), and then use the addition principle to rewrite the dressed measure as a product of a measure supported on scale \({\bar{h}}\) and a measure supported on smaller scales, as in (3.72):
At this point, we integrate the field on scale \({\bar{h}}\) and define:
where, denoting by \(\hat{\phi }_\mathbf{k}\) the two-component column vector with components \(\hat{\phi }_\mathbf{k}^l\) and \(\hat{\phi }_\mathbf{k}^t\),
We now set \(F_{{\bar{h}}-1}=t_{\bar{h}}+\tilde{F}_{\bar{h}}\) and \(S^{({\bar{h}}-1)}={\mathcal {L}}_Q{\mathcal {V}}^{({\bar{h}})}-\tilde{S}_{\bar{h}}\), and then we iterate the same procedure. After the integration of the fields on scales \(\ge k+1\) we rewrite (8.1) as
where one can inductively prove that, for \(k<{\bar{h}}\),
and the vectorial nature of \(\psi \) (i.e., the fact that \(\psi \) has two components, labelled \(l\) and \(t\)) is implicitly understood. Moreover, the functions \(Q^{(k)}\) and \(G^{(k)}\), \(k\le {\bar{h}}+1\), are defined by the iterative relations
with \(Q^{({\bar{h}}+1)}=G^{({\bar{h}}+1)}=0\).
The iteration goes on in the same fashion until we reach scale \(h\), where a small difference from the previous scheme should be taken into account: in fact, by proceeding as described above, one finds that the dressed propagator on the last scale, rather than being equal to \(\tilde{g}^{(h)}\), it is equal to
where \({\tilde{f}}(\mathbf{k})=\chi _h(\mathbf{k})-\chi _{h-1}(\mathbf{k})\), as usual, and
After the integration of the last scale we finally find
where the tildes on the functions at exponent recall the fact that these functions are defined in the same fashion as their analogues without tilde, with the only difference that the single-scale propagator on the last scale, \(\tilde{g}^{(h)}\), wherever it enters the definition of these objects, should be replaced by \(\tilde{\mathfrak g}^{(h)}\).
Equation (8.53) provides us the desired relation between \({\mathcal {W}}^{(h-1)}\) and \({\mathcal {V}}^{(h-1)}\). It shows that the local parts of the kernels of \({\mathcal {W}}^{(h-1)}\) (in the sense of the values of their Fourier transforms at zero external momenta, as well as the zero momenta values of their derivatives w.r.t. \(\mathbf{k}\)), are related in a very simple fashion with the corresponding local parts of the kernels of \({\mathcal {V}}^{(h-1)}\), simply because the local part of the kernels of \(\tilde{\mathcal {B}}^{(h-1)}(\phi ,\phi )\) is equal to \(2\sum _{k=h}^{\bar{h}}{\mathcal {L}}_Q{\mathcal {V}}^{(k)}(\phi )\), due to the compact support properties of \(\tilde{g}^{(k)}\) and of \(G^{(k)}\), and similarly the local part of \(\tilde{S}^{(h-1)}\) is equal to \(\sum _{k=h}^{\bar{h}}{\mathcal {L}}_Q{\mathcal {V}}^{(k)}\). Therefore,
which induces a simple, explicit, connection between the actual renormalization and running coupling constants of the model, introduced in Sect. 3.3.2, and \({\mathcal {L}}{\mathcal {W}}^{(h-1)}(\phi )\).
Appendix 5: Verification of Ward Identities at Lowest Order
In this section, we verify the validity of some of the global and local Ward Identities among the running coupling and the renormalization constants, at lowest non trivial order in perturbation theory. In particular, we shall compute the effect of the infrared cutoff and discuss its role in the different identities worked out below.
1.1 Appendix 5.1: Global Ward Identities
Let us verify the validity of (8.20) at lowest non trivial order in \(\lambda \) and/or in \(\gamma ^{h-{\bar{h}}}\), in the region \(h^*\le h\le {\bar{h}}\). If \(W^{(h)}_{1,1;J^\Delta }(\mathbf{x},\mathbf{y};\mathbf{0})\) is the kernel of \(J^\Delta _{\mathbf{0}}\phi ^t_\mathbf{x}\phi ^l_\mathbf{y}\) in \({\tilde{{\mathcal {W}}}}^{(h)}(\phi ,J^\Delta , {\mathbf {0}})\), and
we can rewrite (8.20) as
Using the beta function Eqs. (3.77)–(3.80), the l.h.s. can be rewritten as
Now, using the explicit expressions of \(Z_{\bar{h}-1}, \nu _{\bar{h}}, \mu _{\bar{h}}\), one can check that the combination \(Z_{\bar{h}-1}-2 \gamma ^{{\bar{h}}} \nu _{\bar{h}}-2\sqrt{2\rho _0}\gamma ^{{\bar{h}}/2}\mu _{\bar{h}}\) is zero at lowest order (i.e., at the order \(\lambda \), as well as at the order \(\bar{\nu }\)) and, therefore, it is (at most) of the order \(O(\lambda ^2|{\bar{h}}|)\). Moreover, using the explicit expression of the beta functions for \(Z_h\), \(\nu _h\) and \(\mu _h\), see (3.82), (3.83), (3.89), as well as the replacements
induced by the Ward Identities (8.20)–(8.21) (which, once inserted in the expressions (3.82), (3.83), (3.89) of the one-loop beta function, induce errors of higher order in \(\lambda \) and/or in \(\gamma ^{h-{\bar{h}}}\), as one can prove inductively in \(h\)), we can rewrite, for \(h^*\le h\le {\bar{h}}\),
modulo higher order correction terms in \(\lambda \) and/or in \(\gamma ^{h-{\bar{h}}}\) (in deriving the expression of the first term in the second line we also used the fact that \(C_kZ_k=1\), up to higher order corrections). Now, the sum over \(k\) of \(-\frac{2\gamma ^k}{A_k C_k}\frac{1}{8\rho _0}(\beta _0^{(1)}-2\beta _0^{(1,\chi )})\) is of the order \(\lambda ^2\), which is subdominant with respect to the other two terms, whose sum over \(k\) gives (using the fact that \(\mu _k=\gamma ^{({\bar{h}}-k)/2} \mu _{\bar{h}}\), up to higher order corrections)
We now want to show that this expression is equal to \(\hat{W}^{(h)}_{1,1;J^\Delta }(\mathbf{0}, \mathbf{0}) \), modulo higher order corrections. In fact, the lowest order contribution to \(\hat{W}^{(h)}_{1,1;J^\Delta }(\mathbf{0}, \mathbf{0}) \) is given by the sum of the diagrams in Fig. 23, with at least one propagator at scale \(h\). After the same considerations made for the calculation of the beta function for \(\mu _{h}\) (see Appendix 3.3) we find that, at lowest order,
Note that, for \(j\ge 0\),
where
Note that, if \(\gamma ^{-1} \le |\mathbf{k}|< 1\), then \({\mathfrak {f}}_1(|\mathbf{k}|)=0\). Therefore,
where we used that \(Z_h=2 \lambda \rho _0 \hat{v} (\vec {0})\), up to higher order corrections.
A similar discussion can be repeated for proving the validity of (8.21), but we will not belabor the details here.
1.2 Appendix 5.2: Local WI Relating \(E_h\) and \(Z_h\)
In this section we prove the validity of the local WI (8.37), that is
at lowest non trivial order. Now, using (3.90), we find that at lowest order
while, using (8.38) and (3.89),
as can also be checked by an explicit calculation of the second diagram on the first line of Fig. 24. Therefore, at lowest order \(Z_{h-2}^{J_0}=\sqrt{2\rho _0}(E_{h-2}-1)\). In fact, it can be checked that the two correction terms in the r.h.s. cancel among each other: at lowest order their values are
which are the same, since \(E_h=1\) up to higher order corrections.
1.3 Appendix 5.3: Local WI Relating \(B_h\) and \(E_h\)
In this section we prove the validity of the local WI (8.39), that is
Let us discuss the various terms involved in this equation separately. Using (3.91) and the fact that \(B_{\bar{h}}=0\), we find
Using (8.40) and (3.90) we find
so that, recalling that at lowest order \(E_h=1\) and \(Z_h=2\lambda \rho _0\hat{v}({\mathbf {0}})\),
We now want to verify that the sum of the two correction terms in the r.h.s. of (9.15) gives exactly the same contribution.
The leading order contribution to the correction term \(\partial ^2_{p_0}\hat{W}_{0,1;J^\Delta }^{(h-1)}({\mathbf {0}})\) is represented on the second line of Fig. 25. Using the same notations as in Appendix 3.6, we find
that is, using also the fact that at lowest order \(\alpha _h=1\) and \(Z_h=2\lambda \rho _0\hat{v} (\vec {0})\),
We now compute the derivatives and pass to polar coordinates, thus finding
The leading order contribution to the correction term \(\partial ^2_{p_0}\hat{W}_{0,1;J^{\delta T}}^{(h-1)}({\mathbf {0}})\) is represented on the third line of Fig. 25, and is equal to
Using the fact that at lowest order \(E_h/(C_hZ_h)=1\), we find that the difference between (9.21) and (9.22) is
Using the first of (9.4), we can rewrite this equation as
that, after an integration by parts, can be easily recognized to be the same as (9.18).
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Cenatiempo, S., Giuliani, A. Renormalization Theory of a Two Dimensional Bose Gas: Quantum Critical Point and Quasi-Condensed State. J Stat Phys 157, 755–829 (2014). https://doi.org/10.1007/s10955-014-1034-7
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DOI: https://doi.org/10.1007/s10955-014-1034-7